Sobre a Aritmética de Curvas Elípticas: O Teorema de Mordell-Weil, a Conjectura de Birch e Swinnerton-Dyer e o Problema dos Números Congruentes

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❯♥✐✈❡rs✐❞❛❞❡ ❋❡❞❡r❛❧ ❞❛ ❇❛❤✐❛ ✲ ❯❋❇❆

■♥st✐t✉t♦ ❞❡ ▼❛t❡♠át✐❝❛ ❡ ❊st❛tíst✐❝❛ ✲ ■▼❊

Pr♦❣r❛♠❛ ❞❡ Pós✲●r❛❞✉❛çã♦ ❡♠ ▼❛t❡♠át✐❝❛ ✲ P●▼❆❚

❉✐ss❡rt❛çã♦ ❞❡ ▼❡str❛❞♦

  

❙♦❜r❡ ❛ ❆r✐t♠ét✐❝❛ ❞❡ ❈✉r✈❛s ❊❧í♣t✐❝❛s✿ ❖

❚❡♦r❡♠❛ ❞❡ ▼♦r❞❡❧❧✲❲❡✐❧✱ ❛ ❈♦♥❥❡❝t✉r❛ ❞❡ ❇✐r❝❤

❡ ❙✇✐♥♥❡rt♦♥✲❉②❡r ❡ ♦ Pr♦❜❧❡♠❛ ❞♦s ◆ú♠❡r♦s

❈♦♥❣r✉❡♥t❡s

  

❨✉r❡ ❈❛r♥❡✐r♦ ❞❡ ❖❧✐✈❡✐r❛

  ❙❛❧✈❛❞♦r✲❇❛❤✐❛ ▼❛rç♦ ❞❡ ✷✵✶✽

  

❙♦❜r❡ ❛ ❆r✐t♠ét✐❝❛ ❞❡ ❈✉r✈❛s ❊❧í♣t✐❝❛s✿ ❖

❚❡♦r❡♠❛ ❞❡ ▼♦r❞❡❧❧✲❲❡✐❧✱ ❛ ❈♦♥❥❡❝t✉r❛ ❞❡ ❇✐r❝❤

❡ ❙✇✐♥♥❡rt♦♥✲❉②❡r ❡ ♦ Pr♦❜❧❡♠❛ ❞♦s ◆ú♠❡r♦s

❈♦♥❣r✉❡♥t❡s

  

❨✉r❡ ❈❛r♥❡✐r♦ ❞❡ ❖❧✐✈❡✐r❛

  ❉✐ss❡rt❛çã♦ ❞❡ ▼❡str❛❞♦ ❛♣r❡s❡♥t❛❞❛ ❛♦ ❈♦❧❡❣✐❛❞♦ ❞❛ Pós✲●r❛❞✉❛çã♦ ❡♠ ▼❛t❡♠át✐❝❛ ❞❛ ❯♥✐✈❡rs✐❞❛❞❡ ❋❡❞❡r❛❧ ❞❛ ❇❛❤✐❛ ❝♦♠♦ r❡q✉✐s✐t♦ ♣❛r❝✐❛❧ ♣❛r❛ ♦❜t❡♥çã♦ ❞♦ tít✉❧♦ ❞❡ ▼❡str❡ ❡♠ ▼❛t❡♠át✐❝❛✳

  ❛ ❛

  ❖r✐❡♥t❛❞♦r❛✿ Pr♦❢ ✳ ❉r ✳ ▼❛♥✉❡❧❛ ❞❛ ❙✐❧✈❛ ❙♦✉③❛✳

  ❙❛❧✈❛❞♦r✲❇❛❤✐❛ ▼❛rç♦ ❞❡ ✷✵✶✽

  ❖❧✐✈❡✐r❛✱ ❨✉r❡ ❈❛r♥❡✐r♦ ❞❡✱ ✶✾✾✺ ❙♦❜r❡ ❛ ❆r✐t♠ét✐❝❛ ❞❡ ❈✉r✈❛s ❊❧í♣t✐❝❛s✿ ❖ ❚❡♦r❡♠❛ ❞❡ ▼♦r❞❡❧❧✲❲❡✐❧✱

❛ ❈♦♥❥❡❝t✉r❛ ❞❡ ❇✐r❝❤ ❡ ❙✇✐♥♥❡rt♦♥✲❉②❡r ❡ ♦ Pr♦❜❧❡♠❛ ❞♦s ◆ú♠❡r♦s

❈♦♥❣r✉❡♥t❡s✴ ❨✉r❡ ❈❛r♥❡✐r♦ ❞❡ ❖❧✐✈❡✐r❛✳ ✕ ❙❛❧✈❛❞♦r✿ ❯❋❇❆✱ ✷✵✶✽✳

  ◗✉❛♥t✐❞❛❞❡ ❞❡ ❢♦❧❤❛s ❢✳ ✶✵✷ ✿ ✐❧✳ ❛ ❛

  ❖r✐❡♥t❛❞♦r❛✿ Pr♦❢ ✳ ❉r ✳ ▼❛♥✉❡❧❛ ❞❛ ❙✐❧✈❛ ❙♦✉③❛✳ ❉✐ss❡rt❛çã♦ ✭♠❡str❛❞♦✮ ✕ ❯♥✐✈❡rs✐❞❛❞❡ ❋❡❞❡r❛❧ ❞❛ ❇❛❤✐❛✱ ■♥st✐t✉t♦ ❞❡ ▼❛t❡♠át✐❝❛✱ Pr♦❣r❛♠❛ ❞❡ Pós✲❣r❛❞✉❛çã♦ ❡♠ ▼❛t❡♠át✐❝❛✱ ✷✵✶✽✳ ❘❡❢❡rê♥❝✐❛s ❜✐❜❧✐♦❣rá✜❝❛s✳

  ✶✳ ❈✉r✈❛s ❊❧í♣t✐❝❛s✳ ✷✳ ▼♦r❞❡❧❧✲❲❡✐❧✳ ✸✳ ❈♦♥❥❡❝t✉r❛ ❞❡ ❇✐r❝❤ ❡

❙✇✐♥♥❡rt♦♥✲❉②❡r✳ ✹✳ ◆ú♠❡r♦s ❈♦♥❣r✉❡♥t❡s✳ ■✳ ❙♦✉③❛✱ ▼❛♥✉❡❧❛ ❞❛ ❙✐❧✈❛✳

■■✳ ❯♥✐✈❡rs✐❞❛❞❡ ❋❡❞❡r❛❧ ❞❛ ❇❛❤✐❛✱ ■♥st✐t✉t♦ ❞❡ ▼❛t❡♠át✐❝❛ ❡ ❊st❛tíst✐❝❛✳

■■■✳ ❚ít✉❧♦✳

  ❈❉❯ ✿ ✺✶✷✳✺✺✷✳✼

  ➚ ❉❡✉s✱ à ♠✐♥❤❛ ❢❛♠✐❧✐❛ ❡ ❛♠✐❣♦s✳ ✧❖ q✉❡ é ♥❛s❝✐❞♦ ❞❡ ❉❡✉s ✈❡♥❝❡ ♦ ♠✉♥❞♦❀ ❡ ❡st❛ é ❛ ✈✐tór✐❛ q✉❡ ✈❡♥❝❡ ♦ ♠✉♥❞♦✿ ❛ ♥♦ss❛ ❢é✳✑ ✭❇❮❇▲■❆✱ ✶ ❏♦ã♦ ✺✿✹✮

  ❆❣r❛❞❡❝✐♠❡♥t♦s

  Pr✐♠❡✐r❛♠❡♥t❡✱ ❛❣r❛❞❡ç♦ à ❉❡✉s ♣♦r t✉❞♦✱ ♣❡❧♦ ❛♣♦✐♦ q✉❡ ❞❡s❞❡ s❡♠♣r❡ t❡♠ ♠❡ ❝♦♥❝❡❞✐❞♦✱ ♣❡❧❛ ❢♦rç❛ q✉❡ t❡♠ ♠❡ ❞❛❞♦ ♥♦s ♠♦♠❡♥t♦s ❞❡ ❢r❛q✉❡③❛✱ ♥♦s ♠♦♠❡♥t♦s ❞❡ ❞❡s❡s♣❡r♦✱ ♠♦♠❡♥t♦s ❡st❡s q✉❡ s❡ ♥ã♦ ❢♦ss❡ ♣❡❧❛ ❢é q✉❡ t❡♥❤♦ ❡♠ ❚✐✱ ♥ã♦ t❡r✐❛ s✉♣♦rt❛❞♦✳ ❆♦ ❙❡♥❤♦r ♦ ♠❡✉ ♠✉✐t♦ ♦❜r✐❣❛❞♦✳

  ❆❣r❛❞❡ç♦ à ♠✐♥❤❛ ❢❛♠í❧✐❛✱ ♠✐♥❤❛ ♠ã❡✱ ♠✐♥❤❛s ✐r♠ãs ❡ ✐r♠ã♦s✱ s❡♠♣r❡ ♠❡ ❞❛♥❞♦ ❛♣♦✐♦ ♥❛s ♠✐♥❤❛s t♦♠❛❞❛s ❞❡ ❞❡❝✐sõ❡s✱ ♣r❡♦❝✉♣❛❞♦s ❡♠ s❡♠♣r❡ ♠❡ ❛❥✉❞❛r✳ ❊✉ ❞♦✉ ❣r❛ç❛s ♣❡❧❛ ❢❛♠í❧✐❛ ❛❜❡♥ç♦❛❞❛ q✉❡ t❡♥❤♦ ❡ ❡♠ ❡s♣❡❝✐❛❧ à ♠✐♥❤❛ ♠ã❡✱ q✉❡ s❡♠♣r❡ ❝♦❧♦❝♦✉ ♦s ✜❧❤♦s à ❢r❡♥t❡ ❞❡ ❞✉❛s ♥❡❝❡ss✐❞❛❞❡s✱ ❡ ♥❡❧❛ ♣✉❞❡ ♣❡r❝❡❜❡r ✉♠ ❛♠♦r ❞✐❢❡r❡♥t❡✱ q✉❡ ❞❡✈❡ s❡r ✈❛❧♦r✐③❛❞♦ ❡ ♠✉✐t❛s ❞❛s ✈❡③❡s ♥ã♦ ♣❛r❛♠♦s ♣❛r❛ ♣❡r❝❡❜❡r ♦s ❞❡t❛❧❤❡s ❞❡ss❡ ❝✉✐❞❛❞♦ ❡s♣❡❝✐❛❧ ❡ ❞✐❢❡r❡♥❝✐❛❞♦ q✉❡ ✉♠❛ ♠ã❡ t❡♠ ♣❡❧♦s ✜❧❤♦s✳ ❆❣r❛❞❡ç♦ t❛♠❜é♠ ❛ P❛✉❧♦ ❡ ❞♦♥❛ ❱❡r❛✱ ♣❡ss♦❛s s✉♣❡r ❡s♣❡❝✐❛✐s q✉❡ ❢❛③❡♠ ♣❛rt❡ ❞❛ ♠✐♥❤❛ ✈✐❞❛✱ às q✉❛✐s só t❡♥❤♦ ❛ ❛❣r❛❞❡❝❡r✳

  ❆❣r❛❞❡ç♦ à ♠✐♥❤❛ ♦r✐❡♥t❛❞♦r❛✱ ❛ ♣r♦❢❡ss♦r❛ ▼❛♥✉❡❧❛✳ ❆❣r❛❞❡ç♦ ✐♠❡♥s❛♠❡♥t❡ ♣♦r ♣♦❞❡r t❡r ❡st✉❞❛❞♦ ❡ s✐❞♦ ♦r✐❡♥t❛❞♦ ♣❡❧❛ s❡♥❤♦r❛✱ ♣♦r t❡r ♠❡ ❛❥✉❞❛❞♦ ❡♠ ✈ár✐♦s ♠♦♠❡♥t♦s✱ ♣❡❧❛s ❝♦♥✈❡rs❛s s♦❜r❡ ♦ ♠❡✉ ❢✉t✉r♦ ♣r♦✜ss✐♦♥❛❧✱ ♣❡❧❛s ❞✐❝❛s ❡ ❝♦♥s❡❧❤♦s✱ ♣♦r t❡r ♠❡ ♦r✐❡♥t❛❞♦ ❡♠ ♠✐♥❤❛ ♠♦♥♦❣r❛✜❛ ❡ ♥❡st❛ ❞✐ss❡rt❛çã♦ ❡ ♣♦r t❡r ❛❝r❡❞✐❞❛t♦ ❡♠ ♠✐♠ ❝♦♠ r❡s♣❡✐t♦ ❛♦ ❝♦♥t❡ú❞♦ ❞❡st❡ tr❛❜❛❧❤♦✳

  ❯♠ ♠✉✐t♦ ♦❜r✐❣❛❞♦ ❛♦s ♣r♦❢❡ss♦r❡s ❞❛ ❜❛♥❝❛✱ à ♣r♦❢❡ss♦r❛ ❈❡❝í❧✐❛ ❙❛❧❣❛❞♦ q✉❡ ♠❡s♠♦ t❡♥❞♦ ♠❡ ❝♦♥❤❡❝✐❞♦ à ♣♦✉❝♦ t❡♠♣♦✱ ❛❝❡✐t♦✉ ♠❡ ♦r✐❡♥t❛r ♥♦ ❢✉t✉r♦ ❞♦✉t♦r❛❞♦ ❡ ❢❛③❡r ♣❛rt❡ ❞❡ss❛ ❜❛♥❝❛✱ t❛♠❜é♠ ♣❡❧❛ ❛❥✉❞❛ q✉❡ ♠❡ ❞❡✉ s♦❜r❡ ❛ t❡♠át✐❝❛ ♣❛r❛ ❡ss❛ ❞✐ss❡rt❛çã♦✳ ❆♦ ♣r♦❢❡ss♦r ▼❛r❝ ❍✐♥❞r②✱ q✉❡ t❛♠❜é♠ ♠❡s♠♦ ♠❡ ❝♦♥❤❡❝❡♥❞♦ à ♣♦✉❝♦ t❡♠♣♦✱ ❛❝❡✐t♦✉ ❢❛③❡r ♣❛rt❡ ❞❡ss❛ ❜❛♥❝❛✳ ➱ ✉♠ ✐♠❡♥s♦ ♣r❛③❡r ❝♦♥t❛r ❝♦♠ ❛ ♣r❡s❡♥ç❛ ❞❡ ✈♦❝ês ♥❛ ❛✈❛❧✐❛çã♦ ❞❡ss❡ tr❛❜❛❧❤♦✳

  ◆ã♦ ♣♦ss♦ ❞❡✐①❛r ❞❡ ❛❣r❛❞❡❝❡r ❛♦ ♣r♦❢❡ss♦r ❙❛♠✉❡❧ ●♦♠❡s✱ q✉❡ ❛❝r❡❞✐t♦✉ ❡♠ ♠✐♠ ❞❡s❞❡ ❛ ♠✐♥❤❛ ❡♥tr❛❞❛ ♥❛ ❣r❛❞✉❛çã♦✱ ♠❡ ❛❥✉❞♦✉ ♠✉✐t♦ ❡ t❡✈❡ ❣r❛♥❞❡ ✐♥✢✉ê♥❝✐❛ s♦❜r❡ ❛ ❢♦r♠❛çã♦ ❞♦ ♠❡✉ r❛❝✐♦❝í♥✐♦ ♠❛t❡♠át✐❝♦✱ ❡♠ q✉❡ ❣r❛♥❞❡ ♣❛rt❡ ❛♣r❡♥❞✐ ❝♦♠ ❡❧❡✳

  ❋♦r❛♠ ♠✉✐t♦s ♦s ♣r♦❢❡ss♦r❡s ❞♦ ■♥st✐t✉t♦ ❞❡ ▼❛t❡♠át✐❝❛ ❞❛ ❯❋❇❆ q✉❡ t✐✈❡r❛♠ ♣❛rt✐❝✐♣❛çã♦ ♥❛ ♠✐♥❤❛ ❢♦r♠❛çã♦ ❡ ♠❡ ❛❥✉❞❛r❛♠ ❡♠ ✈ár✐♦s ♠♦♠❡♥t♦s✱ ❛♦s q✉❛✐s ❞❡✈♦ ❛❣r❛❞❡❝✐♠❡♥t♦✱ ❛♦s ♣r♦❢❡ss♦r❡s ❡ ♣r♦❢❡ss♦r❛s✱ ❈❛r❧♦s ❇❛❤✐❛♥♦✱ ❖s❝❛r✱ ❘✐t❛✱ ❏♦sé ◆❡❧s♦♥✱ ❏♦s❡♣❤✱ P❛✉❧♦ ❱❛r❛♥❞❛s✱ ❱✐t♦r✱ ❏♦✐❧s♦♥✱ ❏✉❛♥ ●♦♥③❛❧❡③✱ ❈❛r♠❡❧❛✱ ❈✐r♦✱ ❚❤✐❛❣♦✱ ❱❛♥❡ss❛✱ ▼❛t❤✐❡✉✱ ❏❡r♦♠❡✱ ❏❛✐♠❡ ❡ ❙❛♠✉❡❧ ❋❡✐t♦s❛✳ ◗✉❡ ❉❡✉s ❛❜❡♥ç♦❡ ❛ ❝❛❞❛ ✉♠ ❞❡ ✈♦❝ês✳

  ❙♦❜r❡ ♦s ❛♠✐❣♦s q✉❡ ✜③ ❞❡♥tr♦ ❡ ❢♦r❛ ❞❛ ❯♥✐✈❡rs✐❞❛❞❡✱ ❡ss❛ ❧✐st❛ ❝♦♠ ❝❡rt❡③❛

  ❝♦♥té♠ ♠✉✐t❛s ♣❡ss♦❛s✱ à t♦❞♦s ✈♦❝ês ✜❝❛♠ ✉♠ ♠✉✐t♦ ♦❜r✐❣❛❞♦ ♣❡❧♦s ♠♦♠❡♥t♦s ❞❡ ❡st✉❞♦ ❡ ❞✐✈❡rsã♦✱ ♣❡❧♦s ♠♦♠❡♥t♦s ♥♦ ❘✳❯✳✱ ♣❡❧❛s ❜♦❛s ❝♦♥✈❡rs❛s q✉❡ t✐✈❡ ❝♦♠ ❝❛❞❛ ✉♠ ❞❡ ✈♦❝ês✳ ◆❡st❡ ♠♦♠❡♥t♦ ♥ã♦ ❝✐t❛r❡✐ ♥♦♠❡s✱ ♣♦✐s r❡❛❧♠❡♥t❡ sã♦ ♠✉✐t♦s ❞❡ ✈♦❝ês✱ ♠❛s ❝r❡✐♦ q✉❡ ❝❛❞❛ ✉♠ s❛✐❜❛ ♦ q✉❛♥t♦ s♦✉ ❣r❛t♦✳

  ❆❣r❛❞❡ç♦ ❛♦s ♠❡✉s ❛♠✐❣♦s ❞♦ ◗✉❛rt❡t♦ ❋❛♥tást✐❝♦✳ ❆♦s ❛♠✐❣♦s q✉❡ ❝♦♥❤❡❝✐ ❡ ♣r♦❢❡ss♦r❡s q✉❡ t✐✈❡ ♥♦ ■❋❇❆✱ ❡♠ ❡s♣❡❝✐❛❧ ❛♦ ♣r♦❢❡ss♦r P❛✉❧♦ ❱✐❝❡♥t❡✱ à ♣r♦❢❡ss♦r❛ ❏✉❛♥✐❝❡ ❡ ❛♦ ♣r♦❢❡ss♦r ❩✐✉❧✱ ♠✉✐t♦ ♦❜r✐❣❛❞♦✳

  ❋✐♥❛❧♠❡♥t❡✱ ❛❣r❛❞❡ç♦ à ❈❆P❊❙ ♣❡❧♦ ❛♣♦✐♦ ✜♥❛♥❝❡✐r♦ ❝♦♥❝❡❞✐❞♦ ❛ ♠✐♠ ❞✉r❛♥t❡ t♦❞♦ ♦ ♠❡✉ ♠❡str❛❞♦✳

  ❘❡s✉♠♦

  ❖ ♦❜❥❡t✐✈♦ ❞❡st❡ tr❛❜❛❧❤♦ é ❡st✉❞❛r ❛s ❝✉r✈❛s ❡❧í♣t✐❝❛s✱ ♠❛✐s ♣r❡❝✐s❛♠❡♥t❡✱ s❡rá ❛♣r❡s❡♥t❛❞❛ ❛ ❞❡♠♦♥str❛çã♦ ❞♦ ❚❡♦r❡♠❛ ❞❡ ▼♦r❞❡❧❧✲❲❡✐❧ ✭❚✳▼✳❲✮✱ r❡s✉❧t❛❞♦ ❡st❡ q✉❡ ❞✐③ q✉❡ s❡ E/K é ✉♠❛ ❝✉r✈❛ ❡❧í♣t✐❝❛ s♦❜r❡ ✉♠ ❝♦r♣♦ ❞❡ ♥ú♠❡r♦s K✱ ❡♥tã♦ ♦ ❣r✉♣♦ E(K)

  r

  , = E(K) tor

  ❞♦s s❡✉s ♣♦♥t♦s K✲r❛❝✐♦♥❛✐s é ✜♥✐t❛♠❡♥t❡ ❣❡r❛❞♦✳ ❆ss✐♠ E(K) ∼ ⊕ Z ♣❛r❛ ❛❧❣✉♠ r ≥ 0✱ ❡♠ q✉❡ ❡ss❡ ✐♥✈❛r✐❛♥t❡ r é ❝❤❛♠❛❞♦ ♦ ♣♦st♦ ❛❧❣é❜r✐❝♦ ❞❡ E✳ P♦r ✜♠✱ s❡rã♦ ❛♣r❡s❡♥t❛❞♦s ❞♦✐s ❢❛♠♦s♦s ♣r♦❜❧❡♠❛s ❛r✐t♠ét✐❝♦s✱ s❡♥❞♦ ❡❧❡s ❛ ❈♦♥❥❡❝t✉r❛ ❞❡ ❇✐r❝❤ ❡ ❙✇✐♥♥❡rt♦♥✲❉②❡r ❡ ♦ Pr♦❜❧❡♠❛ ❞♦s ◆ú♠❡r♦s ❈♦♥❣r✉❡♥t❡s ❡ s❡rá ❞✐s❝✉t✐❞❛ ❛ r❡❧❛çã♦ ❡♥tr❡ ❡ss❡s ❞♦✐s ♣r♦❜❧❡♠❛s✳ P❛❧❛✈r❛s✲❝❤❛✈❡✿ ❈✉r✈❛s ❊❧í♣t✐❝❛s✳ ▼♦r❞❡❧❧✲❲❡✐❧✳ ❈♦♥❥❡❝t✉r❛ ❞❡ ❇✐r❝❤ ❡ ❙✇✐♥♥❡rt♦♥✲ ❉②❡r✳ ◆ú♠❡r♦s ❈♦♥❣r✉❡♥t❡s✳

  ❆❜str❛❝t

  ❚❤❡ ♦❜❥❡❝t✐✈❡ ♦❢ t❤✐s ✇♦r❦ ✐s t♦ st✉❞② t❤❡ ❡❧❧✐♣t✐❝ ❝✉r✈❡s✱ ♠♦r❡ ♣r❡❝✐s❡❧②✱ t❤❡ ❞❡♠♦♥str❛t✐♦♥ ♦❢ t❤❡ ▼♦r❞❡❧❧✲❲❡✐❧ ❚❤❡♦r❡♠ ✭▼✳❲✳❚✮ ✇✐❧❧ ❜❡ ♣r❡s❡♥t❡❞✱ ✇❤✐❝❤ r❡s✉❧ts t❤❛t ✐❢ E/K ✐s ❛ ❡❧❧✐♣t✐❝ ❝✉r✈❡ ♦♥ ❛ ♥✉♠❜❡rs ✜❡❧❞ K✱ t❤❡♥ t❤❡ E(K) ❣r♦✉♣ ♦❢ ✐ts K✲

  r

  , = E(K) tor r❛t✐♦♥❛❧ ♣♦✐♥ts ✐s ✜♥✐t❡❧② ❣❡♥❡r❛t❡❞✳ ❚❤✉s E(K) ∼ ⊕ Z ❢♦r s♦♠❡ r ≥ 0✱ ✇❤❡r❡ t❤❛t ✐♥✈❛r✐❛♥t r ✐s ❝❛❧❧❡❞ t❤❡ ❛❧❣❡❜r❛✐❝ r❛♥❦✳ ❋✐♥❛❧❧②✱ t✇♦ ❢❛♠♦✉s ❛r✐t❤♠❡t✐❝❛❧ ♣r♦❜❧❡♠s ✇✐❧❧ ❜❡ ♣r❡s❡♥t❡❞✱ t❤❡ ❇✐r❝❤ ❛♥❞ ❙✇✐♥♥❡rt♦♥✲❉②❡r ❈♦♥❥❡❝t✉r❡ ❛♥❞ t❤❡ ❈♦♥❣r✉❡♥t ◆✉♠❜❡rs Pr♦❜❧❡♠✱ ❛♥❞ t❤❡ r❡❧❛t✐♦♥s❤✐♣ ❜❡t✇❡❡♥ t❤❡s❡ t✇♦ ♣r♦❜❧❡♠s ✇✐❧❧ ❜❡ ❞✐s❝✉ss❡❞✳ ❑❡②✇♦r❞s✿ ❊❧❧✐♣t✐❝ ❈✉r✈❡s✳ ▼♦r❞❡❧❧✲❲❡✐❧✳ ❇✐r❝❤ ❛♥❞ ❙✇✐♥♥❡rt♦♥✲❉②❡r ❝♦♥❥❡❝t✉r❡✳ ❈♦♥✲ ❣r✉❡♥t ◆✉♠❜❡rs✳

  ❙✉♠ár✐♦

  ■♥tr♦❞✉çã♦ ✶

  ✶ Pr❡❧✐♠✐♥❛r❡s ✼

  ✶✳✶ ❱❛r✐❡❞❛❞❡s ❆✜♥s ❡ Pr♦❥❡t✐✈❛s ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✼ ✶✳✶✳✶ ❱❛r✐❡❞❛❞❡s ❆✜♥s ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✼ ✶✳✶✳✷ ❱❛r✐❡❞❛❞❡s Pr♦❥❡t✐✈❛s ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶✷

  ✶✳✷ ❈✉r✈❛s ❆❧❣é❜r✐❝❛s ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶✺ ✶✳✷✳✶ ▼❛♣❛s ❡♥tr❡ ❈✉r✈❛s ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶✽ ✶✳✷✳✷ ❉✐✈✐s♦r❡s ❞❡ ✉♠❛ ❝✉r✈❛ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✷✵ ✶✳✷✳✸ ❉✐❢❡r❡♥❝✐❛✐s ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✷✹ ✶✳✷✳✹ ❖ ❚❡♦r❡♠❛ ❞❡ ❘✐❡♠❛♥♥✲❘♦❝❤ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✷✺

  ✷ ❈✉r✈❛s ❊❧í♣t✐❝❛s ✸✵

  ✷✳✶ ▲❡✐ ❞❡ ❣r✉♣♦ ♣❛r❛ ✉♠❛ ❝✉r✈❛ ❡❧í♣t✐❝❛ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✸✺ ✷✳✷ ❘❡❞✉çã♦ ❞❡ ✉♠❛ ❝✉r✈❛ ❡❧í♣t✐❝❛ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✹✶

  ✸ ❖ ❚❡♦r❡♠❛ ❞❡ ▼♦r❞❡❧❧✲❲❡✐❧ ✹✺

  ✸✳✶ ❆ ✈❡rsã♦ ❢r❛❝❛ ❞♦ ❚❡♦r❡♠❛ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✹✺ ✸✳✶✳✶ ❖ ❡♠♣❛r❡❧❤❛♠❡♥t♦ ❞❡ ❑✉♠♠❡r ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✹✼ ✸✳✶✳✷ ❆ ❞❡♠♦♥str❛çã♦ ❞❛ ✈❡rsã♦ ❢r❛❝❛ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✹✾

  ✸✳✷ ❖ ❚❡♦r❡♠❛ ❞❛ ❉❡s❝✐❞❛ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✺✷ ✸✳✸ ❆❧t✉r❛s ♥♦ ❊s♣❛ç♦ Pr♦❥❡t✐✈♦ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✺✹ ✸✳✹ ❆❧t✉r❛s ❡♠ ❈✉r✈❛s ❊❧í♣t✐❝❛s ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✻✺

  ✹ ❆ ❈♦♥❥❡❝t✉r❛ ❞❡ ❇✐r❝❤ ❡ ❙✇✐♥♥❡rt♦♥✲❉②❡r ✭❇❙❉✮ ❡ ♦ Pr♦❜❧❡♠❛ ❞♦s ◆ú♠❡r♦s ❈♦♥❣r✉❡♥t❡s ✭P◆❈✮

  ✼✸ ✹✳✶ ❆ ❈♦♥❥❡❝t✉r❛ ❞❡ ❇✐r❝❤ ❡ ❙✇✐♥♥❡rt♦♥✲❉②❡r ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✼✹ ✹✳✷ ❖ Pr♦❜❧❡♠❛ ❞♦s ◆ú♠❡r♦s ❈♦♥❣r✉❡♥t❡s ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✽✶

  ❘❡❢❡rê♥❝✐❛s ✾✷

  ■♥tr♦❞✉çã♦

  ❖ ♦❜❥❡t♦ ❞❡ ❡st✉❞♦ ❞❡st❡ tr❛❜❛❧❤♦ sã♦ ❛s ❝❤❛♠❛❞❛s ❝✉r✈❛s ❡❧í♣t✐❝❛s✳ ❙✉❛s ❛♣❛✲ r✐çõ❡s ✐♥✐❝✐❛✐s r❡♠♦♥t❛♠ à ●ré❝✐❛ ❆♥t✐❣❛✱ ♥♦ ❧✐✈r♦ ❆r✐t❤♠❡t✐❝❛ ❞❡ ❉✐♦♣❤❛♥t✉s✳ ▼✉✐t♦s ♠❛t❡♠át✐❝♦s ❛♦ ❧♦♥❣♦ ❞♦ t❡♠♣♦✱ ❛ ❡①❡♠♣❧♦ ❞❡ P♦✐♥❝❛ré✱ ❇✐r❝❤✱ ❙✇✐♥♥❡rt♦♥✲❉②❡r✱ ▲❡♥str❛ ❞❡♥tr❡ ♦✉tr♦s s❡ ❞❡♣❛r❛r❛♠ ❝♦♠ ♣r♦❜❧❡♠❛s q✉❡ ♦s ❧❡✈❛r❛♠ ❛ ❡st✉❞❛r ❛ ♥❛t✉r❡③❛ ❞❛s ❝✉r✈❛s ❡❧í♣t✐❝❛s✳ ❆t✉❛❧♠❡♥t❡✱ s❛❜❡✲s❡ ❜❛st❛♥t❡ ❛ s❡✉ r❡s♣❡✐t♦ ❡ ❞❡ s✉❛ ✉t✐❧✐❞❛❞❡ ❡♠ ❛♣❧✐❝❛çõ❡s q✉❡ ✈ã♦ ❞❡s❞❡ ❛ ▼❛t❡♠át✐❝❛ ♣✉r❛ à ❋ís✐❝❛ ❡ ❈r✐♣t♦❣r❛✜❛✱ ❡♠❜♦r❛ ♠✉✐t♦s s❡❥❛♠ t❛♠❜é♠ ♦s ♣r♦❜❧❡♠❛s ❡ ❝♦♥❥❡❝t✉r❛s q✉❡ ❛s ❡♥✈♦❧✈❡♠✳ ❆❧❣✉♥s ❞❡st❡s ♣r♦❜❧❡♠❛s s❡rã♦ ❛❜♦r❞❛❞♦s ❛q✉✐✳

  ❆s ❝✉r✈❛s ❡❧í♣t✐❝❛s sã♦ ♦❜❥❡t♦s ❞❡ ♥❛t✉r❡③❛ ❣❡♦♠étr✐❝❛✱ q✉❡ ❛❞♠✐t❡♠ ✉♠❛ ❡s✲ tr✉t✉r❛ ❛❧❣é❜r✐❝❛ ❞❡ ❣r✉♣♦ ❞❡✜♥✐❞❛ ❣❡♦♠❡tr✐❝❛♠❡♥t❡✳ ❊st❡ ❝♦♠♣♦rt❛♠❡♥t♦ ❣❡ô♠❡tr♦✲ ❛❧❣é❜r✐❝♦ ❢❛③ ❞❡st❡s ✉♠ ❞♦s ♦❜❥❡t♦s ♠❛✐s ❢❛s❝✐♥❛♥t❡s ❞❛ ♠❛t❡♠át✐❝❛✱ ❝♦♠ ✈❛r✐❛❞❛s ❛♣❧✐✲ ❝❛çõ❡s✳ ❆q✉✐ K ❞❡♥♦t❛rá ✉♠ ❝♦r♣♦ ❞❡ ♥ú♠❡r♦s ❡ K ✉♠ ❢❡❝❤♦ ❛❧❣é❜r✐❝♦✳

  2

  ❯♠❛ ❝✉r✈❛ ❡❧í♣t✐❝❛ é ✉♠❛ ❝✉r✈❛ ♥♦ ♣❧❛♥♦ ♣r♦❥❡t✐✈♦ P ✱ ❝♦♠ ✉♠ ♣♦♥t♦ ❜❛s❡ ❡s♣❡✲ ❝í✜❝❛❞♦✱ s❛t✐s❢❛③❡♥❞♦ ✉♠❛ ❡q✉❛çã♦ ❤♦♠♦❣ê♥❡❛ ❞❛ ❢♦r♠❛

  2

  2

  

3

  2

  2

  3 Y Z + a

  3 Y Z = X + a

  6 Z ,

  1 XY Z + a

  2 X Z + a

  4 XZ + a

  , . . . , a

  1

  

6

  ♦♥❞❡ O = [0, 1, 0] é ♦ ♣♦♥t♦ ❜❛s❡ ❡ a ∈ K✳ ❆ ❡q✉❛çã♦ ❛❝✐♠❛ é ❝❤❛♠❛❞❛ ❡q✉❛çã♦

  X ❞❡ ❲❡✐❡rstr❛ss✱ ❡ ♣♦❞❡ s❡r ❝♦❧♦❝❛❞❛ ❡♠ ❝♦♦r❞❡♥❛❞❛s ♥ã♦ ❤♦♠♦❣ê♥❡❛s ❢❛③❡♥❞♦ x = ❡

  Z Y y =

  ✱ ♦❜t❡♥❞♦ Z

  2

  3

  2 E : y + a xy + a y = x + a x + a x + a .

  1

  3

  2

  4

  6

  , . . . ,

  1

  ▲❡♠❜r❛♥❞♦ q✉❡ ❞❡✈❡♠♦s ❝♦♥s✐❞❡r❛r ♦ ♣♦♥t♦ ♥♦ ✐♥✜♥✐t♦✱ ❞❡♥♦t❛❞♦ ♣♦r O✳ ❙❡ a a

  6

  ∈ K✱ ❞✐③❡♠♦s q✉❡ E ❡stá ❞❡✜♥✐❞❛ s♦❜r❡ K✱ ❡ ❡s❝r❡✈❡♠♦s E/K✳ P♦❞❡♠♦s ❞❡✜♥✐r t❛♠❜é♠ ✉♠❛ ❝✉r✈❛ ❡❧í♣t✐❝❛ ❝♦♠♦ s❡♥❞♦ ✉♠ ♣❛r (E, O)✱ ♦♥❞❡

  E é ✉♠❛ ❝✉r✈❛ ♣r♦❥❡t✐✈❛ ❞❡ ❣ê♥❡r♦ ✶✱ ❡ O ✉♠ ♣♦♥t♦ ❞❡ E✳ ❆♠❜❛s ❛s ❞❡✜♥✐çõ❡s sã♦

  ❡q✉✐✈❛❧❡♥t❡s✱ ♥♦ s❡♥t✐❞♦ ❞❡ q✉❡ ❝♦♥s✐❞❡r❛♥❞♦ ✉♠❛ ❝✉r✈❛ ❞❛❞❛ ♣❡❧❛ ♣r✐♠❡✐r❛ ❞❡✜♥✐çã♦✱ ❡❧❛ s❛t✐s❢❛③ ❛ s❡❣✉♥❞❛ ❞❡✜♥✐çã♦✱ ❡ q✉❡ s❡ ❝♦♥s✐❞❡r❛r♠♦s ✉♠❛ ❝✉r✈❛ ♣❡❧❛ s❡❣✉♥❞❛ ❞❡✜♥✐çã♦✱

  2

  ❡①✐st✐rá ✉♠ ♠♦❞❡❧♦ ❡♠ P ✐s♦♠♦r❢♦ à ❝✉r✈❛ ❡♠ q✉❡stã♦✳ ❖ ❝♦♥❥✉♥t♦ ❞♦s ♣♦♥t♦s K✲r❛❝✐♦♥❛✐s ❞❡ E✱ ❞❡♥♦t❛❞♦ ♣♦r E(K)✱ é ❞❡✜♥✐❞♦ ❝♦♠♦ ♦

  ✸ ❝♦♥❥✉♥t♦ ❞♦s ♣♦♥t♦s s♦❜r❡ ❛ ❝✉r✈❛ ❝♦♠ ❝♦♦r❞❡♥❛❞❛s ❡♠ K✱ ✐st♦ é✱

  2

  2

  3

  2 E(K) = : y + a xy + a y = x + a x + a x + a

  1

  3

  2

  4

  6

  {(x, y) ∈ K } ∪ {O},

  1 , . . . , a

  6

  ♦♥❞❡ a ∈ K✳ ❊st❛r❡♠♦s ✐♥t❡r❡ss❛❞♦s ❡♠ ❡st✉❞❛r ❛ ❣❡♦♠❡tr✐❛ ❞❡ss❛s ❝✉r✈❛s ♣❛r❛ ❡♥tã♦✱ ❜✉s❝❛r

  ❡♥t❡♥❞❡r ❛ s✉❛ ❛r✐t♠ét✐❝❛ ♥♦ s❡♥t✐❞♦ ❞❡ ❜✉s❝❛r ✐♥❢♦r♠❛çõ❡s s♦❜r❡ ❛ ❡str✉t✉r❛ ❞♦s s❡✉s ♣♦♥t♦s K✲r❛❝✐♦♥❛✐s✳

  ❍❡♥r② P♦✐♥❝❛ré ❝♦♥❥❡❝t✉r♦✉ q✉❡ ♦ ❝♦♥❥✉♥t♦ ❞❡ ♣♦♥t♦s r❛❝✐♦♥❛✐s s❡r✐❛ ✉♠ ❣r✉♣♦ ✜♥✐t❛♠❡♥t❡ ❣❡r❛❞♦ ❡ ▼♦r❞❡❧❧ ♣r♦✈♦✉ ❡ss❡ r❡s✉❧t❛❞♦ ♣❛r❛ ❝✉r✈❛s ❡❧í♣t✐❝❛s r❛❝✐♦♥❛✐s ❡♠ ✶✾✷✷✱ ❡ ❡♠ ✶✾✷✽✱ ❲❡✐❧ ❡st❡♥❞❡✉ ❡ss❡ r❡s✉❧t❛❞♦ ♣❛r❛ ❝✉r✈❛s ❡❧í♣t✐❝❛s s♦❜r❡ ❝♦r♣♦s ❞❡ ♥ú♠❡r♦s✱ t❡♥❞♦ ❤♦❥❡ ♦ r❡s✉❧t❛❞♦ ❛ s❡❣✉✐♥t❡ ❢♦r♠❛✿ ❙❡ E/K é ✉♠❛ ❝✉r✈❛ ❡❧í♣t✐❝❛ s♦❜r❡ ✉♠ ❝♦r♣♦ ❞❡ ♥ú♠❡r♦s✱ ❡♥tã♦ E(K) é ✉♠ ❣r✉♣♦ ✜♥✐t❛♠❡♥t❡ ❣❡r❛❞♦✱ ❝♦♠

  r

  E(K) ∼ = E(K) ,

  tor ⊕ Z tor

  ♣❛r❛ ❛❧❣✉♠ r ≥ 0✳ ❊ss❡ r ❝❤❛♠❛r❡♠♦s ❞❡ ♣♦st♦ ❛❧❣é❜r✐❝♦ ❞❡ E✱ ❡ E(K) ♦ ❣r✉♣♦ ❞❡ t♦rçã♦✱ q✉❡ é ♦ s✉❜❣r✉♣♦ ❞♦s ♣♦♥t♦s K✲r❛❝✐♦♥❛✐s ❞❡ ♦r❞❡♠ ✜♥✐t❛✳ ❖ ❈❛♣ít✉❧♦ ✸ s❡rá ❞❡❞✐❝❛❞♦ à ❞❡♠♦♥str❛çã♦ ❞❡ss❡ r❡s✉❧t❛❞♦✳

  ●r❛♥❞❡ ♣❛rt❡ ❞❛ ♣❡sq✉✐s❛ ❡♠ ❣❡♦♠❡tr✐❛ ❛r✐t♠ét✐❝❛✱ ár❡❛ q✉❡ ✉s❛ ❞❡ ♠ét♦❞♦s ❞❛ ❣❡♦♠❡tr✐❛ ❛❧❣é❜r✐❝❛ ♣❛r❛ ♦❜t❡r r❡s✉❧t❛❞♦s ❛r✐t♠ét✐❝♦s✱ é ♦ ❡st✉❞♦ ❞❡ ♣♦♥t♦s r❛❝✐♦♥❛✐s ❡♠ ✈❛r✐❡❞❛❞❡s✳ ❆ss✐♠✱ é ❞❡ ❣r❛♥❞❡ ✐♠♣♦rtâ♥❝✐❛ s❛❜❡r ♦ q✉❡ ❛❝♦♥t❡❝❡ ❝♦♠ ♦ s✉❜❣r✉♣♦ ❞❡ t♦rçã♦ ❡ ♦ ♣♦st♦ ❞❡ ✉♠❛ ❝✉r✈❛ ❡❧í♣t✐❝❛✳ ❙♦❜r❡ ♦ s✉❜❣r✉♣♦ ❞❡ t♦rçã♦ ✭♣♦♥t♦s ❞❡ ♦r❞❡♠ ✜♥✐t❛✮✱ ▼❛③✉r ♣r♦✈♦✉ ♦ s❡❣✉✐♥t❡ r❡s✉❧t❛❞♦ ♣❛r❛ ❝✉r✈❛s ❡❧í♣t✐❝❛s r❛❝✐♦♥❛✐s✱ q✉❡ ❞❡s❝r❡✈❡ ❛s ♣♦ss✐❜✐❧✐❞❛❞❡s ♣❛r❛ ♦ s✉❜❣r✉♣♦ ❞❡ t♦rçã♦✳

  ❙❡❥❛ E/Q ✉♠❛ ❝✉r✈❛ ❡❧í♣t✐❝❛✳ ❊♥tã♦ E(Q) tor é ✐s♦♠♦r❢♦ ❛ ✉♠ ❞♦s s❡❣✉✐♥t❡s ❣r✉♣♦s

  Z/nZ, ❝♦♠ 1 ≤ n ≤ 10 ♦✉ n = 12;

  Z/2Z ⊕ Z/2nZ, ❝♦♠ 1 ≤ n ≤ 4. ❈❛❧❝✉❧❛r ♦ ♣♦st♦ ❞❡ ✉♠❛ ❝✉r✈❛ ❡❧✐♣t✐❝❛ ❡♠ ❣❡r❛❧ é ✉♠ ♣r♦❜❧❡♠❛ ❞✐❢í❝✐❧✱ ❡ s♦❜r❡

  ✐ss♦✱ ❡♠ ✶✾✻✺✱ ❇✳ ❏✳ ❇✐r❝❤ ❡ ❙✐r ❍✳ P✳ ❋✳ ❙✇✐♥♥❡rt♦♥✲❉②❡r ❝♦♥❥❡❝t✉r❛r❛♠ q✉❡ ♦ ♣♦st♦ ❛❧❣é❜r✐❝♦ ❞❡ ✉♠❛ ❝✉r✈❛ ❡❧í♣t✐❝❛ r❛❝✐♦♥❛❧ s❡r✐❛ ✐❣✉❛❧ ❛ ✉♠ ♦✉tr♦ ✐♥✈❛r✐❛♥t❡✱ ❛❣♦r❛ ❛♥❛❧ít✐❝♦✱ r❡❧❛❝✐♦♥❛❞♦ ❛ ▲✲sér✐❡ ❞❛ ❝✉r✈❛✱ q✉❡ ❞❡✜♥✐r❡♠♦s ❧♦❣♦ ♠❛✐s✳

  2

  3

  = x + ax + b ❙❡❥❛ E : y ✱ ✉♠❛ ❝✉r✈❛ ❡❧í♣t✐❝❛ r❛❝✐♦♥❛❧ ♥❛ ❢♦r♠❛ ❞❡ ❲❡✐❡rstr❛ss

  p

  r❡❞✉③✐❞❛✱ ❝♦♠ a, b ∈ Z ❡ ∆ s❡✉ ❞✐s❝r✐♠✐♥❛♥t❡✳ ❉❛❞♦ p ♣r✐♠♦✱ s❡❥❛ Z → F ✱ z 7→ z✱ ❛ r❡❞✉çã♦ ♠ó❞✉❧♦ p ❡ ❝♦♥s✐❞❡r❡ ❛ ❝✉r✈❛

  2

  

3

E : y = x + ax + b

p ❡

  ✹ a = p + 1 (F ),

  

p p p

  − #E (F )

  p p p p

  ♦♥❞❡ E é ♦ ❝♦♥❥✉♥t♦ ❞❡ ♣♦♥t♦s F ✲r❛❝✐♦♥❛✐s ❞❛ ❝✉r✈❛ E ✱ ❝❤❛♠❛❞❛ ❛ r❡❞✉çã♦ ♠ó❞✉❧♦ p

  p

  ❞❡ E✳ ❉✐③❡♠♦s q✉❡ p é ✉♠ ♣r✐♠♦ ❞❡ ❜♦❛ r❡❞✉çã♦ s❡ ❛ ❝✉r✈❛ E é ♥ã♦ s✐♥❣✉❧❛r✱ ❝❛s♦ ❝♦♥trár✐♦ ❞✐③❡♠♦s q✉❡ p é ❞❡ ♠á r❡❞✉çã♦✳ ❚❡♠♦s q✉❡ ♦s ♣r✐♠♦s ❞❡ ❜♦❛ r❡❞✉çã♦ sã♦

  2

  3

  = x + ax + b ❙❡♥❞♦ E/Q ✉♠❛ ❝✉r✈❛ r❛❝✐♦♥❛❧ ❝♦♠ ❡q✉❛çã♦ E : y ✱ ❝♦♠ a, b ∈ Z✳

  ❙✉❛ L✲sér✐❡ é ❞❡✜♥✐❞❛ ♣❡❧♦ ♣r♦❞✉t♦ ❞❡ ❊✉❧❡r Y Y

  1

  1 L(E, s) = ,

  −s −s 1−2s

  1 p 1 p + p

  p p

  − a − a

  p | ∆ p ∤ ∆

  ❝♦♠ s ∈ C✳ ❚❡♠♦s q✉❡ ♦ ♣r♦❞✉t♦ ❛❝✐♠❛ ❞❡✜♥❡ ✉♠❛ sér✐❡ ❞❡ ❉✐r✐❝❤❧❡t

  ∞

  n

  X b

  L(E, s) = ,

  s

  n

  n=1

  = a

  p p

  ❡ ♥❡ss❡ ❝❛s♦✱ b ✱ ♣❛r❛ t♦❞♦ p ♣r✐♠♦✳ ➱ ♣♦ssí✈❡❧ ♠♦str❛r q✉❡ ❛ sér✐❡ ❛❝✐♠❛ ❝♦♥✈❡r❣❡

  3 ♣❛r❛ t♦❞♦ ♣♦♥t♦ s ❞♦ ♣❧❛♥♦ ❝♦♠ Re(s) > ✳ ●r❛ç❛s ❛♦ ❚❡♦r❡♠❛ ❞❛ ▼♦❞✉❧❛r✐❞❛❞❡ ♣r♦✈❛❞♦

  2 ♣♦r ❲✐❧❡s ❡♠ ❬❲✐❧❡s✷❪✱ é ♣♦ssí✈❡❧ ♠♦str❛r ❛ ❡①✐stê♥❝✐❛ ❞❡ ❝♦♥t✐♥✉❛çã♦ ❛♥❛❧ít✐❝❛ ❞❡ L(E, s) ❛ t♦❞♦ ♦ ♣❧❛♥♦ ❝♦♠♣❧❡①♦✳ ❆ss✐♠ ♣♦❞❡♠♦s ❝♦♥s✐❞❡r❛r ❛ ❡①♣❛♥sã♦ ❞❡ L(E, s) ❡♠ t♦r♥♦ ❞❡ s = 1

  ✳ ❚❡♠♦s ❡♥tã♦ ❛ ✈❡rsã♦ ❢r❛❝❛ ❞❛ ❈♦♥❥❡❝t✉r❛ ❞❡ ❇✐r❝❤ ❡ ❙✇✐♥♥❡rt♦♥✲❉②❡r✳ ❆ ❡①♣❛♥sã♦ ❞❡ ❚❛②❧♦r ❞❡ L(E, s) ❡♠ t♦r♥♦ ❞❡ s = 1 t❡♠ ❛ ❢♦r♠❛

  r r+1

  L(E, s) = c (s + c (s + r r+1 − 1) − 1) t❡r♠♦s ❞❡ ❣r❛✉s ♠❛✐♦r❡s,

  r

  ❡♠ q✉❡ c 6= 0 ❡ r é ♦ ♣♦st♦ ❛❧❣é❜r✐❝♦ ❞❡ E✳ ❉❡s❞❡ ❛ s✉❛ ❢♦r♠✉❧❛çã♦✱ ❛❧❣✉♥s r❡s✉❧t❛❞♦s ♣❛r❝✐❛✐s ❢♦r❛♠ ❡♥❝♦♥tr❛❞♦s✱ ♠❛s ❛té

  ❤♦❥❡ ♦ ♣r♦❜❧❡♠❛ ❝♦♥t✐♥✉❛ ❡♠ ❛❜❡rt♦✱ s❡♥❞♦ q✉❡ ❡♠ ✷✵✵✵ ♦ ❈❧❛② ▼❛t❤❡♠❛t✐❝s ■♥st✐t✉t❡ ♦ ❧✐st♦✉ ❝♦♠♦ ✉♠ ❞♦s ♣r♦❜❧❡♠❛s ❞♦ ♠✐❧ê♥✐♦✳

  ◆♦ ❈❛♣ít✉❧♦ ✹✱ ❛♣r❡s❡♥t❛r❡♠♦s ❛ ❝♦♥❥❡❝t✉r❛✱ q✉❡ s❡♥❞♦ ✈❡r❞❛❞❡✐r❛ t❡rá ❝♦♠♦ ❝♦♥✲ s❡q✉ê♥❝✐❛ ❛ ❡①✐stê♥❝✐❛ ❞❡ ✉♠ ❛❧❣♦r✐t♠♦ ♣❛r❛ ❞❡t❡r♠✐♥❛r s❡ ✉♠ ❞❛❞♦ ✐♥t❡✐r♦ ♣♦s✐t✐✈♦ n é ♦✉ ♥ã♦ ✉♠ ♥ú♠❡r♦ ❝♦♥❣r✉❡♥t❡✳ P♦r ✉♠ ♥ú♠❡r♦ ❝♦♥❣r✉❡♥t❡✱ ❡♥t❡♥❞❡♠♦s ✉♠ r❛❝✐♦♥❛❧ q✉❡ é ár❡❛ ❞❡ ✉♠ tr✐â♥❣✉❧♦ r❡tâ♥❣✉❧♦ ❝♦♠ ❧❛❞♦s r❛❝✐♦♥❛✐s✳ ❆té ♦ ♠♦♠❡♥t♦ ♥ã♦ t❡♠♦s ❝♦♥❤❡✲ ❝✐♠❡♥t♦ ❞❡ ✉♠ t❛❧ ❛❧❣♦r✐t♠♦✱ ❡ ❡ss❡ é ♦ ❝❤❛♠❛❞♦ Pr♦❜❧❡♠❛ ❞♦s ◆ú♠❡r♦s ❈♦♥❣r✉❡♥t❡s✱ ❞♦ q✉❛❧ ❢❛❧❛r❡♠♦s ♥❡ss❡ tr❛❜❛❧❤♦✳

  ❆ r❡❧❛çã♦ ❞♦ ♣r♦❜❧❡♠❛ ❞♦s ♥ú♠❡r♦s ❝♦♥❣r✉❡♥t❡s ❝♦♠ ❛s ❝✉r✈❛s ❡❧í♣t✐❝❛s s❡ ❞❡✈❡ ❛♦ s❡❣✉✐♥t❡✳

  ✺ ❉❛❞♦ n > 1✱ ❞❡✜♥✐♠♦s ❛ ❝✉r✈❛ ❡❧í♣t✐❝❛

  2

  3

  2 E : y = x x. n

  − n

  n

  ❙♦❜r❡ E t❡♠♦s ♦ s❡❣✉✐♥t❡ r❡s✉❧t❛❞♦✳ n

  n

  é ❝♦♥❣r✉❡♥t❡ s❡✱ ❡ s♦♠❡♥t❡ s❡✱ E t❡♠ ✐♥✜♥✐t♦s ♣♦♥t♦s r❛❝✐♦♥❛✐s✳ P❡❧❛ ♣r♦♣♦s✐çã♦ ❛❝✐♠❛✱ n é ✉♠ ♥ú♠❡r♦ ❝♦♥❣r✉❡♥t❡ s❡✱ ❡ s♦♠❡♥t❡ s❡✱ ♦ ♣♦st♦

  n

  ❛❧❣é❜r✐❝♦ ❞❡ E é ♣♦s✐t✐✈♦✳ ❙♦❜r❡ ♦ ♣r♦❜❧❡♠❛ ❞❡ ❞❡❝✐❞✐r s❡ ✉♠ ❞❛❞♦ ✐♥t❡✐r♦ ♣♦s✐t✐✈♦ n é ❝♦♥❣r✉❡♥t❡✱ ❚✉♥♥❡❧❧

  ❡♠ ❬❚✉♥♥❡❧❧❪✱ ♣r♦✈♦✉ ♦ s❡❣✉✐♥t❡ r❡s✉❧t❛❞♦✳ ❙❡❥❛ n ✉♠ ✐♥t❡✐r♦ ♣♦s✐t✐✈♦ ❧✐✈r❡ ❞❡ q✉❛❞r❛❞♦s✳ ❉❡✜♥❛

  

3

  2

  2

  2 A = # : n = 2x + y + 8z n

  {(x, y, z) ∈ Z },

  3

  2

  2

  2 B n = # : n = 2x + y + 32z

  {(x, y, z) ∈ Z },

  3

  2

  2

  2 C n = # : n = 8x + 2y + 16z

  {(x, y, z) ∈ Z },

  3

  2

  2

  2 D = # : n = 8x + 2y + 64z n

  {(x, y, z) ∈ Z }.

  = 2B

  n n

  ❙❡ n é í♠♣❛r ❡ ✉♠ ♥ú♠❡r♦ ❝♦♥❣r✉❡♥t❡✱ ❡♥tã♦ A ✳ ❙❡ n é ♣❛r ❡ ❝♦♥❣r✉❡♥t❡✱ ❡♥tã♦ C = 2D

  n n n

  ✳ P♦r ♦✉tr♦ ❧❛❞♦✱ s❡ ❛ ✈❡rsã♦ ❢r❛❝❛ ❞❡ ❇❙❉ ✈❛❧❡ ♣❛r❛ E ✱ ❡♥tã♦ s❡ n é í♠♣❛r ❡ A = 2B = 2D

  n n n n

  ♦✉ s❡ n é ♣❛r ❡ C ✱ ❡♥tã♦ n é ❝♦♥❣r✉❡♥t❡✳ ◆♦ ❈❛♣ít✉❧♦ ✶✱ ✐♥tr♦❞✉③✐r❡♠♦s ❛❧❣✉♥s ❝♦♥❝❡✐t♦s ❡ r❡s✉❧t❛❞♦s s♦❜r❡ ✈❛r✐❡❞❛❞❡s

  ❛❧❣é❜r✐❝❛s ❛✜♥s ❡ ♣r♦❥❡t✐✈❛s✱ ❝♦♥❝❡✐t♦s ❝♦♠♦ s✉❛✈✐❞❛❞❡ ❡ ❞✐♠❡♥sã♦ s❡rã♦ ❞❡✜♥✐❞♦s✳ ❆✐♥❞❛ ♥❡ss❡ ❝❛♣ít✉❧♦✱ ❞❛r❡♠♦s ❛t❡♥çã♦ às ✈❛r✐❡❞❛❞❛❞❡s ❛❧❣é❜r✐❝❛s ❞❡ ❞✐♠❡♥sã♦ ✶✱ ❛s ❝❤❛♠❛❞❛s ❝✉r✈❛s ❛❧❣é❜r✐❝❛s✳ ❘❡s✉❧t❛❞♦s ❝♦♠♦ ♦ ❚❡♦r❡♠❛ ❞❡ ❇é③♦✉t ❡ ♦ ❚❡♦r❡♠❛ ❞❡ ❘✐❡♠❛♥♥✲❘♦❝❤ s❡rã♦ ❡♥✉♥❝✐❛❞♦s✳

  ◆♦ ❈❛♣ít✉❧♦ ✷✱ ❛s ❝✉r✈❛s ❡❧í♣t✐❝❛s s❡rã♦ ✐♥tr♦❞✉③✐❞❛s✳ ▼♦str❛r❡♠♦s ❝♦♠♦ é ✐♥tr♦✲ ❞✉③✐❞❛ à ❡str✉t✉r❛ ❞❡ ❣r✉♣♦ ❡♠ ✉♠❛ ❝✉r✈❛ ❡❧í♣t✐❝❛✱ ❡ ❡♥❝♦♥tr❛r❡♠♦s ❡①♣r❡ssõ❡s ❛❧❣é❜r✐❝❛s ♣❛r❛ ❡ss❛ ♦♣❡r❛çã♦ ❞❡ ❣r✉♣♦ q✉❡ s❡rá ❞❡✜♥✐❞❛ ❣❡♦♠❡tr✐❝❛♠❡♥t❡✳ ❚❛♠❜é♠ ❡st❛r❡♠♦s ✐♥✲ t❡r❡ss❛❞♦s ❡♠ ❡st✉❞❛r ❛ ❝✉r✈❛ q✉❛♥❞♦ ✈✐st❛ s♦❜r❡ ♦ ❝♦♠♣❧❡t❛♠❡♥t♦ ❞♦ ❝♦r♣♦ K ♣♦r ✉♠❛ ✈❛❧♦r✐③❛çã♦ ❞✐s❝r❡t❛✱ ❛ ✜♠ ❞❡ ❞❡✜♥✐r ❛ r❡❞✉çã♦ ❞❛ ❝✉r✈❛ ❝♦♠ r❡❧❛çã♦ ❛ ✈❛❧♦r✐③❛çã♦✳

  ◆♦ ❈❛♣ít✉❧♦ ✸✱ s❡rá ❛♣r❡s❡♥t❛❞❛ ❛ ❞❡♠♦♥str❛çã♦ ❞♦ ❚❡♦r❡♠❛ ❞❡ ▼♦r❞❡❧❧✲❲❡✐❧✳ ❈♦♠❡ç❛♠♦s ❞❡♠♦♥str❛♥❞♦ ❛ ✈❡rsã♦ ❢r❛❝❛ ❞♦ ❚❡♦r❡♠❛ ❞❡ ▼♦r❞❡❧❧✲❲❡✐❧✱ ♣r♦✈❛♠♦s ♦ ❚❡♦✲ r❡♠❛ ❞❛ ❉❡s❝✐❞❛ q✉❡ ❞✐③ q✉❛♥❞♦ ✉♠❛ ❢✉♥çã♦ r❡❛❧ ❞❡✜♥✐❞❛ ❡♠ ✉♠ ❣r✉♣♦ ❛❜❡❧✐❛♥♦ s❛t✐s❢❛③ ❝❡rt❛s ♣r♦♣r✐❡❞❛❞❡s ♥♦s ♣❡r♠✐t❡ ❞❡❞✉③✐r q✉❡ ❡st❡ ❣r✉♣♦ é ✜♥✐t❛♠❡♥t❡ ❣❡r❛❞♦✳ ❉❡♣♦✐s ❞❡ ♣r♦✈❛❞♦ ♦ ❚❡♦r❡♠❛ ❞❛ ❞❡s❝✐❞❛✱ ❡st❛r❡♠♦s ❡♠ ❜✉s❝❛ ❞❡ ❝♦♥str✉✐r ❢✉♥çõ❡s ❡♠ ❝✉r✈❛s ❡❧í♣✲ t✐❝❛s q✉❡ s❛t✐s❢❛ç❛♠ ❛s ❤✐♣ót❡s❡s ❞♦ ❚❡♦r❡♠❛ ❞❛ ❉❡s❝✐❞❛✱ ♣❛r❛ ❡♥✜♠ ♣r♦✈❛r ♦ ❚❡♦r❡♠❛ ❞❡ ▼♦r❞❡❧❧✲❲❡✐❧✳

  ✻ ◆♦ ❈❛♣ít✉❧♦ ✹✱ ❢❛r❡♠♦s ✉♠❛ ❜r❡✈❡ ❛♣r❡s❡♥t❛çã♦ ❞❛ ❈♦♥❥❡❝t✉r❛ ❞❡ ❇✐r❝❤ ❡ ❙✇✐♥♥❡r✲ t♦♥✲❉②❡r ✭❇❙❉✮ ❡ ❞♦ Pr♦❜❧❡♠❛ ❞♦s ◆ú♠❡r♦s ❈♦♥❣r✉❡♥t❡s✱ ❝♦♠♦ ❡st❡s ♣r♦❜❧❡♠❛s ❛r✐t♠é✲ t✐❝♦s ❡stã♦ r❡❧❛❝✐♦♥❛❞♦s ❝♦♠ ❝✉r✈❛s ❡❧í♣t✐❝❛s✳ ❯♠ ❞❡❧❡s✱ ❛ ❈♦♥❥❡❝t✉r❛ ❇❙❉ ❥á ♥❛s❝❡ ❝♦♠♦ ✉♠ ♣r♦❜❧❡♠❛ s♦❜r❡ ❝✉r✈❛s ❡❧í♣t✐❝❛s✱ ❥á ♦ Pr♦❜❧❡♠❛ ❞♦s ◆ú♠❡r♦s ❈♦♥❣r✉❡♥t❡s é ✉♠ ♣r♦✲ ❜❧❡♠❛ s♦❜r❡ ❛ ❞❡t❡r♠✐♥❛çã♦ ❞♦s ✐♥t❡✐r♦s q✉❡ sã♦ ♠❡❞✐❞❛s ❞❡ ár❡❛s ❞❡ tr✐â♥❣✉❧♦s r❡tâ♥❣✉❧♦s ❝♦♠ ❧❛❞♦s r❛❝✐♦♥❛✐s✳

  ❈❛♣ít✉❧♦ ✶ Pr❡❧✐♠✐♥❛r❡s

  ❊st❡ t❡①t♦ tr❛t❛rá ❞❛s ❝❤❛♠❛❞❛s ❝✉r✈❛s ❡❧í♣t✐❝❛s✱ ❡ ♣❛r❛ ✐ss♦ ♥❡❝❡ss✐t❛♠♦s ❞❡ ❛❧❣✉♥s ❝♦♥❝❡✐t♦s ❡ r❡s✉❧t❛❞♦s ❞❛ ●❡♦♠❡tr✐❛ ❆❧❣é❜r✐❝❛✳ ◆❡st❡ ❝❛♣ít✉❧♦ s❡rã♦ ✐♥tr♦❞✉③✐❞♦s ♥♦çõ❡s ❜ás✐❝❛s ❞❛ ❧✐♥❣✉❛❣❡♠ ❞❛ ●❡♦♠❡tr✐❛ ❆❧❣é❜r✐❝❛ q✉❡ s❡rã♦ ♦ s✉♣♦rt❡ ✐♥✐❝✐❛❧ ♣❛r❛ ♦s ❞❡♠❛✐s ❝❛♣ít✉❧♦s✳ P❛r❛ t❛❧ ❛❜♦r❞❛❣❡♠ ❛ss✉♠✐r❡♠♦s q✉❡ ♦ ❧❡✐t♦r ❡st❡❥❛ ❢❛♠✐❧✐❛r✐③❛❞♦ ❝♦♠ ❛❧❣✉♠❛s ❢❡rr❛♠❡♥t❛s ❞❛ ➪❧❣❡❜r❛ ❈♦♠✉t❛t✐✈❛ ❡ ❚❡♦r✐❛ ❞❡ ●❛❧♦✐s✳ ■♥❞✐❝❛♠♦s ❛s r❡❢❡rê♥❝✐❛s ❬❆t✐②❛❤❪ ❡ ❬▼♦r❛♥❞✐❪✳ ❯s❛r❡♠♦s |X| ♣❛r❛ ❞❡♥♦t❛r ❛ ❝❛r❞✐♥❛❧✐❞❛❞❡ ❞♦ ❝♦♥❥✉♥t♦ X✳

  ❆ ♣r✐♥❝✐♣❛❧ r❡❢❡rê♥❝✐❛ ♣❛r❛ ♦ ❞❡s❡♥✈♦❧✈✐♠❡♥t♦ ❞❡st❡ ❝❛♣ít✉❧♦ é ♦ ❧✐✈r♦ ✭❬❙✐❧✈❡r✲ ♠❛♥❪✮✳ ❉✉r❛♥t❡ ♦ t❡①t♦ K ❞❡♥♦t❛rá ✉♠ ❝♦r♣♦ ♣❡r❢❡✐t♦✱ ❡ K ♦ s❡✉ ❢❡❝❤♦ ❛❧❣é❜r✐❝♦ ❡

G(K/K)

  ♦ ❣r✉♣♦ ❞❡ ●❛❧♦✐s ❞❡ss❛ ❡①t❡♥sã♦✳

  ✶✳✶ ❱❛r✐❡❞❛❞❡s ❆✜♥s ❡ Pr♦❥❡t✐✈❛s

  ◆❡st❛ s❡çã♦ ❢❛r❡♠♦s ✉♠❛ ❜r❡✈❡ ❡①♣♦s✐çã♦ s♦❜r❡ ✈❛r✐❡❞❛❞❡s ❛❧❣é❜r✐❝❛s✱ ❞❡✜♥✐♥❞♦ ❝♦♥❝❡✐t♦s ❝♦♠♦ s✉❛✈✐❞❛❞❡✱ ❞✐♠❡♥sã♦✱ ❛♥❡❧ ❧♦❝❛❧ ❞❡♥tr❡ ♦✉tr♦s✳ ❆❧❣✉♥s r❡s✉❧t❛❞♦s s❡rã♦ ♣r♦✈❛❞♦s✱ ❡ ♦✉tr♦s ❡st❛rã♦ ❛❝♦♠♣❛♥❤❛❞♦s ❞❡ r❡❢❡rê♥❝✐❛s ♣❛r❛ s✉❛ ❞❡♠♦♥str❛çã♦✳

  ✶✳✶✳✶ ❱❛r✐❡❞❛❞❡s ❆✜♥s

  ❉❡✜♥✐çã♦ ✶✳✶✳ ❖ n✲❡s♣❛ç♦ ❛✜♠ ✭s♦❜r❡ K✮ é ♦ ❝♦♥❥✉♥t♦ ❞❛s n✲✉♣❧❛s ❝♦♠ ❡♥tr❛❞❛s ❡♠ K

  n n

  A = A (K) = , . . . , x ) : x

  1 n i {(x ∈ K}. n

  ❙✐♠✐❧❛r♠❡♥t❡✱ ♦ ❝♦♥❥✉♥t♦ ❞♦s ♣♦♥t♦s K✲r❛❝✐♦♥❛✐s ❞❡ A é ♦ ❝♦♥❥✉♥t♦

  n

  A (K) = , . . . , x ) : x

  1 n i {(x ∈ K}. n

  ➱ ✐♠❡❞✐❛t♦ ✈❡r✐✜❝❛r q✉❡ ❣r✉♣♦ ❞❡ ●❛❧♦✐s G(K/K) ❛❣❡ s♦❜r❡ A ✱ ❞❛ s❡❣✉✐♥t❡

  ✽

  n

  ❢♦r♠❛✿ P❛r❛ τ ∈ G(K/K) ❡ P ∈ A ✱

  τ P = (τ (x ), . . . , τ (x )).

  1 n n

  (K) ❆ss✐♠✱ A ♣♦❞❡ s❡r ❝❛r❛❝t❡r✐③❛❞♦ ♣♦r

  n n τ

  A (K) = : P = P, {P ∈ A ∀τ ∈ G(K/K)}.

  , . . . , x ]

  1 n

  ❙❡❥❛ K[X] = K[x ♦ ❛♥❡❧ ❞❡ ♣♦❧✐♥ô♠✐♦s ❡♠ n ✈❛r✐á✈❡✐s ❡ s❡❥❛ I ⊆ K[X]

  n

  ✉♠ ✐❞❡❛❧✳ P❛r❛ ❝❛❞❛ I ❛ss♦❝✐❛♠♦s ✉♠ s✉❜❝♦♥❥✉♥t♦ V (I) ❞❡ A

  n

  V (I) = : f (P ) = 0, {P ∈ A ∀f ∈ I}. ❉✐ss♦✱ s❡❣✉❡ ❛ ❞❡✜♥✐çã♦ ❞❡ ❝♦♥❥✉♥t♦ ❛❧❣é❜r✐❝♦✳ ❉❡✜♥✐çã♦ ✶✳✷✳ ❯♠ ❝♦♥❥✉♥t♦ ❛❧❣é❜r✐❝♦ ✭❛✜♠✮ é q✉❛❧q✉❡r ❝♦♥❥✉♥t♦ ❞❛ ❢♦r♠❛ V (I)✳ ❙❡

  V é ✉♠ ❝♦♥❥✉♥t♦ ❛❧❣é❜r✐❝♦✱ ♦ ✐❞❡❛❧ ❞❡ V é ❞❡✜♥✐❞♦ ❝♦♠♦

  I(V ) = {f ∈ K[X] : f(P ) = 0, ∀P ∈ V }. ❯♠ ❝♦♥❥✉♥t♦ ❛❧❣é❜r✐❝♦ é ❞❡✜♥✐❞♦ s♦❜r❡ K s❡ s❡✉ ✐❞❡❛❧ I(V ) ♣♦❞❡ s❡r ❣❡r❛❞♦ ♣♦r ♣♦❧✐♥ô✲ ♠✐♦s ❡♠ K[X]✱ ❡ ❞❡♥♦t❛r❡♠♦s ♣♦r V/K✳

  ❙❡ V é ❞❡✜♥✐❞♦ s♦❜r❡ K✱ ❡♥tã♦ ♦ ❝♦♥❥✉♥t♦ ❞♦s ♣♦♥t♦s K✲r❛❝✐♦♥❛✐s ❞❡ V é ♦ ❝♦♥❥✉♥t♦

  n V (K) = V (K).

  ∩ A ◆❛ ❞❡✜♥✐çã♦ ❞❡ ❝♦♥❥✉♥t♦ ❛❧❣é❜r✐❝♦✱ ❝♦♥s✐❞❡r❛♠♦s ♦ ❝♦♥❥✉♥t♦ ❞❡ ③❡r♦s ❡♠ ❝♦♠✉♠

  ❞❡ ♣♦❧✐♥ô♠✐♦s ❡♠ ✉♠ ✐❞❡❛❧✱ ♠❛s ♣♦❞❡rí❛♠♦s t❡r t♦♠❛❞♦ ✉♠ ❝♦♥❥✉♥t♦ S q✉❛❧q✉❡r ❞❡ ♣♦❧✐♥ô♠✐♦s ❡ t♦♠❛❞♦ V (S) ❝♦♠♦ ♥❛ ❞❡✜♥✐çã♦ ❞❡ V (I)✳ ❚❡♠♦s q✉❡ V (S) = V (hSi)✱ ♦♥❞❡ hSi é ♦ ✐❞❡❛❧ ❣❡r❛❞♦ ♣♦r S✳

  ❙❛❜❡♠♦s q✉❡ ♣❡❧♦ ❚❡♦r❡♠❛ ❞❛ ❇❛s❡ ❞❡ ❍✐❧❜❡rt✱ t♦❞♦s ♦s ✐❞❡❛✐s ❞❡ K[X] ❡ K[X] sã♦ ✜♥✐t❛♠❡♥t❡ ❣❡r❛❞♦s✱ ❞❡ ♦♥❞❡ s❡❣✉❡ q✉❡ t♦❞♦ ❝♦♥❥✉♥t♦ ❛❧❣é❜r✐❝♦ é ❛ ✐♥t❡rs❡çã♦ ❞❡ ✉♠❛ q✉❛♥t✐❞❛❞❡ ✜♥✐t❛ ❞❡ ❝♦♥❥✉♥t♦s ❛❧❣é❜r✐❝♦s ❣❡r❛❞♦s ♣♦r ❝♦♥❥✉♥t♦s ✉♥✐tár✐♦s ❞❡ ❝❡rt♦s ♣♦❧✐♥ô♠✐♦s✳

  ❯♠ ♦✉tr♦ r❡s✉❧t❛❞♦ ❞❡ ❡①tr❡♠❛ ✐♠♣♦rtâ♥❝✐❛ é ♦ ❚❡♦r❡♠❛ ❞♦s ❩❡r♦s ❞❡ ❍✐❧❜❡rt✳ ❚❡♦r❡♠❛ ✶✳✸ ✭◆✉❧❧st❡❧❧❡♥ss❛t③✮✳ ❙❡❥❛ K ✉♠ ❝♦r♣♦ ❛❧❣❡❜r✐❝❛♠❡♥t❡ ❢❡❝❤❛❞♦✳ ❙❡ I é ✉♠

  1 , . . . , x n ]

  ✐❞❡❛❧ ❞❡ K[x ✱ ❡♥tã♦ √

  I(V (I)) = I, √ √

  m

  I I =

  1 , . . . , x n ] : f

  ♦♥❞❡ ❞❡♥♦t❛ ♦ r❛❞✐❝❛❧ ❞❡ I✱ ♦✉ s❡❥❛✱ {f ∈ K[x ∈ I, ♣❛r❛ ❛❧❣✉♠ m >

  1 }.

  ✾ ❉❡♠♦♥str❛çã♦✳ ❱❡r ✭❈❛♣ít✉❧♦ ✶✱ ❚❡♦r❡♠❛ ✶✳✸❆✱ ❬❍❛rts❤♦r♥❡❪✮✳ ❉❡✜♥✐çã♦ ✶✳✹✳ ❯♠ ❝♦♥❥✉♥t♦ ❛❧❣é❜r✐❝♦ ❛✜♠ V é ❝❤❛♠❛❞❛ ✉♠❛ ✈❛r✐❡❞❛❞❡ ✭❛✜♠✮ s❡ I(V )

  é ✉♠ ✐❞❡❛❧ ♣r✐♠♦ ❡♠ K[X]✳ ❊①❡♠♣❧♦ ✶✳✺✳

  ◆❛ ✜❣✉r❛ t❡♠♦s ❞♦✐s ❡①❡♠♣❧♦s ❞❡ ❝♦♥❥✉♥t♦s ❛❧❣é❜r✐❝♦s✱ ♦♥❞❡ ♠♦str❛♠♦s ❛♣❡♥❛s ♦s ♣♦♥t♦s ❞❡ ❝♦♦r❞❡♥❛❞❛s r❡❛✐s✳ ❊♠ a) t❡♠♦s ✉♠ ❡①❡♠♣❧♦ ❞❡ ✈❛r✐❡❞❛❞❡✱ ❥á ♦ ❝♦♥❥✉♥t♦ ❛❧❣é❜r✐❝♦ ❡♠ b) ♥ã♦ é ✉♠❛ ✈❛r✐❡❞❛❞❡✳ p

  2

  2

  2

  2

  ❈♦♠ ❡❢❡✐t♦✱ ❝♦♠♦ I(V (x −y)) = hx − yi = hx −yi é ♣r✐♠♦✱ ❡♥tã♦ V (x −y) é ✉♠❛ ✈❛r✐❡❞❛❞❡✳ ❆❣♦r❛✱ s❡♥❞♦ f(x, y) = x−y ❡ g(x, y) = x+y✱ t❡♠♦s q✉❡ f(1, −1) = 2 6= 0

  2

  2

  2

  2

  ) )) ❡ g(1, 1) = 2 6= 0✳ ▼❛s ❝♦♠♦ (1, −1), (1, 1) ∈ V (x ❡♥tã♦ x−y, x+y 6∈ I(V (x ✳

  −y −y

  2

  2

  2

  2

  2

  2

  )) ))

  ❆ss✐♠✱ I(V (x − y ♥ã♦ é ♣r✐♠♦✱ ✈✐st♦ q✉❡ (x − y) · (x + y) = x − y ∈ I(V (x − y ✳ ❙❡❥❛ V/K ✉♠❛ ✈❛r✐❡❞❛❞❡✱ ✐st♦ é✱ V é ✉♠❛ ✈❛r✐❡❞❛❞❡ ❞❡✜♥✐❞❛ s♦❜r❡ K✳ ❊♥tã♦✱ ♦

  ❛♥❡❧ ❞❡ ❝♦♦r❞❡♥❛❞❛s ❛✜♥s ❞❡ V/K é ❞❡✜♥✐❞♦ ♣♦r K[X] K[V ] = .

I(V /K)

  ❖ ❛♥❡❧ K[V ] é ✉♠ ❞♦♠í♥✐♦ ❞❡ ✐♥t❡❣r✐❞❛❞❡✳ ❙❡✉ ❝♦r♣♦ q✉♦❝✐❡♥t❡ ✭❝♦r♣♦ ❞❡ ❢r❛çõ❡s✮ é ❞❡♥♦t❛❞♦ ♣♦r K(V ) ❡ é ❝❤❛♠❛❞♦ ♦ ❝♦r♣♦ ❞❡ ❢✉♥çõ❡s ❞❡ V/K✳ ❙✐♠✐❧❛r♠❡♥t❡✱ K[V ] ❡ K(V ) sã♦ ❞❡✜♥✐❞♦s s✉❜st✐t✉✐♥❞♦ K ♣♦r K✳

  ❉❡✜♥✐♠♦s ❛❣♦r❛ ✉♠❛ ♥♦çã♦ ❞❡ ❡①tr❡♠❛ ✐♠♣♦rtâ♥❝✐❛✱ q✉❡ é ♦ ❝♦♥❝❡✐t♦ ❞❡ ❞✐♠❡♥sã♦ ❞❡ ✉♠❛ ✈❛r✐❡❞❛❞❡✳ ❉❡✜♥✐çã♦ ✶✳✻ ✭❉✐♠❡♥sã♦ ❞❡ ✉♠❛ ✈❛r✐❡❞❛❞❡✮✳ ❙❡❥❛ V ✉♠❛ ✈❛r✐❡❞❛❞❡✳ ❆ ❞✐♠❡♥sã♦ ❞❡

  V ✱ ❞❡♥♦t❛❞❛ ♣♦r dim(V )✱ é ♦ ❣r❛✉ ❞❡ tr❛♥s❝❡♥❞ê♥❝✐❛ ❞❡ K(V ) s♦❜r❡ K✳

  ❊①❡♠♣❧♦ ✶✳✼✳

  n

  ) = n

  ✶✵

  • ❙❡ V = V (f)✱ ♣❛r❛ ❛❧❣✉♠ f ∈ K[X] − K✱ ❡♥tã♦ dim(V ) = n − 1✳ ❉✐③❡♠♦s q✉❡ V é ✉♠❛ ❤✐♣❡rs✉♣❡r❢í❝✐❡✳

  ◗✉❛♥❞♦ ❡st✉❞❛♠♦s ♦❜❥❡t♦s ❣❡♦♠étr✐❝♦s✱ ❡st❛♠♦s ✐♥t❡r❡ss❛❞♦s ❡♠ s❛❜❡r s❡ ♣❛r❡❝❡ r❛③♦❛✈❡❧♠❡♥t❡ ✏s✉❛✈❡✑✳ ❆ ❞❡✜♥✐çã♦ q✉❡ s❡❣✉❡✱ ❢♦r♠❛❧✐③❛ ❡ss❛ ✐❞é✐❛ ❡♠ t❡r♠♦s ❞♦ ❝r✐tér✐♦ ❏❛❝♦❜✐❛♥♦ ✉s✉❛❧ ♣❛r❛ ❛ ❡①✐stê♥❝✐❛ ❞❡ ❡s♣❛ç♦ t❛♥❣❡♥t❡✳

  1 , . . . , f m

  ❉❡✜♥✐çã♦ ✶✳✽✳ ❙❡❥❛ V ✉♠❛ ✈❛r✐❡❞❛❞❡✱ P ∈ V ✱ ❡ f ∈ K[X] ✉♠ ❝♦♥❥✉♥t♦ ❞❡ ❣❡r❛❞♦r❡s ♣❛r❛ I(V )✳ ❊♥tã♦ V é ♥ã♦ s✐♥❣✉❧❛r ✭♦✉ s✉❛✈❡✮ ❡♠ P s❡ ❛ ♠❛tr✐③ m × n

  ∂f

  1

  (P ) ∂x

  j 16i6m,16j6n

  t❡♠ ♣♦st♦ n − dim(V )✳ ❙❡ V é ♥ã♦ s✐♥❣✉❧❛r ❡♠ t♦❞♦ ♣♦♥t♦ ❡♥tã♦ ❞✐③❡♠♦s q✉❡ V é ♥ã♦ s✐♥❣✉❧❛r ✭♦✉ s✉❛✈❡✮✳ ❊①❡♠♣❧♦ ✶✳✾✳ ❙❡ V é ❞❛❞❛ ♣♦r ✉♠❛ ú♥✐❝❛ ❡q✉❛çã♦ ♣♦❧✐♥♦♠✐❛❧ ♥ã♦ ❝♦♥st❛♥t❡ f (x

  1 , . . . , x n ) = 0.

  ❊♥tã♦ ❝♦♠♦ dim(V ) = n − 1✱ t❡♠♦s q✉❡ P ∈ V é ♣♦♥t♦ s✐♥❣✉❧❛r s❡✱ ❡ s♦♠❡♥t❡ s❡✱ ∂f ∂f (P ) = (P ) = 0.

  · · · = ∂x ∂x

  1 n

  ❊①✐st❡ ✉♠❛ ♦✉tr❛ ❢♦r♠❛ ❞❡ ❝❛r❛❝t❡r✐③❛r♠♦s ❛ s✉❛✈✐❞❛❞❡ ❞❡ ✉♠ ♣♦♥t♦ ❡♠ ✉♠❛ ✈❛r✐❡❞❛❞❡✳ ■ss♦ s❡ ❞❛rá ❡♠ t❡r♠♦s ❞❛s ❢✉♥çõ❡s ♥❛ ✈❛r✐❡❞❛❞❡ ❡ ♠♦str❛rá q✉❡ ❡st❛ ❞❡✜♥✐çã♦ é ✐♥❞❡♣❡♥❞❡♥t❡ ❞❛ ❡s❝♦❧❤❛ ❞♦s ❣❡r❛❞♦r❡s ❞♦ ✐❞❡❛❧✳ P❛r❛ ❝❛❞❛ ♣♦♥t♦ P ∈ V ✱ ❞❡✜♥✐♠♦s ♦

  P

  s❡❣✉✐♥t❡ ✐❞❡❛❧ M ❞❡ K[V ] ♣♦r M =

  P {f ∈ K[V ] : f(P ) = 0}. P P

  ❈❧❛r❛♠❡♥t❡✱ M ❡stá ❜❡♠ ❞❡✜♥✐❞♦ ❡ ❛❧é♠ ❞✐ss♦✱ M é ✉♠ ✐❞❡❛❧ ♠❛①✐♠❛❧✱ ✈✐st♦ q✉❡ ❡①✐st❡

  P ✉♠ ✐s♦♠♦r✜s♠♦ K[V ]/M → K ❞❛❞♦ ♣♦r f 7→ f(P ).

  2

  /M ❚❡♠♦s t❛♠❜é♠ q✉❡ ♦ q✉♦❝✐❡♥t❡ M P é ✉♠ K✲❡s♣❛ç♦ ✈❡t♦r✐❛❧ ❞❡ ❞✐♠❡♥sã♦

  P

  ✜♥✐t❛✳ ❆♥t❡s ❞❡ ♠♦str❛♠♦s ❛ ❝❛r❛❝t❡r✐③❛çã♦ ❞❡ s✉❛✈✐❞❛❞❡ ❡♠ t❡r♠♦s ❞❛s ❢✉♥çõ❡s ❞❡✜♥✐❞❛s ♥❛ ✈❛r✐❡❞❛❞❡✱ ♣r♦✈❛r❡♠♦s ♦ s❡❣✉✐♥t❡ r❡s✉❧t❛❞♦✳

  1 , . . . , a n P = (x

  1 1 , . . . , x n

  Pr♦♣♦s✐çã♦ ✶✳✶✵✳ ❙❡❥❛ L ❝♦r♣♦ ❡ a ∈ L✳ ❊♥tã♦ ♦ ✐❞❡❛❧ ❆ − a − a n )

  1 , . . . , x n ]

  ∈ L[X] = L[x é ♠❛①✐♠❛❧✳

  P

  ❉❡♠♦♥str❛çã♦✳ P❛r❛ ✈❡r✐✜❝❛r♠♦s ✐ss♦✱ ❜❛st❛ ♠♦str❛r q✉❡ ❛ ❛♣❧✐❝❛çã♦ L[X]/❆ → L ❞❛❞❛ , . . . , a )

  P 1 n P

  ♣♦r f + ❆ 7→ f(a é ✉♠ ✐s♦♠♦r✜s♠♦ ❞❡ ❛♥é✐s✳ ❆ss✐♠ t❡r❡♠♦s q✉❡ L[X]/❆ é

  P

  ✉♠ ❝♦r♣♦✱ ❞❡ ♦♥❞❡ s❡❣✉❡ q✉❡ ❆ é ♠❛①✐♠❛❧ ❡♠ L[X]✳

  ✶✶ ◆ã♦ é ❞✐❢í❝✐❧ ✈❡r✐✜❝❛r q✉❡ ❡st❛ ❛♣❧✐❝❛çã♦ ❡stá ❜❡♠ ❞❡✜♥✐❞❛✱ é ✉♠ ❤♦♠♦♠♦r✜s♠♦

  ❞❡ ❛♥é✐s ❡ q✉❡ é s♦❜r❡❥❡t✐✈❛✳ ❆ ♠❛✐♦r ❞✐✜❝✉❧❞❛❞❡ ✈❡♠ ❞❛ s✉❛ ✐♥❥❡t✐✈✐❞❛❞❡✳ , . . . , a ) = 0

  ❙❡❥❛ f ∈ L[X] t❛❧ q✉❡ f(a

  1 n ✱ ♠♦str❡♠♦s q✉❡ f ∈ ❆ P ✱ ❞❡ ♦♥❞❡

  ❝♦♥❝❧✉✐r❡♠♦s q✉❡ ❡st❛ ❛♣❧✐❝❛çã♦ t❡♠ ♥ú❝❧❡♦ tr✐✈✐❛❧✱ ✐♠♣❧✐❝❛♥❞♦ ♥❛ s✉❛ ✐♥❥❡t✐✈✐❞❛❞❡✳ , . . . , a , x ) ]

  ] ❚❡♠♦s q✉❡ f(a

  1 n−1 n ∈ L[x n t❡♠ a n ❝♦♠♦ r❛í③✳ ❆ss✐♠✱ ❡①✐st❡ f n ∈ L[x n

  t❛❧ q✉❡ f (a

  1 , . . . , a n−1 , x n ) = f n n n ).

  · (x − a

  1 , . . . , x n−1 , x n ) n n n ) n ])[x n−1 ] n−1

  ❆❣♦r❛✱ t❡♠♦s q✉❡ f(a − f · (x − a ∈ (L[x t❡♠ a , x ]

  n−1 n−1 n

  ❝♦♠♦ r❛í③✳ ❆ss✐♠✱ ❡①✐st❡ ✉♠ ♣♦❧✐♥ô♠✐♦ f ∈ L[x ♣❛r❛ ♦ q✉❛❧ t❡♠♦s f (a , . . . , x , x ) ) = f ).

  

1 n−1 n n n n n−1 n−1 n−1

  − f · (x − a · (x − a , . . . , x )

  1 n

  Pr♦❝❡❞❡♥❞♦ ❞❡ ❢♦r♠❛ r❡❝✉rs✐✈❛✱ ♦❜t❡r❡♠♦s q✉❡ f(x ♣♦❞❡ s❡r ❡s❝r✐t♦ ❞❛ ❢♦r♠❛

  n

  X f = f i i i ), · (x − a

  i=1 1 , . . . , f n P

  ♣❛r❛ ❛❧❣✉♠ f ∈ L[X]✳ ❉❡ ♦♥❞❡ s❡❣✉❡ q✉❡ f ∈ ❆ ✳ Pr♦♣♦s✐çã♦ ✶✳✶✶✳ ❙❡❥❛ V ✉♠❛ ✈❛r✐❡❞❛❞❡✳ ❯♠ ♣♦♥t♦ P ∈ V é ♥ã♦ s✐♥❣✉❧❛r s❡✱ ❡ s♦♠❡♥t❡ s❡✱

  

2

dim M /M = dim(V ). P K P

  , . . . , a ) = (x , . . . , x )

  1 n P

  1 1 n n

  ❉❡♠♦♥str❛çã♦✳ ❙❡❥❛ P = (a ∈ V ✱ ❡ s❡❥❛ ❆ − a − a ♦ ❝♦rr❡s✲ ♣♦♥❞❡♥t❡ ✐❞❡❛❧ ♠❛①✐♠❛❧ ❞❡ K[X]✳ ❉❡✜♥✐♠♦s ❛ ❛♣❧✐❝❛çã♦ ❧✐♥❡❛r

  n

  θ : K[X] → K

  ∂f ∂f θ(f ) = (P ), . . . , (P ) .

  ∂x ∂x

  1

  1 n

  2

  ) ] =

  i i

  ❚❡♠♦s q✉❡ θ(x −a ✱ i = 1, . . . , n✱ ❢♦r♠❛♠ ✉♠❛ ❜❛s❡ ♣❛r❛ K ✱ ❡ q✉❡ θ[❆ P {0}✳ ❆ss✐♠✱

  n

  2 ′

  θ : /

  P

  ✐♥❞✉③ ✉♠ ✐s♦♠♦r✜s♠♦ ❧✐♥❡❛r θ ❆ ❆ P → K ✳ , . . . , f

  1 t

  ❆❣♦r❛ s❡❥❛ I(V ) ♦ ✐❞❡❛❧ ❞❡ V ❡♠ K[X]✱ ❡ s❡❥❛♠ f ✉♠ ❝♦♥❥✉♥t♦ ❞❡ ❣❡r❛✲ ∂f

  1

  (P ) ❞♦r❡s ❞❡ I(V )✳ ❊♥tã♦ ♦ ♣♦st♦ ❞❛ ♠❛tr✐③ ❏❛❝♦❜✐❛♥❛ J = é ✐❣✉❛❧ ❛

  ∂x j

  16i6t,16j6n

n

  ❞✐♠❡♥sã♦ ❞❡ θ[I(V )] ❝♦♠♦ s✉❜❡s♣❛ç♦ ❞❡ K ✳ ❯s❛♥❞♦ ♦ ✐s♦♠♦r✜s♠♦ θ ✱ ❡ss❛ ❞✐♠❡♥sã♦ é

  2

  2

  2

  )/ / ✐❣✉❛❧ ❛ ❞✐♠❡♥sã♦ ❞♦ s✉❜❡s♣❛ç♦ (I(V ) + ❆ ❆ ❞❡ ❆ P ❆ ✳

  P P P

  ❚❡♠♦s t❛♠❜é♠ q✉❡

  2

  2

  ∼ M /M /(I(V ) + ).

  P = P P ❆ ❆ P

  2

  2

  2

  2 ( / ) = dim ( /(I(V ) + )) + dim ((I(V ) + )/ ). P P

  ❆❧é♠ ❞✐ss♦✱ dim K ❆ ❆ P K ❆ ❆ P K ❆ P ❆ P

  ✶✷ ❆ss✐♠✱

  2

  2

  ((I(V ) + )/ ) P♦st♦ (J) = dim ❆ ❆

  K P P

  2

  2

  = dim ( P / ) ( P /(I(V ) + )) ❆ ❆ − dim ❆ ❆

  K P K P

  2 = n (M P /M ).

  − dim

  K P

  2 M /M = dim(V ) P

  P♦rt❛♥t♦✱ P é ♥ã♦ s✐♥❣✉❧❛r s❡✱ ❡ s♦♠❡♥t❡ s❡✱ dim K P ✳ ❆❣♦r❛ ❞❡✜♥✐r❡♠♦s ♦ ❛♥❡❧ ❧♦❝❛❧ ❞❡ ✉♠❛ ✈❛r✐❡❞❛❞❡ ❡♠ ✉♠ ♣♦♥t♦✳

  P

  ❉❡✜♥✐çã♦ ✶✳✶✷✳ ❖ ❛♥❡❧ ❧♦❝❛❧ ❞❡ V ❡♠ P ✱ ❞❡♥♦t❛❞♦ ♣♦r K[V ] ✱ é ❛ ❧♦❝❛❧✐③❛çã♦ ❞❡ K[V ]

  P

  ❡♠ M ✱ ♦✉ s❡❥❛✱ K[V ] =

  P {F ∈ K(V ) : F = f/g ♣❛r❛ ❛❧❣✉♠ f, g ∈ K[V ], ❝♦♠ g(P ) 6= 0}.

  P

  ◆♦t❡ q✉❡ s❡ F = f/g ∈ K[V ] ✱ ❡♥tã♦ F (P ) = f(P )/g(P ) ❡stá ❜❡♠ ❞❡✜♥✐❞♦✳ ❆s ❢✉♥çõ❡s

  P

  ❡♠ K[V ] sã♦ ❝❤❛♠❛❞❛s r❡❣✉❧❛r❡s ✭♦✉ ❞❡✜♥✐❞❛s✮ ❡♠ P ✳

  ✶✳✶✳✷ ❱❛r✐❡❞❛❞❡s Pr♦❥❡t✐✈❛s

  ❖s ❡s♣❛ç♦s ♣r♦❥❡t✐✈♦s ❢♦r❛♠ ♣❡♥s❛❞♦s ❝♦♠♦ ✉♠ ♣r♦❝❡ss♦ ❞❡ ❛❞✐❝✐♦♥❛r ✏♣♦♥t♦s ♥♦ ✐♥✜♥✐t♦✑ ❛♦ ❡s♣❛ç♦ ❛✜♠✳ ❉❡✜♥✐♠♦s ♦ ❡s♣❛ç♦ ♣r♦❥❡t✐✈♦ ❝♦♠♦ ❛ ❝♦❧❡çã♦ ❞❡ r❡t❛s ♣❛ss❛♥❞♦ ♣❡❧❛ ♦r✐❣❡♠ ♥♦ ❡s♣❛ç♦ ❛✜♠ ❞❡ ❞✐♠❡♥sã♦ ♠❛✐♦r q✉❡ ✉♠✳

  n n

  (K) ❉❡✜♥✐çã♦ ✶✳✶✸✳ ❖ n✲❡s♣❛ç♦ ♣r♦❥❡t✐✈♦ ✭s♦❜r❡ K✮✱ ❞❡♥♦t❛❞♦ ♣♦r P ♦✉ P ✱ é ♦

  n+1

  , x , . . . , x ) ❝♦♥❥✉♥t♦ ❞❡ t♦❞❛s ❛s (n + 1)✲✉♣❧❛s (x

  1 n t❛❧ q✉❡✱ ❛♦ ♠❡♥♦s ✉♠ x i é ♥ã♦

  ∈ A ♥✉❧♦✱ ♠ó❞✉❧♦ ❛ r❡❧❛çã♦ ❞❡ ❡q✉✐✈❛❧ê♥❝✐❛

  (x , x , . . . , x ) , y , . . . , y )

  1 n 1 n

  ∼ (y

  ∗

  = λy

  i i

  s❡ ❡①✐st❡ ✉♠ λ ∈ K t❛❧ q✉❡ x ♣❛r❛ t♦❞♦ i✳ ❯♠❛ ❝❧❛ss❡ ❞❡ ❡q✉✐✈❛❧ê♥❝✐❛

  ∗

  , λx , . . . , λx ) : λ

  1 n

  {(λx ∈ K } , x , . . . , x ] , x , . . . , x

  é ❞❡♥♦t❛❞❛ ♣♦r [x

  1 n ✱ ❡ ♦s x 1 n ✐♥❞✐✈✐❞✉❛❧♠❡♥t❡ sã♦ ❝❤❛♠❛❞❛s ❝♦♦r✲ n

  ❞❡♥❛❞❛s ❤♦♠♦❣ê♥❡❛s ♣❛r❛ ♦ ❝♦rr❡♣♦♥❞❡♥t❡ ♣♦♥t♦ ❡♠ P ✳ ❖ ❝♦♥❥✉♥t♦ ❞♦s ♣♦♥t♦s

  n

  K ✲r❛❝✐♦♥❛✐s ❡♠ P é ♦ ❝♦♥❥✉♥t♦

  n n

  P (K) = , x , . . . , x ] : , . . . , y , x , . . . , x ] = [y , y , . . . , y ]

  1 n n 1 n 1 n {[x ∈ P ❡①✐st❡♠ y ∈ K, [x }. n

  P♦❞❡✲s❡ ♥♦t❛r q✉❡ ♦ ❣r✉♣♦ ❞❡ ●❛❧♦✐s G(K/K) ❛❣❡ s♦❜r❡ P ♣♦r ❛çã♦ s♦❜r❡ ❛s

  ✶✸ ❝♦♦r❞❡♥❛❞❛s ❤♦♠♦❣ê♥❡❛s

  τ τ τ τ [x , x , . . . , x ] = [x , x , . . . , x ]. 1 n 1 n

  ❊st❛ ❛çã♦ ❡stá ❜❡♠ ❞❡✜♥✐❞❛✱ ✐♥❞❡♣❡♥❞❡♥t❡ ❞❛ ❡s❝♦❧❤❛ ❞❡ ❝♦♦r❞❡♥❛❞❛s ❤♦♠♦❣ê♥❡❛s✱ ❞❛❞♦ q✉❡

  τ τ τ τ τ τ τ τ τ τ [λx , λx , . . . , λx ] = [λ x , λ x , . . . , λ x ] = [x , x , . . . , x ]. 1 n 1 n

  1 n

  ❆ ✜♠ ❞❡ ❞❡✜♥✐r♠♦s ❛s ✈❛r✐❡❞❛❞❡s ♣r♦❥❡t✐✈❛s✱ s❡❣✉✐♥❞♦ ❛ ✐❞é✐❛ ❞❡ t♦♠❛r ♣♦♥t♦s ❞♦ ❡s♣❛ç♦ ♣r♦❥❡t✐✈♦ q✉❡ s❡❥❛♠ ③❡r♦s ❞❡ ♣♦❧✐♥ô♠✐♦s✱ ♣r❡❝✐s❛♠♦s q✉❡ ❡st❛ ❞❡✜♥✐çã♦ ✐♥❞❡✲ ♣❡♥❞❛ ❞❛s ❝♦♦r❞❡♥❛❞❛s ❤♦♠♦❣ê♥❡❛s ❛ss♦❝✐❛❞❛ ❛♦ ♣♦♥t♦✳ P❛r❛ ✐ss♦ ♣r❡❝✐s❛♠♦s ❞❛ s❡❣✉✐♥t❡ ❞❡✜♥✐çã♦✳

  , x , . . . , x ]

  1 n

  ❉❡✜♥✐çã♦ ✶✳✶✹✳ ❯♠ ♣♦❧✐♥ô♠✐♦ f ∈ K[X] = K[x é ❤♦♠♦❣ê♥❡♦ ❞❡ ❣r❛✉ d s❡

  d

  f (λx , λx , . . . , λx ) = λ , x , . . . , x ),

  1 n 1 n · f(x ∀λ ∈ K.

  ❯♠ ✐❞❡❛❧ I ⊆ K[X] é ❤♦♠♦❣ê♥❡♦ s❡ é ❣❡r❛❞♦ ♣♦r ♣♦❧✐♥ô♠✐♦s ❤♦♠♦❣ê♥❡♦s✳

  n

  ❙❡❥❛ f ✉♠ ♣♦❧✐♥ô♠✐♦ ❤♦♠♦❣ê♥❡♦ ❡ s❡❥❛ P ∈ P ✳ ❆❣♦r❛ ❢❛③ s❡♥t✐❞♦ ♣❡r❣✉♥t❛r s❡ f (P ) = 0 ✱ ✉♠❛ ✈❡③ q✉❡ ❛ r❡s♣♦st❛ é ✐♥❞❡♣❡♥❞❡♥t❡ ❞❛ ❡s❝♦❧❤❛ ❞❛s ❝♦♦r❞❡♥❛❞❛s ❤♦♠♦❣ê♥❡❛s

  ♣❛r❛ P ✳

  n

  P❛r❛ ❝❛❞❛ ✐❞❡❛❧ ❤♦♠♦❣ê♥❡♦ I✱ ❛ss♦❝✐❛♠♦s ✉♠ s✉❜❝♦♥❥✉♥t♦ ❞❡ P ❞❛❞♦ ♣♦r

  n

  V (I) = : f (P ) = 0, {P ∈ P ♣❛r❛ t♦❞♦ ♣♦❧✐♥ô♠✐♦ ❤♦♠♦❣ê♥❡♦ f ∈ I}. ❉❡✜♥✐çã♦ ✶✳✶✺✳ ❯♠ ❝♦♥❥✉♥t♦ ❛❧❣é❜r✐❝♦ ✭♣r♦❥❡t✐✈♦✮ é q✉❛❧q✉❡r ❝♦♥❥✉♥t♦ ❞❛ ❢♦r♠❛ V (I)

  ♣❛r❛ ❛❧❣✉♠ ✐❞❡❛❧ ❤♦♠♦❣ê♥❡♦ I✳ ❙❡ V é ✉♠ ❝♦♥❥✉♥t♦ ❛❧❣é❜r✐❝♦ ♣r♦❥❡t✐✈♦✱ ♦ ✐❞❡❛❧ ✭❤♦♠♦❣ê♥❡♦✮ ❞❡ V ✱ ❞❡♥♦t❛❞♦ ♣♦r I(V )✱ é ♦ ✐❞❡❛❧ ❞❡ K[X] ❣❡r❛❞♦ ♣♦r {f ∈ K[X] : f é ❤♦♠♦❣ê♥❡♦ ❡ f(P ) = 0, ∀P ∈ V }.

  ❉✐③❡♠♦s q✉❡ V é ❞❡✜♥✐❞♦ s♦❜r❡ K✱ ❞❡♥♦t❛❞♦ ♣♦r V/K✱ s❡ s❡✉ ✐❞❡❛❧ I(V ) ♣♦❞❡ s❡r ❣❡r❛❞♦ ♣♦r ♣♦❧✐♥ô♠✐♦s ❤♦♠♦❣ê♥❡♦s ❡♠ K[X]✳ ❙❡ V é ❞❡✜♥✐❞♦ s♦❜r❡ K✱ ❡♥tã♦ ♦ ❝♦♥❥✉♥t♦ ❞♦s ♣♦♥t♦s K✲r❛❝✐♦♥❛✐s ❞❡ V é ♦ ❝♦♥❥✉♥t♦

  n V (K) = V (K).

  ∩ P ❈♦♠♦ ✉s✉❛❧✱ V (K) ♣♦❞❡ t❛♠❜é♠ s❡r ❞❡s❝r✐t♦ ❝♦♠♦

  τ

  V (K) = = P, {P ∈ V : P ∀τ ∈ Gal(K/K)}. ❆ss✐♠ ❝♦♠♦ ❢♦✐ ❢❡✐t♦ ♣❛r❛ ♦s ❝♦♥❥✉♥t♦s ❛❧❣é❜r✐❝♦s ❛✜♥s✱ ♣♦❞❡♠♦s ❞❡✜♥✐r ❛ ♥♦çã♦

  ✶✹ ❞❡ ✈❛r✐❡❞❛❞❡ ♣r♦❥❡t✐✈❛✳ ❉❡✜♥✐çã♦ ✶✳✶✻✳ ❯♠ ❝♦♥❥✉♥t♦ ❛❧❣é❜r✐❝♦ ♣r♦❥❡t✐✈♦ V é ❝❤❛♠❛❞♦ ✉♠❛ ✈❛r✐❡❞❛❞❡ ✭♣r♦❥❡✲ t✐✈❛✮ s❡ s❡✉ ✐❞❡❛❧ ❤♦♠♦❣ê♥❡♦ I(V ) é ✉♠ ✐❞❡❛❧ ♣r✐♠♦ ❡♠ K[X]✳

  n

  ❆❣♦r❛✱ ❞❡ ❢♦r♠❛ ♥❛t✉r❛❧ ♣♦❞❡♠♦s ❡♥①❡r❣❛r ♦ ❡s♣❛ç♦ ❛✜♠ A ❝♦♠♦ s✉❜❝♦♥❥✉♥t♦

  n

  ❞♦ ❡s♣❛ç♦ ♣r♦❥❡t✐✈♦ P ✱ ❞❛ s❡❣✉✐♥t❡ ❢♦r♠❛✳ ❋✐①❛❞❛ ✉♠❛ ❝♦♦r❞❡♥❛❞❛ 0 6 i 6 n ❝♦♥s✐❞❡r❛♠♦s ❛ ✐♥❝❧✉sã♦

  n n

  ϕ : A

  i

  −→ P (x , . . . , x ) , . . . , x , 1, x , . . . , x ].

  1 n 1 i−1 i n

  7−→ [x

  n

  = 0

  i i

  ❆❣♦r❛ ❞❡♥♦t❛♠♦s ♣♦r H ❛♦ ❤✐♣❡r♣❧❛♥♦ ❡♠ P ❞❡✜♥✐❞♦ ♣❡❧❛ ❡q✉❛çã♦ x ✱ ✐st♦ é✱

  n

  H = , . . . , x ] : x = 0

  i n i {P = [x ∈ P }. n

  ❚❡♠♦s q✉❡ ❡①✐st❡ ✉♠❛ ❜✐❥❡çã♦ ♥❛t✉r❛❧ ❡♥tr❡ A ❡ ♦ ❝♦♠♣❧❡♠❡♥t♦ ❞❡ H i ✱ ❞❡♥♦t❛❞♦

  

n

  = , . . . , x ] : x ♣♦r U i ✱ ♦✉ s❡❥❛✱ s❡♥❞♦ U i

  1 n i

  {[x ∈ P 6= 0} t❡♠♦s ❛ ❝♦rr❡s♣♦♥❞ê♥❝✐❛

  −1 n

  ϕ : U i −→ A

  i

  x x x x

  i−1 i+1 n [x , . . . , x ] , . . . , , , . . . , . n

  7−→ x x x x

  i i i i n n i

  ❆ss✐♠✱ ♣❛r❛ ✉♠ i ✜①❛❞♦✱ ♣♦❞❡♠♦s ✐❞❡♥t✐✜❝❛r A ❝♦♠ U ❡♠ P ✳ ❙❡ V é ✉♠❛

  n n −1

  (V i ) ✈❛r✐❡❞❛❞❡ ♣r♦❥❡t✐✈❛ ❡♠ P t❡♠♦s q✉❡ V ∩ A ✱ q✉❡ ♥❛❞❛ ♠❛✐s é ❞♦ q✉❡ ϕ ∩ U ♣❛r❛

  i n

  ) ❛❧❣✉♠ i ✜①❛❞♦✱ é ✉♠❛ ✈❛r✐❡❞❛❞❡ ❛✜♠ ❝✉❥♦ ✐❞❡❛❧ I(V ∩ A ⊆ K[Y ] é ❞❛❞♦ ♣♦r

  n

  I(V ) = , y , . . . , y , 1, y , . . . , y ) : f (x , . . . , x )

  1 i−1 i+1 n n

  ∩ A {f(y ∈ I(V )}.

  n i

  ◗✉❛♥❞♦ ❡st✐✈❡r♠♦s ❢❛③❡♥❞♦ r❡❢❡rê♥❝✐❛ à ✐♥❞❡♥t✐✜❝❛çã♦ A ❝♦♠ U ✱ ✉s❛r❡♠♦s ❛

  n

  ♥♦t❛çã♦ A i ✳

  n

  , . . . , U

  n

  ❈♦♠♦ ♦s ❝♦♥❥✉♥t♦s U ❝♦❜r❡♠ ♦ ❡s♣❛ç♦ ♣r♦❥❡t✐✈♦ P ✱ t♦❞❛ ✈❛r✐❡❞❛❞❡ ♣r♦✲

  n n

  , . . . , V ❥❡t✐✈❛ V é ❝♦❜❡rt❛ ♣❡❧❛ ❝♦❧❡çã♦ ❞❡ ✈❛r✐❡❞❛❞❡s ❛✜♥s V ∩ A ✱ ✈✐❛ ❛s ❛♣❧✐❝❛çõ❡s

  ∩ A n ϕ

  i ✳

  ❙❡ ✜①❛r♠♦s i✱ ♣♦❞❡♠♦s ❝♦♥s✐❞❡r❛r ✉♠❛ ✈❛r✐❡❞❛❞❡ ♣r♦❥❡t✐✈❛ V ❝♦♠♦ ❛ ✉♥✐ã♦ ❞❡

  n i

  s✉❛ ♣❛rt❡ ❛✜♠ V ∩A ❝♦♠ ♦ ❝♦♥❥✉♥t♦ ❞♦s s❡✉s ♣♦♥t♦s ♥♦ ✐♥✜♥✐t♦ V ∩H ✱ ❡ ❞❡♥♦t❛r❡♠♦s

  i

  H i ∞ ♣♦r H ✳

  , y , . . . , y , 1, y , . . . , y )

  1 i−1 i+1 n

  ❊ss❡ ♣r♦❝❡ss♦ ❞❡ ❝♦♥s✐❞❡r❛r ♦ ♣♦❧✐♥ô♠✐♦ f(y ♥♦ ❧✉❣❛r , . . . , x )

  n i

  ❞❡ f(x é ❝❤❛♠❛❞♦ ❞❡s♦♠♦❣❡♥❡✐③❛çã♦ ❝♦♠ r❡s♣❡✐t♦ ❛ x ✳ ❊ ❛ ✐♥✈❡rsã♦ ❞❡st❡ , . . . , y ) , . . . , y ]

  1 n 1 n

  ♣r♦❝❡ss♦ é ❞❡✜♥✐❞❛ ❞❛ s❡❣✉✐♥t❡ ❢♦r♠❛✱ ❞❛❞♦ f(y ∈ K[y ✱ ❞❡✜♥✐♠♦s

  ∗ d

  f (x , . . . , x ) = x /x , . . . , x /x , x /x , . . . , x /x ),

  n i i−1 i i+1 i n i i · f(x

  ✶✺

  ∗ ∗

  ♦♥❞❡ d = deg(f) é ♦ ♠❡♥♦r ✐♥t❡✐r♦ ♣♦s✐t✐✈♦ t❛❧ q✉❡ f é ✉♠ ♣♦❧✐♥ô♠✐♦✳ ❈❤❛♠❛r❡♠♦s f

  i

  ❞❡ ❛ ❤♦♠♦❣❡♥❡✐③❛çã♦ ❞❡ f ❝♦♠ r❡s♣❡✐t♦ à x ✳

  n

  ❉❡✜♥✐çã♦ ✶✳✶✼✳ ❙❡❥❛ V ✉♠❛ ✈❛r✐❡❞❛❞❡ ♣r♦❥❡t✐✈❛ ❡ ❡s❝♦❧❤❛ i t❛❧ q✉❡ V ∩A

  i 6= ∅✳ ❉❡✜♥✐♠♦s n

  ❛ ❞✐♠❡♥sã♦ ❞❡ V ❝♦♠♦ ❛ ❞✐♠❡♥sã♦ ❞❛ ✈❛r✐❡❞❛❞❡ ❛✜♠ V ∩ A ✳

  i n

  ❚❡♠♦s q✉❡ ❡st❛ ❞❡✜♥✐çã♦ ✐♥❞❡♣❡♥❞❡ ❞❛ ❡s❝♦❧❤❛ ❞♦ j ♣❛r❛ ♦ q✉❛❧ V ∩A 6= ∅✳ P❛r❛

  j

  ✐ss♦ ♣♦❞❡♠♦s t❛♠❜é♠ ❞❡✜♥✐r ♦ ❝♦r♣♦ ❞❡ ❢✉♥çõ❡s ❞❡ V ❝♦♠♦ s❡♥❞♦ ♦ ❝♦r♣♦ ❞❛s ❢✉♥çõ❡s f (X) r❛❝✐♦♥❛✐s F (X) = t❛✐s q✉❡ g(X)

  ✭✐✮ f ❡ g sã♦ ❤♦♠♦❣ê♥❡♦s ❞❡ ♠❡s♠♦ ❣r❛✉✳ ✭✐✐✮ g 6∈ I(V )✳ f f

  1

  2

  g g

  1

  2

  2

  1

  ✭✐✐✐✮ ❡ ✐❞❡♥t✐✜❝❛♠♦s ❞✉❛s ❢✉♥çõ❡s ❡ s❡ f − f ∈ I(V )✳ g

  1 g

  2

  ❊①✐st❡ ✉♠ ✐s♦♠♦r✜s♠♦ ❡♥tr❡ ♦s ❝♦r♣♦s ❞❛❞♦s ♣♦r ❡ss❛s ❞✉❛s ❞❡✜♥✐çõ❡s✳ ❆ss✐♠

  n

  t❡♠♦s q✉❡ ❛ ❞❡✜♥✐çã♦ ❞❡ ❞✐♠❡♥sã♦ ❞❡ V ✐♥❞❡♣❡♥❞❡ ❞❛ ❡s❝♦❧❤❛ ❞♦ j t❛❧ q✉❡ V ∩ A j 6= ∅✳

  ✶✳✷ ❈✉r✈❛s ❆❧❣é❜r✐❝❛s

  ❈♦♠♦ ❥á ❢♦✐ ♠❡♥❝✐♦♥❛❞♦ ♥♦ ✐♥í❝✐♦ ❞❡st❡ ❝❛♣ít✉❧♦✱ ♣r❡t❡♥❞❡♠♦s ❞❡✜♥✐r ♦s ♣r✐♥❝✐♣❛✐s ❝♦♥❝❡✐t♦s ❡ r❡s✉❧t❛❞♦s q✉❡ s❡rã♦ ♥❡❝❡ssár✐♦s ♥♦s ❞❡♠❛✐s ❝❛♣ít✉❧♦s✳ ◆❡st❛ s❡çã♦✱ ❛♣r❡s❡♥✲ t❛♠♦s ❛s ❞❡✜♥✐çõ❡s ❞❡ ❝✉r✈❛s ❛❧❣é❜r✐❝❛s✱ ♠❛♣❛s ❡♥tr❡ ❝✉r✈❛s✱ ❞✐✈✐s♦r❡s ❡ ❞✐❢❡r❡♥❝✐❛✐s✱ ❡ ❡♥✉♥❝✐❛♠♦s ♦ ❚❡♦r❡♠❛ ❞❡ ❘✐❡♠❛♥♥✲❘♦❝❤✳

  ❆♦ ❛❞♦t❛r♠♦s ❛ ❧✐♥❣✉❛❣❡♠ ❡ ♥♦t❛çã♦ ✉s❛❞❛s ❡♠ ❬❙✐❧✈❡r♠❛♥❪✱ ♦♣t❛♠♦s ♣♦r ♥ã♦ ❞❡✲ ✜♥✐r ❝♦♥❝❡✐t♦s ❝♦♠♦ ❛ ❚♦♣♦❧♦❣✐❛ ❞❡ ❩❛r✐s❦✐✱ ❡s♣❛ç♦s ♠✉❧t✐❤♦♠♦❣ê♥❡♦s ❡ ✈❛r✐❡❞❛❞❡s q✉❛s❡✲ ♣r♦❥❡t✐✈❛s✳ ◆❡ss❛ ❛❜♦r❞❛❣❡♠ ♠❛✐s ❣❡r❛❧✱ ♠✉♥✐♠♦s ♦s ❡s♣❛ç♦s ❛✜♥s✱ ♣r♦❥❡t✐✈♦s ❡ ♠❛✐s ❣❡r❛❧♠❡♥t❡ ♦s ♠✉❧t✐♣r♦❥❡t✐✈♦s✱ ❝♦♠ ✉♠❛ t♦♣♦❧♦❣✐❛ ♥❛ q✉❛❧ ♦s ❢❡❝❤❛❞♦s sã♦ ♦s ❝♦♥❥✉♥t♦s ❛❧❣é❜r✐❝♦s✱ ❡ ❛s ✈❛r✐❡❞❛❞❡s s❡rã♦ ♦s ❢❡❝❤❛❞♦s ✐rr❡❞✉tí✈❡✐s✱ s❡♥❞♦ q✉❡ ✉♠ s✉❜❝♦♥❥✉♥t♦ Y ❞❡ ✉♠ ❡s♣❛ç♦ t♦♣♦❧ó❣✐❝♦ X é ❞✐t♦ ✐rr❡❞✉tí✈❡❧✱ q✉❛♥❞♦ ♥ã♦ ♣✉❞❡r s❡r ❡s❝r✐t♦ ❝♦♠♦ ❛ ✉♥✐ã♦

  1

  2

  1

  2

  ❞❡ ❞♦✐s s✉❜❝♦♥❥✉♥t♦s ❢❡❝❤❛❞♦s r❡❧❛t✐✈♦s ❡ ♣ró♣r✐♦s✱ ✐st♦ é✱ s❡ Y = Y ∪ Y ❝♦♠ Y ❡ Y

  1

  2

  ❢❡❝❤❛❞♦s ❡♠ Y ✱ ❡♥tã♦ Y = Y ♦✉ Y = Y ✳ ❆ss✐♠✱ ❞❡✜♥✐❞❛ ❛ t♦♣♦❧♦❣✐❛ ♣♦r ♠❡✐♦ ❞♦s ❢❡❝❤❛❞♦s✱ ♣♦❞❡♠♦s ❢❛❧❛r ❞❡ ❛❜❡rt♦s ❡ ❛ss✐♠ t❡♠♦s ✉♠❛ ♥♦çã♦ ♣r❡❝✐s❛ ❞❡ ✈✐③✐♥❤❛♥ç❛ ❞❡ ✉♠ ♣♦♥t♦ ♣♦r ♠❡✐♦ ❞♦s ❛❜❡rt♦s q✉❡ ❝♦♥té♠ ❡st❡ ♣♦♥t♦✳ ❚♦❞❛s ❡ss❛s ♥♦çõ❡s ♣♦❞❡♠ s❡r ❡♥❝♦♥tr❛❞❛s ❡♠ ❧✐✈r♦s ❝♦♠♦ ❬❍❛rts❤♦r♥❡❪✱ ❬❙❤❛❢❛r❡✈✐❝❤❪ ❡ ❬❋✉❧t♦♥❪✳

  ❊♥tr❛♠♦s ♥❡ss❡ ❛ss✉♥t♦ ♣❛r❛ ❥✉st✐✜❝❛r q✉❡ ♠✉✐t♦s ❞♦s ❝♦♥❝❡✐t♦s ❡ r❡s✉❧t❛❞♦s ❛q✉✐ ❡♥✉♥❝✐❛❞♦s✱ ♥ã♦ s❡rã♦ ❞❡♠♦♥str❛❞♦s✱ ♣♦✐s ♣❛r❛ ✐ss♦ ♥❡❝❡ss✐t❛rí❛♠♦s ❞❡ ✉♠ ♣♦✉❝♦ ♠❛✐s ❞❡ t❡♦r✐❛ ♥ã♦ ✈✐st❛ ❛q✉✐✳ ❆ss✐♠✱ ♥♦ q✉❡ s❡❣✉❡✱ ✈❡r❡♠♦s r❡s✉❧t❛❞♦s q✉❡ sã♦ ♥❡❝❡ssár✐♦s ♣❛r❛ ♦s ❢✉t✉r♦s ❝❛♣ít✉❧♦s✱ ❝✉❥❛s ❞❡♠♦♥str❛çõ❡s s❡rã♦ ♦♠✐t✐❞❛s✱ ♠❛s q✉❡ s❡rã♦ ❛❝♦♠♣❛♥❤❛❞♦s ❝♦♠ ❛s ❞❡✈✐❞❛s r❡❢❡rê♥❝✐❛s✳

  ✶✻ ❉❡✜♥✐çã♦ ✶✳✶✽✳ ❆s ❝✉r✈❛s ❛❧❣é❜r✐❝❛s sã♦ ✈❛r✐❡❞❛❞❡s ♣r♦❥❡t✐✈❛s ❞❡ ❞✐♠❡♥sã♦ ✶✳

  2

  ❊①❡♠♣❧♦ ✶✳✶✾✳ ❙❡❥❛ V ⊆ P ❛ ✈❛r✐❡❞❛❞❡ ♣r♦❥❡t✐✈❛ ❢♦r♠❛❞❛ ♣❡❧♦s ♣♦♥t♦s q✉❡ s❛t✐s❢❛③❡♠

  2 yz = x .

  ❚♦♠❛♥❞♦ z = 1✱ ♦❜t❡♠♦s

  2

  2 V = ∩ A {(x, y) : y = x }.

  3

  2

  2

  ), y + (y )) ❆ss✐♠✱ K(V ) = K(x + (y − x − x ✱ ♦✉ s❡❥❛✱ K(V ) = K(˜x, ˜y) ♦♥❞❡ ˜y

  2

  ❡ ˜x sã♦ ❢✉♥çõ❡s s❛t✐s❢❛③❡♥❞♦ ˜y = ˜x ✳ ❚❡♠♦s✿

  ❛❧❣é❜r✐❝❛ tr❛♥s❝❡♥❞❡♥t❡

  K ֒ K(˜ x) ֒ K(˜ x, ˜ y).

  → → ❉❡ ♦♥❞❡ t❡♠♦s q✉❡ ♦ ❣r❛✉ ❞❡ tr❛♥s❝❡♥❞ê♥❝✐❛ ❞❛ ❡①t❡♥sã♦ K(V )/K é ✐❣✉❛❧ ❛ ✶✱

  ❞❡ ♦♥❞❡ s❡❣✉❡ q✉❡ V é ✉♠❛ ❝✉r✈❛✳ ❉❡ ♠♦❞♦ ❣❡r❛❧✱ t❡♠♦s q✉❡ ♣❛r❛ q✉❛❧q✉❡r f ∈ K[x, y, z]

  2

  : f (P ) = 0 ♣r✐♠♦ ❡ ❤♦♠♦❣ê♥❡♦✱ ❛ ✈❛r✐❡❞❛❞❡ V (f) = {P ∈ P } é ✉♠❛ ❝✉r✈❛✳

  ❆❣♦r❛ ♠♦str❛r❡♠♦s ✉♠❛ ✐♠♣♦rt❛♥t❡ ♣r♦♣r✐❡❞❛❞❡ ❞♦s ❛♥é✐s ❧♦❝❛✐s ❞❡ ✉♠❛ ❝✉r✈❛ ❡♠ ✉♠ ♣♦♥t♦ ♥ã♦ s✐♥❣✉❧❛r✳ ❊st❛ ♣r♦♣r✐❡❞❛❞❡ é ❛ ❞❡ q✉❡ ❡st❡s sã♦ ❞♦♠í♥✐♦s ❞❡ ✈❛❧♦r✐③❛çã♦ ❞✐s❝r❡t❛✳ ❉❡✜♥✐çã♦ ✶✳✷✵✳ ❙❡❥❛ K ✉♠ ❝♦r♣♦✳ ❈❤❛♠❛r❡♠♦s ❞❡ ✈❛❧♦r✐③❛çã♦ ❞✐s❝r❡t❛ ❡♠ K ❛ ✉♠❛ ❛♣❧✐❝❛çã♦ s♦❜r❡❥❡t✐✈❛

  

  v : K → Z s❛t✐s❢❛③❡♥❞♦✿

  ✐✮ v(xy) = v(x) + v(y), ♦✉ s❡❥❛✱ v é ✉♠ ❤♦♠♦♠♦r✜s♠♦✳ ✐✐✮ v(x + y) > min{v(x), v(y)}.

  ➱ ❝♦♥✈❡♥✐❡♥t❡ ❡st❡♥❞❡r v à K ❝♦❧♦❝❛♥❞♦ v(0) = +∞✱ ♦♥❞❡ +∞ é t❛❧ q✉❡ a+∞ =

  • ∞ ❡ +∞ > a, ∀a ∈ Z ∪ {+∞}.

  ❚❡♠♦s q✉❡ ♦ ❝♦♥❥✉♥t♦ ❞♦s x ∈ K t❛✐s q✉❡ v(x) > 0 é ✉♠ ❞♦♠í♥✐♦✱ ❝❤❛♠❛❞♦ ♦ ❞♦♠í♥✐♦ ❞❡ ✈❛❧♦r✐③❛çã♦ ❞❡ v✳ ❊①❡♠♣❧♦ ✶✳✷✶✳ ❙❡❥❛ K = Q✳ ❉❛❞♦ p ♣r✐♠♦ ✜①❛❞♦✱ t❡♠♦s q✉❡ q ∈ Q ♣♦❞❡ s❡r ❡s❝r✐t♦ ❞❡

  a

  y ❢♦r♠❛ ú♥✐❝❛ ❞❛ ❢♦r♠❛ q = p ✱ ♦♥❞❡ a ∈ Z ❡ t❛♥t♦ ♦ ♥✉♠❡r❛❞♦r q✉❛♥t♦ ♦ ❞❡♥♦♠✐♥❛❞♦r

  (q) = a

  p

  ❞❡ y ♥ã♦ ❞✐✈✐❞❡♠ p✳ ❉❡✜♥✐♠♦s v ✳ ❊st❛ é ❝❤❛♠❛❞❛ ❞❡ ✈❛❧♦r✐③❛çã♦ p✲á❞✐❝❛✳ ❉✐③❡♠♦s q✉❡ ✉♠ ❞♦♠í♥✐♦ D é ✉♠ ❞♦♠í♥✐♦ ❞❡ ✈❛❧♦r✐③❛çã♦ ❞✐s❝r❡t❛ s❡ ❡①✐st❡

  ✉♠❛ ✈❛❧♦r✐③❛çã♦ ❞✐s❝r❡t❛ v ❞❡ s❡✉ ❝♦r♣♦ ❞❡ ❢r❛çõ❡s K t❛❧ q✉❡ D é ♦ ❞♦♠í♥✐♦ ❞❡ ✈❛❧♦r✐③❛çã♦

  ✶✼ Pr♦♣♦s✐çã♦ ✶✳✷✷✳ ❙❡ D é ✉♠ ❞♦♠í♥✐♦ ❞❡ ✈❛❧♦r✐③❛çã♦ ❞✐s❝r❡t❛✱ ❡♥tã♦ D é ✉♠ ❛♥❡❧ ❧♦❝❛❧✱ ❡ s❡✉ ✐❞❡❛❧ ♠❛①✐♠❛❧ m é ♦ ❝♦♥❥✉♥t♦ ❞♦s x ∈ D✱ t❛✐s q✉❡ v(x) > 0✳ ❉❡♠♦♥str❛çã♦✳ ❱❡r ✭Pr♦♣♦s✐çã♦ ✺✳✶✽✱ ❬❆t✐②❛❤❪✮ ♣❛r❛ ♠♦str❛r q✉❡ D é ❧♦❝❛❧✳ ❊ ❜❛st❛ ♠♦str❛r q✉❡ ♦ ✐❞❡❛❧ ♠❛①✐♠❛❧ ❞❡ D ❝♦✐♥❝✐❞❡ ❝♦♠ ♦ ✐❞❡❛❧ m = {x ∈ D : v(x) > 0}. Pr♦♣♦s✐çã♦ ✶✳✷✸✳ ❙❡❥❛ D ✉♠ ❞♦♠í♥✐♦ ❧♦❝❛❧ ◆♦❡t❤❡r✐❛♥♦ q✉❡ ♥ã♦ é ✉♠ ❝♦r♣♦✱ s❡❥❛ m s❡✉ ✐❞❡❛❧ ♠❛①✐♠❛❧ ❡ k = D/m s❡✉ ❝♦r♣♦ r❡s✐❞✉❛❧✳ ❆s s❡❣✉✐♥t❡s ❛✜r♠❛çõ❡s sã♦ ❡q✉✐✈❛❧❡♥t❡s✳

  ✐✮ D é ✉♠ ❞♦♠í♥✐♦ ❞❡ ✈❛❧♦r✐③❛çã♦ ❞✐s❝r❡t❛✳ ✐✐✮ m é ✉♠ ✐❞❡❛❧ ♣r✐♥❝✐♣❛❧✳

  2

  (m/m ) = 1

  k

  ✐✐✐✮ dim ✳ ❉❡♠♦♥str❛çã♦✳ ❱❡r ✭Pr♦♣♦s✐çã♦ ✾✳✷✱ ❬❆t✐②❛❤❪✮✳ Pr♦♣♦s✐çã♦ ✶✳✷✹✳ ❙❡❥❛♠ C ✉♠❛ ❝✉r✈❛ ❡ P ∈ C ✉♠ ♣♦♥t♦ ♥ã♦ s✐♥❣✉❧❛r✳ ❚❡♠♦s q✉❡

K[C]

  P é ✉♠ ❞♦♠í♥✐♦ ❞❡ ✈❛❧♦r✐③❛çã♦ ❞✐s❝r❡t❛✳ P

  ❉❡♠♦♥str❛çã♦✳ ❚❡♠♦s q✉❡ K[C] é ❧♦❝❛❧✱ ◆♦❡t❤❡r✐❛♥♦ ❡ ♥ã♦ é ✉♠ ❝♦r♣♦✳ ❙❡✉ ✐❞❡❛❧ = M

  /m

  P P P = K[C] P P

  ♠❛①✐♠❛❧ é m ✭✈✐st♦ ♥♦ ❛♥❡❧ ❧♦❝❛❧✐③❛❞♦ K[C] ✮✳ ❆ss✐♠✱ ❝♦♠♦ K ∼ ❡

  2

  dim M /M = dim(C) = 1

  P P

K P ✱ t❡♠♦s ♣❡❧❛ ♣r♦♣♦s✐çã♦ ❛♥t❡r✐♦r✱ q✉❡ K[C] é ✉♠ ❞♦♠í♥✐♦

  ❞❡ ✈❛❧♦r✐③❛çã♦ ❞✐s❝r❡t❛✳ ❊ss❛ ✈❛❧♦r✐③❛çã♦ ❞✐s❝r❡t❛ ✭✉♥✐❢♦r♠✐③❛❞❛✮ é ❞❛❞❛ ♣♦r ord : K[C]

  P P

  −→ {0, 1, 2, . . .} ∪ +∞

  d

  f 7−→ sup{d : f ∈ m P }. (f /g) = ord (f ) (g),

  P P P P

  ❊ ✉s❛♥❞♦ ord − ord ♣♦❞❡♠♦s ❡st❡♥❞❡r ord à K(C)✳

  P

  ❉✐③❡♠♦s q✉❡ ✉♠ ✉♥✐❢♦r♠✐③❛♥t❡ ♣❛r❛ C ❡♠ P ✱ é ✉♠ ❣❡r❛❞♦r ❞❡ m ✱ ✐st♦ é✱ ✉♠ t (t) = 1

  P ✳

  ∈ K(C) ♣❛r❛ ♦ q✉❛❧ ord ❉❡✜♥✐çã♦ ✶✳✷✺✳ ❙❡❥❛♠ C ❝✉r✈❛ ❡ P ∈ C ♥ã♦ s✐♥❣✉❧❛r✳ ❈❤❛♠❛♠♦s ❞❡ ♦r❞❡♠ ❞❡ f ❡♠ P (f ) (f ) > 0

  (f ) < 0

  P P P

  à ord ✳ ❙❡ ord ✱ ❞✐③❡♠♦s q✉❡ f t❡♠ ✉♠ ③❡r♦ ❡♠ P ✱ ❡ s❡ ord ✱ ❡♥tã♦ f t❡♠ ✉♠ ♣ó❧♦ ❡♠ P ✳ Pr♦♣♦s✐çã♦ ✶✳✷✻✳ ❙❡❥❛ C ✉♠❛ ❝✉r✈❛ s✉❛✈❡ ❡ f ∈ K(C) ❝♦♠ f 6= 0✳ ❊♥tã♦ ❡①✐st❡♠ ❛♣❡♥❛s ✉♠❛ q✉❛♥t✐❞❛❞❡ ✜♥✐t❛ ❞❡ ♣♦♥t♦s P ❞❡ C t❛✐s q✉❡ f t❡♠ ③❡r♦ ♦✉ ♣ó❧♦ ❡♠ P ✳ ▼❛✐s ❞♦ q✉❡ ✐ss♦✱ s❡ f ♥ã♦ t❡♠ ♣ó❧♦s✱ ❡♥tã♦ f ∈ K✳ ❉❡♠♦♥str❛çã♦✳ P❛r❛ ❛ ✜♥✐t✉❞❡ ❞♦ ❝♦♥❥✉♥t♦ ❞❡ ♣ó❧♦s✱ ✈❡r ✭❈❛♣ít✉❧♦ ✶✱ ▲❡♠❛ ✶✳✺✱ ❬❍❛rts❤♦r♥❡❪✮✳ ❆ss✐♠✱ ♣❛r❛ ♦❜t❡r ❛ ✜♥✐t✉❞❡ ❞♦ ❝♦♥❥✉♥t♦ ❞❡ ③❡r♦s✱ ❜❛st❛ ✉s❛r ❛ ✜♥✐t✉❞❡ ❞♦s ♣ó❧♦s ♣❛r❛ 1/f

  ✳ P❛r❛ ú❧t✐♠❛ ♣❛rt❡ ❞❛ ♣r♦♣♦s✐çã♦✱ ✈❡r ✭❈❛♣ít✉❧♦ ✶✱ ❚❡♦r❡♠❛ ✸✳✹ ❛✮✱ ❬❍❛rts❤♦r♥❡❪✮✳

  ✶✽

  ✶✳✷✳✶ ▼❛♣❛s ❡♥tr❡ ❈✉r✈❛s m n

  1

  2

  ❉❡✜♥✐çã♦ ✶✳✷✼✳ ❙❡❥❛♠ C ⊆ P ❡ C ⊆ P ❝✉r✈❛s ❛❧❣é❜r✐❝❛s✳ ❯♠ ♠❛♣❛ r❛❝✐♦♥❛❧ ❞❡ C

  1

  2

  ♣❛r❛ C é ✉♠❛ ❛♣❧✐❝❛çã♦ ϕ : C

  

1

  2

  → C : F : ] ) (P ) : F (P ) : (P )]

  1 n i

  1 1 n

  2

  ❞❛ ❢♦r♠❛ ϕ = [F · · · : F ✱ ♦♥❞❡ F ∈ K(C ❡ [F · · · : F ∈ C

  1 i

  ♣❛r❛ t♦❞♦ P ∈ C ♦♥❞❡ t♦❞❛s ❛s F ❡stã♦ ❞❡✜♥✐❞❛s✳ ❉✐r❡♠♦s q✉❡ ϕ ❡stá ❞❡✜♥✐❞❛ s♦❜r❡

  ∗

  K )

  i

  1

  ✱ s❡ ❡①✐st❡ ❛❧❣✉♠ λ ∈ K ✱ t❛❧ q✉❡ λF ∈ K(C ✱ ♣r❛ t♦❞♦ i = 0, . . . , n✳ ❉✐r❡♠♦s q✉❡ ϕ é ✉♠ ♠❛♣❛ ❜✐rr❛❝✐♦♥❛❧ s❡ ϕ ♣♦ss✉✐ ✉♠ ♠❛♣❛ r❛❝✐♦♥❛❧ ✐♥✈❡rs♦✳

  1

  2

  ❉✐r❡♠♦s ❡♥tã♦ q✉❡ C ❡ C sã♦ ❜✐rr❛❝✐♦♥❛✐s✳ ❊ ❞✐r❡♠♦s q✉❡ ✉♠❛ ❝✉r✈❛ C é r❛❝✐♦♥❛❧✱

  1

  (K) s❡ é ❜✐rr❛❝✐♦♥❛❧ ❝♦♠ P ✳

  1

  ❉❡✜♥✐çã♦ ✶✳✷✽✳ ◆❛s ❤✐♣ót❡s❡s ❞❛ ❞❡✜♥✐çã♦ ❛♥t❡r✐♦r✳ ❉❛❞♦ P ∈ C ✱ ❞✐r❡♠♦s q✉❡ ϕ é )

  (P )

  1 i i

  r❡❣✉❧❛r ❡♠ P ✱ s❡ ❡①✐st✐r g ∈ K(C t❛❧ q✉❡ gF é r❡❣✉❧❛r ❡♠ P ✱ ♣❛r❛ t♦❞♦ i✱ ❡ gF 6= 0 ♣❛r❛ ❛❧❣✉♠ i✳ ◗✉❛♥❞♦ ϕ é r❡❣✉❧❛r ❡♠ t♦❞♦s ♦s ♣♦♥t♦s ❞❡ C

  1 ✱ ❞✐r❡♠♦s q✉❡ é ✉♠ ♠♦r✲

  ✜s♠♦✳ ❈❛s♦ ❡①✐st❛ ✉♠ ♠♦r✜s♠♦ ❞❡ C

  2 ♣❛r❛ C 1 ❝✉❥❛s ❝♦♠♣♦st❛s ❝♦♠ ϕ s❡❥❛♠ ♦s ♠❛♣❛s

  ✐❞❡♥t✐❞❛❞❡s✱ ❞✐r❡♠♦s q✉❡ ϕ é ✉♠ ✐s♦♠♦r✜s♠♦✱ q✉❡ q✉❡ C

  1 ❡ C 2 sã♦ ✐s♦♠♦r❢❛s✳ n

  Pr♦♣♦s✐çã♦ ✶✳✷✾✳ ❙❡❥❛♠ C ✉♠❛ ❝✉r✈❛✱ V ⊆ P ✉♠❛ ✈❛r✐❡❞❛❞❡✱ P ∈ C ✉♠ ♣♦♥t♦ ♥ã♦ s✐♥❣✉❧❛r ❡ ϕ : C → V ✉♠ ♠❛♣❛ r❛❝✐♦♥❛❧✱ ❡♥tã♦ ϕ é r❡❣✉❧❛r ❡♠ P ✳ ❊♠ ♣❛rt✐❝✉❧❛r✱ s❡ C é s✉❛✈❡✱ ❡♥tã♦ ϕ é ✉♠ ♠♦r✜s♠♦✳ ❉❡♠♦♥str❛çã♦✳ ❱❡r ✭❈❛♣ít✉❧♦ ✷✱ Pr♦♣♦s✐çã♦ ✷✳✶✱ ❬❙✐❧✈❡r♠❛♥❪✮✳

  1

  2

  ❚❡♦r❡♠❛ ✶✳✸✵✳ ❙❡❥❛ ϕ : C → C ✉♠ ♠♦r✜s♠♦ ❡♥tr❡ ❝✉r✈❛s✳ ❊♥tã♦ ϕ é ❝♦♥st❛♥t❡ ♦✉ é s♦❜r❡❥❡t✐✈❛✳ ❉❡♠♦♥str❛çã♦✳ ❱❡r ✭❈❛♣ít✉❧♦ ✶✱ Pr♦♣♦s✐çã♦ ✻✳✽✱ ❬❍❛rts❤♦r♥❡❪✮✳

  ❆❣♦r❛ ❞❛❞❛s ❞✉❛s ❝✉r✈❛s C

  1 ❡ C 2 ❞❡✜♥✐❞❛s s♦❜r❡ K✱ ❡ ✉♠ ♠♦r✜s♠♦ ϕ : C

  1

  2

  → C ❞❡✜♥✐❞♦ s♦❜r❡ K✳ ❉❡✜♥✐♠♦s ✉♠❛ ❛♣❧✐❝❛çã♦

  ∗

  ϕ : K(C

  

2 )

1 )

  → K(C f 7→ f ◦ ϕ q✉❡ s❡rá ♥ã♦ ❝♦♥st❛♥t❡ s❡✱ ❡ s♦♠❡♥t❡ s❡✱ ϕ ❢♦r ♥ã♦ ❝♦♥st❛♥t❡✱ ♦✉ s❡❥❛✱ q✉❛♥❞♦ ϕ ❢♦r

  ∗

  s♦❜r❡❥❡t✐✈❛✱ ❡ ♥❡ss❡ ❝❛s♦ ϕ s❡rá ✉♠ ❤♦♠♦♠♦r✜s♠♦ ✐♥❥❡t♦r ❞❡ ❝♦r♣♦s✱ q✉❡ ✜①❛ K✳ /K /K

  1

  2

  ❚❡♦r❡♠❛ ✶✳✸✶✳ ❙❡❥❛♠ C ❡ C ❝✉r✈❛s s✉❛✈❡s s♦❜r❡ K✳ ❊♥tã♦✿

  1

  2

  ✭❛✮ ❙❡❥❛ ϕ : C → C ✉♠ ♠♦r✜s♠♦ ♥ã♦ ❝♦♥st❛♥t❡ ❞❡✜♥✐❞♦ s♦❜r❡ K✳ ❊♥tã♦

  ∗

K(C )/ϕ (K(C ))

  1

  2

  é ✉♠❛ ❡①t❡♥sã♦ ✜♥✐t❛✳

  ✶✾ ) )

  2

  1

  ✭❜✮ ❙❡❥❛ ι : K(C → K(C ✉♠❛ ✐♥❥❡çã♦ ❞❡ ❝♦r♣♦s q✉❡ ✜①❛ ♦s ❡❧❡♠❡♥t♦s ❞❡ K✳ ❊♥tã♦

  ∗

  = ι

  1

  2

  ❡①✐st❡ ✉♠ ú♥✐❝♦ ♠❛♣❛ ♥ã♦ ❝♦♥st❛♥t❡ ϕ : C → C ❞❡✜♥✐❞♦ s♦❜r❡ K t❛❧ q✉❡ ϕ ✳ )/L

  ✭❝✮ ❙❡❥❛ K(C

  1 ✉♠❛ ❡①t❡♥sã♦ ✜♥✐t❛ ❝♦♥t❡♥❞♦ K✳ ❊♥tã♦ ❡①✐st❡ ✉♠❛ ❝✉r✈❛ s✉❛✈❡

  C ✱ ❞❡✜♥✐❞❛ s♦❜r❡ K✱ ú♥✐❝❛ ❛ ♠❡♥♦s ❞❡ K✲✐s♦♠♦r✜s♠♦✱ ❡ ✉♠ ♠❛♣❛ ♥ã♦ ❝♦♥st❛♥t❡

  

  ϕ : C

  1 (K(C)) = L

  → C ❞❡✜♥✐❞♦ s♦❜r❡ K t❛❧ q✉❡ ϕ ✳ ❉❡♠♦♥str❛çã♦✳ ❱❡r ✭❈❛♣ít✉❧♦ ✷✱ ❚❡♦r❡♠❛ ✷✳✹✱ ❬❙✐❧✈❡r♠❛♥❪✮✳

  1

  2

  ❉❡✜♥✐çã♦ ✶✳✸✷✳ ❙❡❥❛ ϕ : C → C ✉♠ ♠♦r✜s♠♦✳ ❙❡ ϕ ❢♦r s♦❜r❡❥❡t♦r✱ ❝❤❛♠❛♠♦s ♦

  ∗

  ) : ϕ (K(C ))]

  1

  2

  ♥ú♠❡r♦ [K(C ❞❡ ❣r❛✉ ❞❡ ϕ✱ ❞❡♥♦t❛❞♦ ♣♦r deg(ϕ)✳ ❈♦♥✈❡♥❝✐♦♥❛♠♦s q✉❡ ♦ ❣r❛✉ ❞♦ ♠♦r✜s♠♦ ♥✉❧♦ é ③❡r♦✳ ❈❛s♦ ❛ ❡①t❡♥sã♦ ❞❡ ❝♦r♣♦s s❡❥❛ s❡♣❛rá✈❡❧✱ ✐♥s❡♣❛rá✈❡❧ ♦✉ ♣✉r❛♠❡♥t❡ ✐♥s❡♣❛rá✈❡❧ ❞✐r❡♠♦s q✉❡ ϕ é s❡♣❛rá✈❡❧✱ ✐♥s❡♣❛rá✈❡❧ ❡ ♣✉r❛♠❡♥t❡ ✐♥✲ s❡♣❛rá✈❡❧✱ r❡s♣❡❝t✐✈❛♠❡♥t❡✳ ❊ ❞❡♥♦t❛♠♦s ♦ ❣r❛✉ ❞❡ s❡♣❛r❛❜✐❧✐❞❛❞❡ ❡ ✐♥s❡♣❛r❛❜✐❧✐❞❛❞❡ ❞❛

  (ϕ) (ϕ)

  s i

  ❡①t❡♥sã♦ ♣♦r deg ❡ deg ✱ r❡s♣❡❝t✐✈❛♠❡♥t❡✳

  1

  2

  1

  2

  ❈♦r♦❧ár✐♦ ✶✳✸✸✳ ❙❡❥❛♠ C ❡ C ❞✉❛s ❝✉r✈❛s s✉❛✈❡s✱ ❡ s❡❥❛ ϕ : C → C ✉♠ ♠♦r✜s♠♦ ❞❡ ❣r❛✉ ✶✳ ❊♥tã♦ ϕ é ✉♠ ✐s♦♠♦r✜s♠♦✳ ❉❡♠♦♥str❛çã♦✳ ❱❡r ✭❈❛♣ít✉❧♦ ✷✱ ❈♦r♦❧ár✐♦ ✷✳✹✳✶✱ ❬❙✐❧✈❡r♠❛♥❪✮✳

  1

  2

  

1

  2

  1

  ❙❡❥❛♠ C ❡ C ❝✉r✈❛s s✉❛✈❡s✱ ϕ : C → C ✉♠ ♠❛♣❛ ♥ã♦ ❝♦♥st❛♥t❡ ❡ P ∈ C ✳

  ϕ(P ) 2 )

  2

  ❈♦♥s✐❞❡r❛♥❞♦ t ∈ K(C ✉♠ ✉♥✐❢♦r♠✐③❛♥t❡ ♣❛r❛ C ❡♠ ϕ(P )✱ ❝❤❛♠❛r❡♠♦s ❞❡ ✐♥❞í❝❡ (P )

  

ϕ

  ❞❡ r❛♠✐✜❝❛çã♦ ❞❡ ϕ ❡♠ P ✱ ❞❡♥♦t❛❞♦ ♣♦r e ✱ ❛♦ ♥ú♠❡r♦

  ∗ e (P ) = ord (ϕ t ).

ϕ P ϕ(P )

  (P ) > 1

  ϕ

  ❊ss❡ ♥ú♠❡r♦ ✐♥❞❡♣❡♥❞❡ ❞❛ ❡s❝♦❧❤❛ ❞♦ ♣❛râ♠❡tr♦ ❡ ❛❧é♠ ❞✐ss♦ e ✳ ❉✐r❡♠♦s q✉❡ ϕ (P ) = 1

  ϕ

  é ♥ã♦ r❛♠✐✜❝❛❞♦ ❡♠ P s❡ e ✱ ❡ ϕ é ❞✐t♦ ♥ã♦ r❛♠✐✜❝❛❞♦ s❡ ❡❧❡ ♥ã♦ ❢♦r r❛♠✐✜❝❛❞♦ ❡♠ t♦❞♦ ♣♦♥t♦ P ∈ C

  1 ✳

  ❚❡♦r❡♠❛ ✶✳✸✹✳ ❙❡❥❛ ϕ : C

  1 2 ✉♠ ♠♦r✜s♠♦ ♥ã♦ ❝♦♥st❛♥t❡ ❡♥tr❡ ❝✉r✈❛s s✉❛✈❡s✳

  → C ❊♥tã♦✿

  2

  ✭❛✮ P❛r❛ t♦❞♦ Q ∈ C ✱

  X e ϕ (P ) = deg(ϕ).

1 P ∈ϕ (Q)

  2

  ✭❜✮ P❛r❛ t♦❞♦s ♦s ♣♦♥t♦s Q ∈ C ✱ ❝♦♠ ❡①❝❡ssã♦ ❞❡ ♥♦ ♠á①✐♠♦ ✉♠ ❝♦♥❥✉♥t♦ ✜♥✐t♦✱

  −1 (Q) (ϕ). s

  |ϕ | = deg

  ✷✵

  2

  3

  1

  ✭❝✮ ❙❡❥❛ φ : C → C ✉♠ ♦✉tr♦ ♠❛♣❛ ♥ã♦ ❝♦♥st❛♥t❡✳ ❊♥tã♦ ♣❛r❛ t♦❞♦ P ∈ C ✱ e (P ) = e (P ) (ϕ(P )).

  

φ◦ϕ ϕ φ

  · e ❉❡♠♦♥str❛çã♦✳ ❱❡r ✭❈❛♣ít✉❧♦ ✷✱ Pr♦♣♦s✐çã♦ ✷✳✻✱ ❬❙✐❧✈❡r♠❛♥❪✮✳

  1

  

2

  ❈♦r♦❧ár✐♦ ✶✳✸✺✳ ❯♠ ♠♦r✜s♠♦ ϕ : C → C é ♥ã♦ r❛♠✐✜❝❛❞♦ s❡✱ ❡ s♦♠❡♥t❡ s❡✱

  −1 (Q) .

  |ϕ | = deg(ϕ), ∀Q ∈ C

  2

  ❉❡♠♦♥str❛çã♦✳ ❱❡r ✭❈❛♣ít✉❧♦ ✷✱ ❈♦r♦❧ár✐♦ ✷✳✼✱ ❬❙✐❧✈❡r♠❛♥❪✮✳

  ✶✳✷✳✷ ❉✐✈✐s♦r❡s ❞❡ ✉♠❛ ❝✉r✈❛

  ❖s ❞✐✈✐s♦r❡s ❞❡ ❝✉r✈❛ ❛ ♣r✐♥❝í♣✐♦ ♥❛❞❛ ♠❛✐s sã♦ ❞♦ q✉❡ s♦♠❛s ❢♦r♠❛✐s s♦❜r❡ ♦s ♣♦♥t♦s ❞❛ ❝✉r✈❛✱ q✉❡ s❡rã♦ ✐♠♣♦rt❛♥t❡s ♣❛r❛ ♦ ❝❤❛♠❛❞♦ ❚❡♦r❡♠❛ ❞❡ ❘✐❡♠❛♥♥✲❘♦❝❤ ♥♦ ❝♦♥t❡①t♦ ❞❡ ❝✉r✈❛s ♣r♦❥❡t✐✈❛s ♥ã♦ s✐♥❣✉❧❛r❡s✳ ❉❡✜♥✐çã♦ ✶✳✸✻✳ ❯♠ ❞✐✈✐s♦r D ❞❡ ✉♠❛ ❝✉r✈❛ C✱ é ✉♠❛ s♦♠❛ ❢♦r♠❛❧

  X D = n (P ),

  P P ∈C

  = 0

  P P

  ♦♥❞❡ n ∈ Z✱ ❡ n ♣❛r❛ t♦❞♦ P ❝♦♠ ❡①❝❡ssã♦ ❞❡ ♥♦ ♠á①✐♠♦ ✉♠ ♥ú♠❡r♦ ✜♥✐t♦ ❞❡ ♣♦♥t♦s✳ ❉❡♥♦t❛r❡♠♦s ♣♦r Div(C) ❛♦ ❣r✉♣♦ ❛❜❡❧✐❛♥♦ ❧✐✈r❡ ❣❡r❛❞♦ ♣❡❧♦s ❞✐✈✐s♦r❡s ❞❡ C✳

  ❖ ❣r❛✉ ❞❡ D ∈ Div(C) é ❞❡✜♥✐❞♦ ❝♦♠♦

  X deg(D) = n P .

  P ∈C

  ❯♠ s✉❜❣r✉♣♦ ✐♠♣♦rt❛♥t❡ ❞❡ Div(C) é ♦ s✉❜❣r✉♣♦ ❢♦r♠❛❞♦ ♣❡❧♦s ❞✐✈✐s♦r❡s ❞❡ ❣r❛✉ ✵✱ ❞❡♥♦t❛❞♦ ♣♦r

  Div (C) := {D ∈ Div(C) : deg(D) = 0}.

  (C) ◆ã♦ é ❞✐❢í❝✐❧ ❛ ✈❡r✐✜❝❛çã♦ ❞❡ q✉❡ ❞❡ ❢❛t♦ Div é s✉❜❣r✉♣♦ ❞❡ Div(C)✳

  ❈❧❛r❛♠❡♥t❡ ♣♦❞❡♠♦s ✈❡r t❛♠❜é♠ q✉❡ G(K/K) ❛❣❡ s♦❜r❡ ♦ ❣r✉♣♦ Div(C) ❞❡ ❢♦r♠❛ ♥❛t✉r❛❧ ❞❛❞❛ ♣♦r

  X

  τ τ

  D = n (P ),

  P

  ∀τ ∈ G(K/K),

  P ∈C

τ

  = D

  ✷✶ >

  

P

  ❯♠ ❞✐✈✐s♦r s❡rá ❞✐t♦ ♣♦s✐t✐✈♦ s❡ n ✱ ♣❛r❛ t♦❞♦ P ∈ C✱ ❡ ❞❡♥♦t❛r❡♠♦s ♣♦r >

  D > 0 D

  1

  2

  1

  2

  ✳ ❆ss✐♠✱ ❡s❝r❡✈❡r❡♠♦s D ♣❛r❛ ✐♥❞✐❝❛r q✉❡ D − D é ♣♦s✐t✐✈♦✳

  ∗

  ❙✉♣♦♥❞♦ C ✉♠❛ ❝✉r✈❛ s✉❛✈❡✱ t♦♠❡♠♦s f ∈ K(C) ❡ ❝♦♥s✐❞❡r❡♠♦s ♦ ❞✐✈✐s♦r ❛ss♦❝✐❛❞♦ à f ❞❛❞♦ ♣♦r

  X div(f ) = ord (f )(P ).

  P P ∈C

  ❉❡✜♥✐çã♦ ✶✳✸✼✳ ❉❛❞♦ D ∈ Div(C)✱ ❞✐r❡♠♦s q✉❡ D é ♣r✐♥❝✐♣❛❧ s❡ D = div(f) ♣❛r❛

  ∗ ′ ′

  ❛❧❣✉♠ f ∈ K(C) ✱ ❡ q✉❡ D ❡ D sã♦ ❧✐♥❡❛r♠❡♥t❡ ❡q✉✐✈❛❧❡♥t❡s s❡ D − D é ♣r✐♥❝✐♣❛❧✳ ❉❡✜♥✐çã♦ ✶✳✸✽✳ ❖ ❣r✉♣♦ ❞❡ P✐❝❛r❞✱ ❞❡♥♦t❛❞♦ ♣♦r P ic(C)✱ é ❞❡✜♥✐❞♦ ❝♦♠♦ ♦ q✉♦❝✐❡♥t❡ ❡♥tr❡ Div(C) ❡ P rinc(C)✱ ✐st♦ é✱

  Div(C) P ic(C) = ,

  P rinc(C)

  ∗

  ♦♥❞❡ P rinc(C) = {div(f) : f ∈ K(C) } é ♦ ❣r✉♣♦ ❞♦s ❞✐✈✐❞♦r❡s ♣r✐♥❝✐♣❛✐s ❞❡ C✳ ❉❡ ❢♦r♠❛ ❛♥á❧♦❣❛ ❞❡✜♥✐♠♦s

  Div (C) P ic (C) = . P rinc(C)

  (C) ❆ ❞❡✜♥✐çã♦ ❞❡ P ic ❢❛③ s❡♥t✐❞♦ ❣r❛ç❛s à s❡❣✉✐♥t❡ ♣r♦♣♦s✐çã♦✱ q✉❡ ❛✜r♠❛ q✉❡ P rinc(C)

  (C) é s✉❜❣r✉♣♦ ❞❡ Div ✳

  P

  ❊♠ ✈✐st❛ ❞❡ q✉❡ ord é ✉♠❛ ✈❛❧♦r✐③❛çã♦ ❞✐s❝r❡t❛✱ t❡♠♦s q✉❡

  

  div : K(C) → Div(C) f 7→ div(f)

  é ✉♠ ❤♦♠♦♠♦r✜s♠♦ ❞❡ ❣r✉♣♦s✳ ❙❡❣✉❡ ❛ ♣r♦♣♦s✐çã♦✳

  ∗

  Pr♦♣♦s✐çã♦ ✶✳✸✾✳ ❙❡❥❛ C ✉♠❛ ❝✉r✈❛ s✉❛✈❡ ❡ f ∈ K(C) ✱ ❡♥tã♦✿

  ∗

  ✭❛✮ div(f) = 0 ⇔ f ∈ K ✳ ✭❜✮ deg(div(f)) = 0✳

  ❉❡♠♦♥str❛çã♦✳ ❱❡r ✭❈❛♣ít✉❧♦ ✷✱ Pr♦♣♦s✐çã♦ ✸✳✶✱ ❬❙✐❧✈❡r♠❛♥❪✮✳

  1 , C

  2

  ❉❡✜♥✐çã♦ ✶✳✹✵✳ ❙❡❥❛♠ C ❝✉r✈❛s ❛❧❣é❜r✐❝❛s ♣r♦❥❡t✐✈❛s ♣❧❛♥❛s✱ ❞❡✜♥✐❞❛s ♣♦r ♣♦❧✐♥ô✲

  1 , x 2 , x 3 ]

  ♠✐♦s ❤♦♠♦❣ê♥❡♦s f, g ∈ K[x ✳ ❙❡❥❛ t❛♠❜é♠

  2 C = : f (P ) = g(P ) = 0

  1

  2 ∩ C {P ∈ P }.

  ✷✷

  1

  2

  ❉❡✜♥✐♠♦s ❛ ♠✉❧t✐♣❧✐❝✐❞❛❞❡ ❞❡ ✐♥t❡rs❡çã♦ ❞❡ C ❡ C ❡♠ P ❝♦♠♦

  2 K[A ] P

  (C , C ) = dim ,

  1

  2 P K

  2

  (f , g ) ]

  D D P

  · K[A

  D D

  ♦♥❞❡ f ❡ g sã♦ ♦s r❡s♣❡❝t✐✈♦s ♣♦❧✐♥ô♠✐♦s ❞❡s♦♠♦❣❡♥❡✐③❛❞♦s ❡♠ r❡❧❛çã♦ à ❛❧❣✉♠ i✱ ♦✉

  2

  2

  ) = C ) = C

  D

1 D

  2

  s❡❥❛✱ V (f ∩ A i ❡ V (g ∩ A i ✳ ❊ ♦ ❞✐✈✐s♦r ❞❡ ✐♥t❡rs❡çã♦ é ❞❡✜♥✐❞♦ ❝♦♠♦

  X C = (C , C ) (P ).

  1

  2

  1

  2 P

  · C

  P ∈C

1 ∩C

  2

  ❆❣♦r❛ ❡♥✉♥❝✐❛♠♦s ♦ ❚❡♦r❡♠❛ ❞❡ ❇é③♦✉t✱ q✉❡ ♣❡r♠✐t❡ ❝♦♥t❛r ♦ ♥ú♠❡r♦ ❞❡ ♣♦♥t♦s ❞❡ ✐♥t❡rs❡çã♦ ❡♥tr❡ ❞✉❛s ❝✉r✈❛s ♣r♦❥❡t✐✈❛s ♣❧❛♥❛s✳

  1

  2

  ❚❡♦r❡♠❛ ✶✳✹✶ ✭❇é③♦✉t✮✳ ❙❡❥❛♠ C ✱ C ❝✉r✈❛s ♣r♦❥❡t✐✈❛s ♣❧❛♥❛s✱ ❞❡✜♥✐❞❛s ♣♦r ♣♦❧✐♥ô♠✐♦s ✐rr❡❞✉tí✈❡✐s ❞✐st✐♥t♦s f ❡ g✱ r❡s♣❡❝t✐✈❛♠❡♥t❡✳ ❊♥tã♦ ♦ ♥ú♠❡r♦ t♦t❛❧ ❞❡ ♣♦♥t♦s ❞❛ ✐♥t❡rs❡çã♦

  2

  1

  2

  ❞❡ C ❡ C ❡♠ P ✱ ❝♦♥t❛❞♦s ❝♦♠ ♠✉❧t✐♣❧✐❝✐❞❛❞❡✱ é ✐❣✉❛❧ ❛♦ ♣r♦❞✉t♦ ❞♦s ❣r❛✉s ❞❡ f ❡ g✳ ❉❡♠♦♥str❛çã♦✳ ❱❡r ✭❇é③♦✉t✬s ❚❤❡♦r❡♠✱ Pá❣✳ ✺✼✱ ❬❋✉❧t♦♥❪✮✳ ❊①❡♠♣❧♦ ✶✳✹✷✳ ◆❡ss❡ ❡①❡♠♣❧♦✱ ❝♦♥s✐❞❡r❛♠♦s ❛s ❝✉r✈❛s

  2 C : yz = x ,

  1

  ❡

  2

  3

  2 C : y z = x .

  2

  − 2xz

  1

  2

  ❱❡♠♦s q✉❡ ♦s ❣r❛✉s ❞♦s ♣♦❧✐♥ô♠✐♦s q✉❡ ❞❡✜♥❡♠ ❛s ❝✉r✈❛s C ❡ C sã♦✱ r❡s♣❡❝t✐✲ ✈❛♠❡♥t❡✱ ✷ ❡ ✸✱ ❞❡ ♦♥❞❡ s❡❣✉❡ q✉❡✱ ♣❡❧♦ ❚❡♦r❡♠❛ ❞❡ ❇é③♦✉t✱ ♦ ♥ú♠❡r♦ t♦t❛❧ ❞❡ ♣♦♥t♦s ❞❛

  1

  2

  ✐♥t❡rs❡çã♦ ❡♥tr❡ C ❡ C ❞❡✈❡ s❡r ✐❣✉❛❧ ❛ 6 = 2 · 3✳

  ✷✸

  2

  2

  1

  2

  ◆❛ ✜❣✉r❛ ❛✮✱ ✈❡♠♦s ❛s ❝✉r✈❛s ❛✜♥s C ∩A

  1 ❡ C ∩A 1 ✱ ♣❛r❛ ❛s q✉❛✐s ❝♦♥s✐❞❡r❛♠♦s

  x = 1 ✳

  ◆❛s ✜❣✉r❛s ❜✮ ❡ ❝✮✱ ✈❡♠♦s ✉♠ ♣r♦❝❡ss♦ ❛♥á❧♦❣♦ ❛♦ ❢❡✐t♦ ♥❛ ✜❣✉r❛ ❛✮✱ só q✉❡ ❛❣♦r❛ ❝♦♥s✐❞❡r❛♥❞♦ y = 1 ❡ z = 1✱ r❡s♣❡❝t✐✈❛♠❡♥t❡✳ ❊♠ ❛♠❜❛s ❛s ✜❣✉r❛s✱ ✈❡♠♦s ❛♣❡♥❛s ♦s ♣♦♥t♦s ❞❡ ❝♦♦r❞❡♥❛❞❛s r❡❛✐s✳

  ❱❡♠♦s t❛♠❜é♠ q✉❡ ♦s ♣♦♥t♦s P✱ ❉ ❡ ❊ ❛♣❛r❡❝❡♠ ♥❛s ✸ ✜❣✉r❛s✱ ✐st♦ ♣♦✐s✱ ♦ ♣♦♥t♦ P t❡♠ ❝♦♦r❞❡♥❛❞❛s ❤♦♠♦❣ê♥❡❛s ❬✶✱ ✲✶✱ ✲✶❪ ❂ ❬✲✶✱ ✶✱ ✶❪✱ ❥á ♦s ♣♦♥t♦s ❉ ❡ ❊ t❡♠ ❝♦♦r❞❡♥❛❞❛s

  1−i 1−i −i 1+i 1+i i

  ❬✶✱ ✶✰✐✱ ❪ ❂ ❬ ✱ ✶✱ ❪ ❂ ❬✶✰✐✱ ✷✐✱ ✶❪ ❡ ❬✶✱ ✶✲✐✱ ❪ ❂ ❬ ✱ ✶✱ ❪ ❂ ❬✶✲✐✱ ✲✷✐✱ ✶❪✳

  2

  2

  2

  2

  2

  2

  ❖s ♣♦♥t♦s ❉ ❡ ❊ ❛♣❛r❡❝❡♠ ❡♠ ❞❡st❛q✉❡ ❡ s❡♣❛r❛❞♦s ❞❛s ❝✉r✈❛s ♣♦✐s ♥ã♦ ❛♣r❡s❡♥t❛♠ ❝♦♦r❞❡♥❛❞❛s r❡❛✐s✳ ❆❧é♠ ❞✐ss♦✱ t❡♠♦s ♦s ♣♦♥t♦s ◗ ❂ ❬✵✱ ✵✱ ✶❪ ❡ ❙ ❂ ❬✵✱ ✶✱ ✵❪✳ ❖ ♣♦♥t♦ ❙ t❡♠ ♠✉❧t✐♣❧✐❝✐❞❛❞❡ ✷✳

  ❊♠ r❡s✉♠♦✱ ♦s ♣♦♥t♦s ❞❛ ✐♥t❡rs❡çã♦ sã♦ ❙ ✭❝♦♠ ♠✉❧t✐♣❧✐❝✐❞❛❞❡ ✷✮✱ P✱ ◗✱ ❉ ❡ ❊✱ t♦t❛❧✐③❛♥❞♦ ✻ ♣♦♥t♦s ❝♦♥t❛❞❛s ❛s s✉❛s ♠✉❧t✐♣❧✐❝✐❞❛❞❡s✳

  ✷✹

  ✶✳✷✳✸ ❉✐❢❡r❡♥❝✐❛✐s C

  ❉❛❞❛ C ✉♠❛ ❝✉r✈❛✱ ❞❡✜♥✐♠♦s ♦ ❡s♣❛ç♦ ❞❛s ❢♦r♠❛s ❞✐❢❡r❡♥❝✐❛✐s ❞❡ C✱ Ω ✱ ❝♦♠♦ ♦ K(C)✲❡s♣❛ç♦ ✈❡t♦r✐❛❧ ❣❡r❛❞♦ ♣❡❧♦s sí♠❜♦❧♦s ❞❛ ❢♦r♠❛ dx✱ ♣❛r❛ t♦❞♦ x ∈ K(C)✱ s❛t✐s❢❛✲ ③❡♥❞♦ ❛s s❡❣✉✐♥t❡s ♣r♦♣r✐❡❞❛❞❡s ❞✐❢❡r❡♥❝✐❛✐s✳

  ✭✐✮ d(x + y) = dx + dy, ∀ x, y ∈ K(C)✳ ✭✐✐✮ d(xy) = xdy + ydx, ∀ x, y ∈ K(C)✳

  ✭✐✐✐✮ da = 0, ∀ a ∈ K✳

  C

  ❙♦❜r❡ ♦ ❡s♣❛ç♦ Ω t❡♠♦s ♦ s❡❣✉✐♥t❡ r❡s✉❧t❛❞♦✳

  ∗

  Pr♦♣♦s✐çã♦ ✶✳✹✸✳ ❙❡❥❛ C ✉♠❛ ❝✉r✈❛ ❡ P ∈ C✳ ❙❡ t ∈ K(C) é ✉♠ ✉♥✐❢♦r♠✐③❛♥t❡ ♣❛r❛ C

  ❡♠ P ✱ ❡♥tã♦✿

  C

  ✭❛✮ P❛r❛ t♦❞♦ ω ∈ Ω ❡①✐st❡ ✉♠❛ ú♥✐❝❛ ❢✉♥çã♦ g ∈ K(C) q✉❡ ❞❡♣❡♥❞❡ ❞❡ ω ❡ t✱ t❛❧ q✉❡ w = gdt.

  ❉❡♥♦t❛r❡♠♦s g ♣♦r ω/dt✳ ✭❜✮ ❙❡❥❛ f ∈ K(C) ❢✉♥çã♦ r❡❣✉❧❛r ❡♠ P ✳ ❊♥tã♦ df/dt t❛♠❜é♠ é r❡❣✉❧❛r ❡♠ P ✳

  (ω/dt)

  C P

  ✭❝✮ ❙❡❥❛ ω ∈ Ω ✱ ω 6= 0✳ ❖ ♥ú♠❡r♦ ord é ✐♥❞❡♣❡♥❞❡♥t❡ ❞❛ ❡s❝♦❧❤❛ ❞♦ ✉♥✐❢♦r♠✐✲ (ω)

  P

  ③❛♥t❡ t✳ ❈❤❛♠❛r❡♠♦s ❡ss❡ ♥ú♠❡r♦ ❞❡ ♦r❞❡♠ ❞❡ ω ❡♠ P ✱ ❞❡♥♦t❛❞♦ ♣♦r ord ✳ (ω) = 0

  C P

  ✭❞✮ ❙❡❥❛ ω ∈ Ω ✱ ω 6= 0✳ ❊♥tã♦ ord ♣❛r❛ t♦❞♦ P ∈ C ❝♦♠ ❡①❝❡ssã♦ ❞❡ ♥♦ ♠á①✐♠♦ ✉♠ ♥ú♠❡r♦ ✜♥✐t♦ ❞❡ ♣♦♥t♦s✳

  ❉❡♠♦♥str❛çã♦✳ ❱❡r ✭❈❛♣ít✉❧♦ ✷✱ Pr♦♣♦s✐çã♦ ✹✳✸✱ ❬❙✐❧✈❡r♠❛♥❪✮✳ ❈♦♠ ✐ss♦✱ ♣♦❞❡♠♦s ❛ss♦❝✐❛r ❛ ω ∈ Ω C ✉♠ ❞✐✈✐s♦r ❞❛❞♦ ♣♦r

  X div(ω) = ord (ω)(P ).

  P P ∈C C

  ❉✐r❡♠♦s q✉❡ ω ∈ Ω é ❤♦❧♦♠♦r❢♦ s❡ div(ω) é ♣♦s✐t✐✈♦✱ ❡ q✉❡ ω ♥ã♦ s❡ ❛♥✉❧❛ s❡ div(ω) 6 0 ✳

  , ω ◆♦t❡ q✉❡ s❡ ω

  1

2 C sã♦ ❞♦✐s ❞✐❢❡r❡♥❝✐❛✐s ♥ã♦ ♥✉❧♦s✱ ❡♥tã♦ ♣❡❧❛ ♣r♦♣♦s✐çã♦

  ∈ Ω

  ∗

  ❛♥t❡r✐♦r✱ ❡①✐st❡ ✉♠❛ ❢✉♥çã♦ f ∈ K(C) t❛❧ q✉❡ ω

  

1 = f ω

2 ,

  ) = div(f ) + div(ω ). ) )

  1

  2

  1

  2

  ❡ ♣♦rt❛♥t♦ div(ω ❆ss✐♠✱ div(ω − div(ω é ✉♠ ❞✐✈✐s♦r ♣r✐♥❝✐♣❛❧✱ ❞❡

  ✷✺ ❞❡t❡r♠✐♥❛♥❞♦ ♣♦rt❛♥t♦ ❛♣❡♥❛s ✉♠❛ ❝❧❛ss❡ ♥♦ ❣r✉♣♦ ❞❡ P✐❝❛r❞ ❞❡ C✱ ❡ ❛ ❡ss❛ ❝❧❛ss❡ ❞❛♠♦s ♦ ♥♦♠❡ ❞❡ ❝❧❛ss❡ ❝❛♥ô♥✐❝❛ ❡ q✉❛❧q✉❡r ❞✐✈✐s♦r D ♥❡st❛ ❝❧❛ss❡ é ❞✐t♦ ✉♠ ❞✐✈✐s♦r ❝❛♥ô♥✐❝♦ ❞❡ C✳

  ✶✳✷✳✹ ❖ ❚❡♦r❡♠❛ ❞❡ ❘✐❡♠❛♥♥✲❘♦❝❤

  ❖ ❚❡♦r❡♠❛ ❞❡ ❘✐❡♠❛♥♥✲❘♦❝❤ é ✉♠ t❡♦r❡♠❛ ❞❡ ❣r❛♥❞❡ ✐♠♣♦rtâ♥❝✐❛ ♣❛r❛ ❛ t❡♦r✐❛ ❞❡ ❝✉r✈❛s ❛❧❣é❜r✐❝❛s✳ ◆♦ss♦ ♦❜❥❡t✐✈♦ é ❞❡ ❛♣r❡s❡♥tá✲❧♦ ❛✜♠ ❞❡ ♣♦❞❡r♠♦s ❞❡✜♥✐r ♦ ❣ê♥❡r♦ ❞❡ ✉♠❛ ❝✉r✈❛ ❡ ♦❜t❡r ❛❧❣✉♥s ❞❡ s❡✉s ❝♦r♦❧ár✐♦s q✉❡ s❡rã♦ ✐♠♣♦rt❛♥t❡s ❢✉t✉r❛♠❡♥t❡✳

  ❉❛❞♦ D ∈ Div(C) ❞❡✜♥✐♠♦s ♦ s❡❣✉✐♥t❡ ❝♦♥❥✉♥t♦ ❞❡ ❢✉♥çõ❡s

  

  : div(f ) > L(D) = {f ∈ K(C) −D} ∪ 0. ❊ss❡s ❝♦♥❥✉♥t♦s ❛♣r❡s❡♥t❛♠ ❛s s❡❣✉✐♥t❡s ♣r♦♣r✐❡❞❛❞❡s✳

  Pr♦♣♦s✐çã♦ ✶✳✹✹✳ ❙❡❥❛ D ∈ Div(C)✳ ❊♥tã♦✿ ✭❛✮ L(D) é ✉♠ K−❡s♣❛ç♦ ✈❡t♦r✐❛❧✳

  ✭❜✮ L(0) = K ❡ L(D) = {0} s❡ deg(D) < 0✳

  ′ ′

  ) ✭❝✮ ❙❡ D 6 D ❡♥tã♦ L(D) ⊆ L(D ❡

  ′ ′

  dim ( )/ K L(D L(D)) 6 deg(D − D). ✭❞✮ L(D) t❡♠ ❞✐♠❡♥sã♦ ✜♥✐t❛ ♣❛r❛ t♦❞♦ D✳ ❉❡✜♥✐♠♦s ❛ ❞✐♠❡♥sã♦ ❞❡ L(D) s♦❜r❡ K

  ♣♦r ℓ(D)✳ ❙❡ deg(D) > 0✱ ❡♥tã♦ ℓ(D) 6 deg(D) + 1✳

  ′ ′

  ) ✭❡✮ ❙❡ D ❡ D sã♦ ❧✐♥❡❛r♠❡♥t❡ ❡q✉✐✈❛❧❡♥t❡s✱ ❡♥tã♦ ℓ(D) = ℓ(D ✳

  ❉❡♠♦♥str❛çã♦✳ ✭❛✮ ❙❡❥❛ λ ∈ K, f ∈ L(D)✳ ❙❡ λ = 0✱ ❡♥tã♦ λf ∈ L(D)✱ ✈✐st♦ q✉❡ ∈ L(D) ♣♦r ❞❡✜♥✐çã♦✳

  ❙❡ λ 6= 0✱ ❡♥tã♦ div(λf) = div(f) > −D✳ P♦rt❛♥t♦✱ λf ∈ L(D)✳ P n (P ) (f ), ord (g) >

  P P P

  ❆❣♦r❛✱ s❡ f, g ∈ L(D) ❡ D = ✱ ❡♥tã♦ t❡♠♦s q✉❡ ord

  P ∈C

  ,

  P

  −n ♣❛r❛ t♦❞♦ P ∈ C✳ P♦rt❛♥t♦✱ ord (f + g) > min (f ), ord (g) ,

  P P P P

  {ord } > −n ❞❡ ♦♥❞❡ s❡❣✉❡ q✉❡ f + g ∈ L(D)✳

  ∗

  : div(f ) > 0 ✭❜✮ L(0) = {f ∈ K(C) } ∪ {0}.

  ✷✻

  ∗ P

  ❉❛❞♦ f ∈ K(C) t❛❧ q✉❡ div(f) > 0✱ ❡♥tã♦ f ∈ K[C] ♣❛r❛ t♦❞♦ P ∈ C✳ ❉❛í✱ ) = λ

  ) > 0 s❡ f(P ∈ K ♣❛r❛ ❛❧❣✉♠ P ∈ C✱ ❡♥tã♦ div(f − λ ✱ ♣♦✐s ❡st❛ ❢✉♥çã♦ )(P ) = 0

  ❝♦♥t✐♥✉❛ ❞❡✜♥✐❞❛ ❡♠ t♦❞♦s ♦s ♣♦♥t♦s ❞❡ C✳ ▼❛s ❝♦♠♦ (f − λ ✱ ❡♥tã♦

  ∗

  ord (f ) > 1 )) > 0

  P ✱ ❡ ❛ss✐♠ deg(div(f − λ ✱ ♠❛s s❡ f − λ ✱ ❡♥tã♦

  − λ ∈ K(C)

  )) = 0 = 0 ❞❡✈❡rí❛♠♦s t❡r deg(div(f − λ ✱ ❞❡ ♦♥❞❡ s❡❣✉❡ q✉❡ f − λ ❡ ♣♦rt❛♥t♦ f = λ

  ∈ K✱ ❧♦❣♦ L(0) ⊆ K✳ ❆❣♦r❛ s❡ f ∈ K ✱ ❡♥tã♦ div(f) = 0 > 0 ❡ f ∈ L(0)✱ ❞❡ ♦♥❞❡ s❡❣✉❡ q✉❡ L(0) = K. ❆❣♦r❛ s❡❥❛ D ❞✐✈✐s♦r t❛❧ q✉❡ deg(D) < 0 ❡ ❝♦♥s✐❞❡r❡ f ∈ L(D) − {0}✳ ❈♦♠♦

  ∗

  f ∈ K(C) ✱ ❡♥tã♦ 0 = deg(div(f))✳ ❈♦♠♦ div(f) > −D✱ ❡♥tã♦ 0 = deg(div(f )) > deg(

  −D) = −deg(D), ❞❡ ♦♥❞❡ s❡❣✉❡ q✉❡ deg(D) > 0✱ ❝♦♥tr❛❞✐çã♦✳ ❆ss✐♠✱ s❡ f ∈ L(D) ❡♥tã♦ f = 0✳ P♦rt❛♥t♦✱ L(D) = {0}✳

  ′ ′ ′ ).

  ✭❝✮ ❙❡ D 6 D ❡ f ∈ L(D)−{0}✱ ❡♥tã♦ div(f) > −D > −D ✱ ❧♦❣♦ f ∈ L(D P♦rt❛♥t♦✱

  ′ ).

  L(D) ⊆ L(D

  ′ ′

  = D + P ❆❣♦r❛✱ s✉♣♦♥❤❛ ✐♥✐❝✐❛❧♠❡♥t❡ q✉❡ D ♣❛r❛ P ∈ C✳ ❈♦♠♦ deg(D − D) = 1✱ ❡♥tã♦ ❞❡✈❡♠♦s ♠♦str❛r q✉❡

  

  dim ( )/ K L(D L(D)) 6 1.

  P P

  ❙❡❥❛ t ∈ K[C] ✉♠ ✉♥✐❢♦r♠✐③❛♥t❡ ♣❛r❛ C ❡♠ P ✱ r = n ♦ ❝♦❡✜❝✐❡♥t❡ ❞❡ P ❡♠ D ❡ ❝♦♥s✐❞❡r❡ ❛ ❛♣❧✐❝❛çã♦

  ϕ : L(D + P ) → K,

  r+1 r+1

  f )(P ) (f ) > f ❞❡✜♥✐❞❛ ♣♦r ϕ(f) = (t ✳ ❈♦♠♦ ord P ❡stá

  −(r + 1)✱ ❡♥tã♦ t ❞❡✜♥✐❞❛ ❡♠ P ✱ ❡ ❛ss✐♠ ❛ ❛♣❧✐❝❛çã♦ ϕ ❡stá ❜❡♠ ❞❡✜♥✐❞❛✳ ▼❛✐s ❞♦ q✉❡ ✐ss♦✱ ϕ é ❧✐♥❡❛r ❡ Ker(ϕ) = L(D)✳ P♦rt❛♥t♦✱ dim ( K = 1.

  L(D + P )/L(D)) 6 dim

  

K K

′ ′

  = D + P +

  1 s

  ❆❣♦r❛ ❣❡♥❡r❛❧✐③❛♥❞♦✱ s❡ D · · · + P ✱ ❡♥tã♦ deg(D − D) = s ❡

  • dim ( )/ )) 6 1,

  1 m 1 m−1 K L(D + P · · · + P L(D + P · · · + P

  ♣❛r❛ t♦❞♦ m = 1, . . . , s✳

  1 s )

  • 1 )

  ❈♦♠♦ L(D) ⊆ L(D + P ⊆ · · · ⊆ L(D + P · · · + P ✱ ❡♥tã♦

  s

  z }| {

  ′ 1 s )/ 1 +

  K

  • dim ( L(D + P · · · + P L(D)) 6 · · · + 1 = s = deg(D − D),

  ✷✼ ❝♦♠♦ q✉❡rí❛♠♦s✳

  ✭❞✮ ❙❡ deg(D) < 0✱ ❡♥tã♦ L(D) = {0} t❡♠ ❞✐♠❡♥sã♦ ✜♥✐t❛✳ ❆❣♦r❛✱ s❡ deg(D) = n > 0✱

  ′ ′

  = D ) = ❡♥tã♦ ❡s❝♦❧❤❛ P ∈ C✱ ❡ ❝♦♥s✐❞❡r❡ D

  − (n + 1)P ✳ ❊♥tã♦ deg(D −1 < 0✱ ❞❡

  ′

  ) = ♦♥❞❡ s❡❣✉❡ q✉❡ L(D

  {0}✳ ) = n + 1

  ❈♦♠♦ D > D ❡ deg(D − D ✱ ❡♥tã♦

  ′ ′ dim ( ( )) 6 deg(D ) = n + 1 = deg(D) + 1.

  L(D)) = dim L(D)/L(D − D

  K K

  P♦rt❛♥t♦ L(D) t❡♠ ❞✐♠❡♥sã♦ ✜♥✐t❛ ❡ ℓ(D) 6 deg(D) + 1✳

  

∗ ′ ′

  ✭❡✮ ❙❡❥❛ h ∈ K(C) t❛❧ q✉❡ D − D = div(h)✱ ❡ss❡ h ❡①✐st❡ ♣♦✐s D ❡ D sã♦ ❧✐♥❡❛r♠❡♥t❡

  ′

  ) ❡q✉✐✈❛❧❡♥t❡s✳ ❈♦♥s✐❞❡r❡ ψ : L(D → L(D) ❞❛❞❛ ♣♦r ψ(f) = fh✱ q✉❡ ❡stá ❜❡♠

  ′

  • div(h) = ❞❡✜♥✐❞❛ ♣❡❧♦ ❢❛t♦ ❞❡ q✉❡ s❡ f 6= 0✱ ❡♥tã♦ div(fh) = div(f)+div(h) > −D −D✱ ❧♦❣♦ fh ∈ L(D)✳ ❚❡♠♦s q✉❡ ψ é ❧✐♥❡❛r ❡ Ker(ψ) = {0}✳ ❆❣♦r❛✱ s❡ g ∈ L(D) − {0}✱ ❡♥tã♦ g

  ′

  div( ) + D = div(g) − div(h) + D + div(h) = div(g) + D > 0. h g g

  ′

  ) ) = g P♦rt❛♥t♦ ❡ ψ( ✱ ♠♦str❛♥❞♦ q✉❡ ψ t❛♠❜é♠ é s♦❜r❡❥❡t✐✈❛✱ ❧♦❣♦ ✉♠

  ∈ L(D h h

  ′ ).

  ✐s♦♠♦r✜s♠♦ ❧✐♥❡❛r✱ ❞❡ ♦♥❞❡ s❡❣✉❡ q✉❡ ℓ(D) = ℓ(D ❆❣♦r❛ ❡♥✉♥❝✐❛♠♦s ♦ ❚❡♦r❡♠❛ ❞❡ ❘✐❡♠❛♥♥✲❘♦❝❤✱ ♦ q✉❛❧ ♥♦s ♣❡r♠✐t✐rá ❞❡✜♥✐r ♦

  ❣ê♥❡r♦ ❞❡ ✉♠❛ ❝✉r✈❛✳

  C

  ❚❡♦r❡♠❛ ✶✳✹✺ ✭❘✐❡♠❛♥♥✲❘♦❝❤✮✳ ❙❡❥❛ C ✉♠❛ ❝✉r✈❛ s✉❛✈❡ ❡ K ✉♠ ❞✐✈✐s♦r ❝❛♥ô♥✐❝♦ ❡♠ C✳ ❊♥tã♦ ❡①✐st❡ ✉♠ ✐♥t❡✐r♦ g > 0✱ q✉❡ só ❞❡♣❡♥❞❡ ❞❡ C✱ t❛❧ q✉❡ ♣❛r❛ ❝❛❞❛ ❞✐✈✐s♦r D

  ∈ Div(C)✱ ℓ(D)

  C − ℓ(K − D) = deg(D) − g + 1.

  ❆ ❡ss❡ ✐♥✈❛r✐❛♥t❡ g ❞❛♠♦s ♦ ♥♦♠❡ ❞❡ ❣ê♥❡r♦ ❞❛ ❝✉r✈❛ C✳ ❉❡♠♦♥str❛çã♦✳ ❱❡r ✭❘✐❡♠❛♥♥✲❘♦❝❤ ❚❤❡♦r❡♠✱ Pá❣✳ ✶✵✽✱ ❬❋✉❧t♦♥❪✮✳

  ❈♦♠♦ ❝♦♥s❡q✉ê♥❝✐❛s ❞♦ ❚❡♦r❡♠❛ ❞❡ ❘✐❡♠❛♥♥✲❘♦❝❤ t❡♠♦s✳ ) = g

  ❈♦r♦❧ár✐♦ ✶✳✹✻✳ ✭❛✮ ℓ(K C ✳

  C ) = 2g

  ✭❜✮ deg(K − 2✳ ✭❝✮ ❙❡ deg(D) > 2g − 2✱ ❡♥tã♦ ℓ(D) = deg(D) − g + 1.

  ✷✽ ❉❡♠♦♥str❛çã♦✳ ✭❛✮ ❇❛st❛ t♦♠❛r D = 0 ♥♦ t❡♦r❡♠❛✳ ❆ss✐♠✱ 1 ) =

  

C

  − ℓ(K −g + 1,

  C ) = g

  ❧♦❣♦ ℓ(K ✳

  C

  ✭❜✮ ❇❛st❛ t♦♠❛r D = K ♥♦ t❡♦r❡♠❛✳ ❆ss✐♠✱ ℓ(K ) )

  

C C

  − 1 = deg(K − g + 1, ) = 2g

  C

  ❧♦❣♦ deg(K − 2✳ ✭❝✮ ❙❡ deg(D) > 2g − 2✱ ❡♥tã♦

  ℓ(K

  C − D) − ℓ(D) = deg(K C − D) − g + 1 C

  −ℓ(K − D) + ℓ(D) = deg(D) − g + 1

  C

  ⇒deg(K − D) − g + 1 = −deg(D) + g − 1

  C ⇒deg(K − D) = 2g − 2 − deg(D) < 0.

  ▲♦❣♦✱ L(K C C − D) = {0} ❡ ℓ(K − D) = 0✳ P♦rt❛♥t♦✱ ℓ(D) = deg(D) − g + 1✳

  ❈♦r♦❧ár✐♦ ✶✳✹✼✳ ❙❡ ❞✉❛s ❝✉r✈❛s C

  1 ❡ C 2 sã♦ ✐s♦♠♦r❢❛s✱ ❡♥tã♦ ❛♠❜❛s t❡♠ ♠❡s♠♦ ❣ê♥❡r♦✳

  ❉❡♠♦♥str❛çã♦✳ ❱❡r ✭❈♦r♦❧ár✐♦ ✶✳✺✳✺✱ ❬❈♦♥❝❡✐çã♦❪✮✳ ❆ s❡❣✉✐r ❡♥✉♥❝✐❛♠♦s ✉♠❛ ❢ór♠✉❧❛ q✉❡ ❢❛❝✐❧✐t❛ ♦ ❝á❧❝✉❧♦ ❞♦ ❣ê♥❡r♦ ❞❡ ✉♠❛ ❝✉r✈❛

  ♣❧❛♥❛ s✉❛✈❡✳ Pr♦♣♦s✐çã♦ ✶✳✹✽ ✭❋ór♠✉❧❛ ❞♦ ●r❛✉✲●ê♥❡r♦✮✳ ❙❡❥❛ C ✉♠❛ ❝✉r✈❛ ♣❧❛♥❛ ❡ s✉❛✈❡ ❞❡✜♥✐❞❛ ♣♦r ✉♠ ♣♦❧✐♥ô♠✐♦ ❤♦♠♦❣ê♥❡♦ ❞❡ ❣r❛✉ d > 1✳ ❊♥tã♦ s❡✉ ❣ê♥❡r♦ é ❞❛❞♦ ♣❡❧❛ ❡①♣r❡ssã♦

  (d − 1)(d − 2) g(C) = .

  2 ❉❡♠♦♥str❛çã♦✳ ❱❡r ✭❈♦r♦❧ár✐♦ ✶✳✺✳✼✱ ❬❈♦♥❝❡✐çã♦❪✮✳ ❊①❡♠♣❧♦ ✶✳✹✾✳ P❛r❛ ❛ ❝✉r✈❛ C ❞♦ ❊①❡♠♣❧♦ ✶✳✶✾✱ ❞❡✜♥✐❞❛ ♣❡❧♦ ♣♦❧✐♥ô♠✐♦ ❤♦♠♦❣ê♥❡♦ ❞❡

  2

  = 0 ❣r❛✉ ✷✱ f(x, y, z) = yz − x ✱ t❡♠♦s q✉❡

  (2 − 1) · (2 − 2) g(C) = = 0.

  2 ❖❜s❡r✈❡♠ q✉❡ [1, 1, 1] ∈ C(Q), ♦✉ s❡❥❛✱ ❛ ❝✉r✈❛ C t❡♠ ❛♦ ♠❡♥♦s ✉♠ ♣♦♥t♦ r❛❝✐♦♥❛❧✳

  2

  , 1] ❆❣♦r❛ ♠❛✐s ❞♦ q✉❡ ✐ss♦✱ t❡♠♦s q✉❡ [t, t ∈ C(Q)✱ ♣❛r❛ t♦❞♦ t ∈ Q✱ ♦✉ s❡❥❛✱ C t❡♠

  ✷✾ ❊ss❡ é ✉♠ ❝❛s♦ ♣❛rt✐❝✉❧❛r ❞♦ q✉❡ ❛❝♦♥t❡❝❡ ❝♦♠ ❝✉r✈❛s ❞❡ ❣ê♥❡r♦ ✵✳ ❙❡✉ ❝♦♥❥✉♥t♦

  ❞❡ ♣♦♥t♦s r❛❝✐♦♥❛✐s ♦✉ é ✈❛③✐♦ ♦✉ é ✐♥✜♥✐t♦✳ ◆♦ ✜♥❛❧ ❞♦ ❈❛♣ít✉❧♦ ✸✱ s❡rá ❛♣r❡s❡♥t❛❞❛ ✉♠❛ t❛❜❡❧❛ ❡sq✉❡♠❛t✐③❛♥❞♦ ❛ ❡str✉t✉r❛ ❞♦s ♣♦♥t♦s r❛❝✐♦♥❛✐s ❞❡ ✉♠❛ ❝✉r✈❛✱ ❞❡ ❛❝♦r❞♦ ❝♦♠ ♦ s❡✉ ❣ê♥❡r♦✳ Pr♦♣♦s✐çã♦ ✶✳✺✵✳ ❙❡❥❛ C/K ✉♠❛ ❝✉r✈❛ s✉❛✈❡ ❡ D ✉♠ ❞✐✈✐s♦r ❞❡✜♥✐❞♦ s♦❜r❡ K✳ ❊♥tã♦ L(D) ♣♦ss✉✐ ✉♠❛ ❜❛s❡ ❝♦♥s✐st✐♥❞♦ ❞❡ ❢✉♥çõ❡s ❡♠ K(C)✳ ❉❡♠♦♥str❛çã♦✳ ❱❡r ✭❈❛♣ít✉❧♦ ✷✱ Pr♦♣♦s✐çã♦ ✺✳✽✱ ❬❙✐❧✈❡r♠❛♥❪✮✳

  ❈❛♣ít✉❧♦ ✷ ❈✉r✈❛s ❊❧í♣t✐❝❛s

  ◆❡st❡ ❝❛♣ít✉❧♦ s❡rã♦ ✐♥tr♦❞✉③✐❞♦s ❝♦♥❝❡✐t♦s ❞❛ ❚❡♦r✐❛ ❞❡ ❈✉r✈❛s ❊❧í♣t✐❝❛s✱ ❝♦♠ ♦ ♦❜❥❡t✐✈♦ ❞❡ ❡st✉❞❛r ❛ ❛r✐t♠ét✐❝❛ ❞❡ss❡s ♦❜❥❡t♦s✳ ❆ ♣r✐♥❝✐♣❛❧ r❡❢❡rê♥❝✐❛ ♣❛r❛ ❡st❡ ❈❛♣ít✉❧♦✱ é ♦ ❧✐✈r♦ ❬❙✐❧✈❡r♠❛♥❪✳ ❆❧❣✉♥s ❝♦♥❝❡✐t♦s ❞❛ ❚❡♦r✐❛ ❆❧❣é❜r✐❝❛ ❞♦s ◆ú♠❡r♦s s❡rã♦ ✉t✐❧✐③❛❞♦s✱ ♣❛r❛ ✉♠❛ ❧❡✐t✉r❛ s♦❜r❡ ♦ ❛ss✉♥t♦ r❡❝♦♠❡♥❞❛♠♦s ❛s r❡❢❡rê♥❝✐❛s ❬◆❡✉❦✐r❝❤❪ ❡ ❬▲❛♥❣❪✳ ❉❡✜♥✐çã♦ ✷✳✶✳ ❯♠❛ ❝✉r✈❛ ❡❧í♣t✐❝❛ é ✉♠ ♣❛r (E, O)✱ ♦♥❞❡ E é ✉♠❛ ❝✉r✈❛ s✉❛✈❡ ❞❡ ❣ê♥❡r♦ ✶ ❡ O ∈ E✳ ❉✐③❡♠♦s q✉❡ E ❡stá ❞❡✜♥✐❞❛ s♦❜r❡ K ❡ ❞❡♥♦t❛r❡♠♦s ♣♦r E/K✱ s❡ E ❡stá ❞❡✜♥✐❞❛ s♦❜r❡ K ❝♦♠♦ ✉♠❛ ✈❛r✐❡❞❛❞❡ ❡ ❛❧é♠ ❞✐ss♦✱ O ∈ E(K)✳

  ❉✐r❡♠♦s q✉❡ O é ♦ ♣♦♥t♦ ❜❛s❡ ❞❡ E✳ P❛r❛ ❡❢❡✐t♦ ❞❡ s✐♠♣❧✐❝❛çã♦✱ ♥❛ ♠❛✐♦r✐❛ ❞❛s ✈❡③❡s ♦ ♣♦♥t♦ ❜❛s❡ s❡rá ♦♠✐t✐❞♦✱ ✉s❛♥❞♦

  ❛♣❡♥❛s E ♣❛r❛ ❞❡♥♦t❛r ❛ ❝✉r✈❛ ❡❧í♣t✐❝❛✳ ❆ s❡❣✉✐r✱ ♠♦str❛r❡♠♦s q✉❡ ❞❛❞❛ ✉♠❛ ❝✉r✈❛ ❡❧í♣t✐❝❛ E✱ ♣♦❞❡♠♦s ❡♥❝♦♥tr❛r ✉♠❛

  2

  ❝✉r✈❛ ❡♠ P ✱ ❞❡✜♥✐❞❛ ♣♦r ✉♠ ♣♦❧✐♥ô♠✐♦ ❞❡ ❣r❛✉ ✸✱ q✉❡ é ✐s♦♠♦r❢❛ à E✳ ▼❛s ❛♥t❡s ❞✐ss♦ ♣r❡❝✐s❛r❡♠♦s ❞♦ s❡❣✉✐♥t❡ r❡s✉❧t❛❞♦✳ ▲❡♠❛ ✷✳✷✳ ❙❡❥❛ C ✉♠❛ ❝✉r✈❛ ❞❡✜♥✐❞❛ ♣♦r ✉♠ ♣♦❧✐♥ô♠✐♦

  2

  

3

  2

  f (x, y) = y + a xy + a y x x

  1

  3

  2

  4

  6 − x − a − a − a ∈ K[x, y].

  1

  ❙❡ C é s✐♥❣✉❧❛r ❡♥tã♦ ❡①✐st❡ ✉♠ ♠❛♣❛ r❛❝✐♦♥❛❧ φ : C → P ❞❡✜♥✐❞♦ s♦❜r❡ K ❞❡ ❣r❛✉ ✶✳ ❉❡♠♦♥str❛çã♦✳ ❋❛③❡♥❞♦ ✉♠❛ ♠✉❞❛♥ç❛ ❞♦ t✐♣♦ x 7→ x + a✱ y 7→ y + b✱ ♣♦❞❡♠♦s s✉♣♦r q✉❡

  ∂f = (0, 0) = 0

  ♦ ♣♦♥t♦ (0, 0) é s✐♥❣✉❧❛r✱ ♣❛r❛ ❡❢❡✐t♦ ❞❡ s✐♠♣❧✐✜❝❛çã♦✳ ❚❡♠♦s ❡♥tã♦ a

  4 ❡

  ∂x ∂f a = (0, 0) = 0

  3

  ✳ P♦rt❛♥t♦✱ ♦❜t❡♠♦s ∂y

  2

  3

  2 y + a xy = x + a x .

  1

  2

  ✸✶ ❆❣♦r❛ ❝♦♥s✐❞❡r❛♥❞♦ ♦ ♠❛♣❛ r❛❝✐♦♥❛❧

  1

  φ : C → P

  (x, y) 7→ [x, y]. (K(P )) = K(x, y) = K(C)

  ❚❡♠♦s q✉❡ φ ✱ ❞❡ ♦♥❞❡ deg(φ) = 1✳ Pr♦♣♦s✐çã♦ ✷✳✸✳ ❙❡❥❛ E ✉♠❛ ❝✉r✈❛ ❡❧í♣t✐❝❛ ❞❡✜♥✐❞❛ s♦❜r❡ K✳ ❊♥tã♦ ❡①✐st❡ ✉♠❛ ❝✉r✈❛ ♣❧❛♥❛ C ❞❡ ❡q✉❛çã♦

  2

  3

  2 C : y + a xy + a y = x + a x + a x + a ,

  1

  3

  2

  4

  6

  , a , a , a , a ❝♦♠ a

  1

  2

  3

  4

  6

  ∈ K✱ ❡ ❢✉♥çõ❡s x, y ∈ K(E) t❛✐s q✉❡ ♦ ♠❛♣❛

  

2

  ϕ : E → P

  P 7→ [x(P ), y(P ), 1]

  é ✉♠ ✐s♦♠♦r✜s♠♦ ❞❡✜♥✐❞♦ s♦❜r❡ K ❞❡ E s♦❜r❡ C✱ s❛t✐s❢❛③❡♥❞♦ ϕ(O) = [0, 1, 0]✳ ❘❡❝✐♣r♦❝❛♠❡♥t❡✱ q✉❛❧q✉❡r ❝✉r✈❛ s✉❛✈❡ C ❞❡✜♥✐❞❛ ♣♦r ✉♠❛ ❡q✉❛çã♦ ❝♦♠♦ ❛ ❛♥t❡✲ r✐♦r✱ é ✉♠❛ ❝✉r✈❛ ❡❧í♣t✐❝❛ ❞❡✜♥✐❞❛ s♦❜r❡ K✱ ❝♦♠ O = [0, 1, 0] ∈ C(K) s❡♥❞♦ s❡✉ ♣♦♥t♦

  ❜❛s❡✳ ❉❡♠♦♥str❛çã♦✳ ❈♦♠❡ç❛♠♦s ❝♦♥s✐❞❡r❛♥❞♦ ♦s ❡s♣❛ç♦s ✈❡t♦r✐❛✐s L(n(O))✱ ♣❛r❛ n = 1, 2, . . .✳

  ❱✐st♦ q✉❡ deg(n(O)) = n > 0 = 2g − 2✱ ♣❡❧♦ ❚❡♦r❡♠❛ ❞❡ ❘✐❡♠❛♥♥✲❘♦❝❤✱ t❡♠♦s q✉❡ ℓ(n( O)) = dim L(n(O)) = deg(n(O)) − g + 1 = n.

  K

  ❈♦♠♦ ♦s ❞✐✈✐❞♦r❡s n(O) sã♦ ❞❡✜♥✐❞♦s s♦❜r❡ K✱ ♣♦❞❡♠♦s ❡s❝♦❧❤❡r ❢✉♥çõ❡s x, y ∈

K(E)

  ♣❛r❛ ❛s q✉❛✐s {1, x} é ✉♠❛ ❜❛s❡ ♣❛r❛ L(2(O)) ❡ {1, x, y} ❢♦r♠❛♠ ✉♠❛ ❜❛s❡ ♣❛r❛ L(3(O))✳ ❚❡♠♦s q✉❡ x t❡♠ ✉♠ ♣ó❧♦ ❞❡ ♦r❞❡♠ ✷ ❡♠ O ❡ y ✉♠ ♣ó❧♦ ❞❡ ♦r❞❡♠ ✸✱ ✐st♦ é✱ ord (x) = (y) =

  (x) >

  O O O

  −2 ❡ ord −3✳ ❈♦♠ ❡❢❡✐t♦✱ ✈✐st♦ q✉❡ x ∈ L(2(O))✱ ❡♥tã♦ ord −2✳ (x) > (x) >

  O O

  ❆❣♦r❛ s❡ ord −2✱ ❡♥tã♦ ord −1✱ ♦ q✉❡ ✐♠♣❧✐❝❛r✐❛ q✉❡ x ∈ L((O)) = K ❡ 1 (y) =

  O

  ❡ x s❡r✐❛♠ ❧✐♥❡❛r♠❡♥t❡ ❞❡♣❡♥❞❡♥t❡s✳ ❉❡ ❢♦r♠❛ ❛♥á❧♦❣❛✱ ♠♦str❛♠♦s q✉❡ ord −3✳

  2

  3

  2

  (x ) = (x ) = (xy) = (y ) =

  O O O O

  ❚❡♠♦s t❛♠❜é♠ q✉❡ ord −4✱ ord −6✱ ord −5 ❡ ord

  2

  3

  2

  , x , xy, y −6✳ ❆ss✐♠✱ 1, x, y, x ∈ L(6(O))✳

  2

  3

  2

  , x , xy, y ❈♦♠♦ ℓ(n(O)) = 6✱ ♦ ❝♦♥❥✉♥t♦ {1, x, y, x } é ❧✐♥❡❛r♠❡♥t❡ ❞❡♣❡♥❞❡♥t❡✱

  , . . . ,

  1

  ❡①✐st✐♥❞♦ ♣♦rt❛♥t♦ ✉♠❛ K✲❞❡♣❡♥❞ê♥❝✐❛ ❧✐♥❡❛r ♥ã♦ tr✐✈✐❛❧ ❡♥tr❡ ❡❧❛s✱ ✐st♦ é✱ ❡①✐st❡♠ A A

  7

  ∈ K ♥ã♦ t♦❞♦s ♥✉❧♦s s❛t✐s❢❛③❡♥❞♦

  2

  2

  3 A 1 + A 2 x + A 3 y + A 4 x + A 5 xy + A 6 y + A 7 x = 0.

  ✸✷

  2

  2

  = 0 = 0 +A x+A y+A x +A xy+A y = 0

  6

  7

  1

  2

  3

  4

  5

  6

  ❙❡ t✐✈éss❡♠♦s A ♦✉ A ✱ ♦❝♦rr❡r✐❛ A

  2

  3

  • A x + A y + A x + A xy + A x = 0

  1

  2

  3

  4

  5

  7

  ♦✉ A ✳ ▼❛s ❝♦♠♦ ❡♠ ❛♠❜♦s ♦s ❝❛s♦s ❛s ❢✉♥çõ❡s

  ❡♥✈♦❧✈✐❞❛s t❡♠ ♦r❞❡♠ ❞✐❢❡r❡♥t❡s ❡♠ O✱ ❡❧❛s sã♦ ❧✐♥❡❛r♠❡♥t❡ ✐♥❞❡♣❡♥❞❡♥t❡s ❡♠ L(6(O))✳

  3

  6

  7

  ❡ y ❞❡✈❡♠ ❛♣❛r❡❝❡r✳ 6= 0 ❡ A 6= 0✱ ♠♦str❛♥❞♦ q✉❡ ♥❛ r❡❧❛çã♦ ❛❝✐♠❛✱ ♦ x

  2 P♦rt❛♥t♦ A

  2 A x A y

  ❋❛③❡♥❞♦ ❛s ♠✉❞❛♥ç❛s x 7→ −A

  6 7 ❡ y 7→ A 6 ♦❜t❡♠♦s

  7

  2

  2

  2

  3

  2

  3

  2

  2

  3

  3

  4

  2 A x + A A A x A A xy + A A y = 0.

  A

  1

  2 A

  6 A 7 x + A

  4 A A x

  3

  6

  4

  − A

  6 7 − A

  6

  7 7 − A

  6

  7

  6

  7

  3

4 A

  ❉✐✈✐❞✐♥❞♦ ♣♦r A

  6 7 ✱ ♦❜t❡♠♦s ✉♠❛ ❡q✉❛çã♦ ❝♦♠♦ ❞❡s❡❥❛❞♦

  2

  3

  2

  y + a xy + a y = x + a x + a x + a , a , a , a , a , a

  1

  3

  2

  4

  6

  1

  2

  3

  4

  6 ∈ K.

  ❊ss❛ ❡q✉❛çã♦ é ❝♦♥❤❡❝✐❞❛ ❝♦♠♦ ❡q✉❛çã♦ ❞❡ ❲❡✐❡rstr❛ss✳ ❆❣♦r❛ ❝♦♥s✐❞❡r❡♠♦s ♦ ♠❛♣❛

  2

  ϕ : E → P

  P 7→ [x(P ), y(P ), 1],

  ❝✉❥❛ ✐♠❛❣❡♠ ❡stá ❝♦♥t✐❞❛ ♥❛ ❝✉r✈❛ C ❞❛❞❛ ♣❡❧❛ ❡q✉❛çã♦ ❛❝✐♠❛✳ ◆♦t❡ q✉❡ ϕ : E → C é ✉♠ ♠❛♣❛ r❛❝✐♦♥❛❧ ♥ã♦ ❝♦♥st❛♥t❡ ❡♥tr❡ ✉♠❛ ❝✉r✈❛ s✉❛✈❡ ❡ ✉♠❛ ✈❛r✐❡❞❛❞❡✱ s❡♥❞♦ ♣♦rt❛♥t♦ ✉♠ ♠♦r✜s♠♦ s♦❜r❡❥❡t✐✈♦✳

  ❆❣♦r❛ ❝♦♥s✐❞❡r❡♠♦s t ✉♠ ✉♥✐❢♦r♠✐③❛♥t❡ ♣❛r❛ E ❡♠ O✱ ❛ss✐♠ ❡①✐st❡♠ ❢✉♥çõ❡s

  −2 −3

  u , u ( ( t t

  x y x y x ❡ y = u y ✳

  ∈ K(E) t❛✐s q✉❡ u O) 6= 0 ❡ u O) 6= 0 s❛t✐s❢❛③❡♥❞♦✱ x = u

  

−2 −3

  3

  t , u t , 1] = [u t, u , t ] ❚❡♠♦s q✉❡ ϕ = [x, y, 1] = [u x y x y ✳ P♦rt❛♥t♦✱ ϕ(O) =

  3

  [u ( ( ] = [0, 1, 0] (

  x O)t(O), u y O), t(O) ✱ ✈✐st♦ q✉❡ t(O) = 0 ❡ u y O) 6= 0✳

  ❆❣♦r❛ ♠♦str❛r❡♠♦s q✉❡ ϕ t❡♠ ❣r❛✉ ✶ ❡ q✉❡ C é s✉❛✈❡✱ ♣♦✐s ❛ss✐♠ t❡r❡♠♦s q✉❡ ϕ é ✉♠ ✐s♦♠♦r✜s♠♦ ✭❈♦r♦❧ár✐♦ ✶✳✸✸✮✳

  P❛r❛ ♠♦str❛r q✉❡ ϕ t❡♠ ❣r❛✉ ✶✱ t❡♠♦s q✉❡ ♣r♦✈❛r q✉❡ [K(E) : K(x, y)] = 1✱ ❥á q✉❡ K(C) = K(x, y)✳ ❚❡♠♦s q✉❡ x ♥ã♦ t❡♠ ♠❛✐s ♣ó❧♦s ❛❧é♠ ❞❡ O✱ ✈✐st♦ q✉❡ ❝♦♠♦ x ∈ L(2(O)) ❡♥tã♦ dif (x) > 2( (x) > 0

  P

  O) ❛ss✐♠✱ ord ♣❛r❛ t♦❞♦ P 6= O✳ ❉❛ ♠❡s♠❛ ❢♦r♠❛ ♦ ú♥✐❝♦ ♣ó❧♦ ❞❡ y é ❡♠ O✳ ❈♦♥s✐❞❡r❡♠♦s ❡♥tã♦ ♦s ♠❛♣❛s

  1

  ϕ : E P

  1

  → [1, 0]

  O 7→ P

  7→ [x(P ), 1], P 6= O

  ✸✸ ❡

  1

  ϕ : E P

  2

  → [1, 0]

  O 7→ P 7→ [y(P ), 1], P 6= O.

  −1

  ([1, 0]) = ❈♦♠♦ ϕ

  1 {O}✱ ❡♥tã♦

  X deg(ϕ ) = e (P ) = e (

  1 ϕ ϕ O) = 2.

1 P ∈ϕ ([1,0])

  1

  2 P♦rt❛♥t♦ [K(E) : K(x)] = 2✳ ❉❡♠♦❞♦ ❛♥á❧♦❣♦✱ t❡r❡♠♦s q✉❡ ♦ ♠❛♣❛ ϕ t❡♠ ❣r❛✉ ✸✱ ❞❡

  ❢♦r♠❛ q✉❡ [K(E) : K(x, y)]

  · [K(x, y) : K(x)] = [K(x, y) : K(x)] = 2 ❡

  [K(E) : K(x, y)] · [K(x, y) : K(y)] = [K(x, y) : K(y)] = 3. ❉❡ ♦♥❞❡ s❡❣✉❡ q✉❡ [K(E) : K(x, y)] = 1✱ ❡ ♣♦rt❛♥t♦ ϕ t❡♠ ❣r❛✉ ✶✳

  ❆❣♦r❛ s✉♣♦♥❤❛ q✉❡ C ♥ã♦ s❡❥❛ s✉❛✈❡✱ ♣❡❧♦ ❧❡♠❛ ❛♥t❡r✐♦r ❡①✐st✐r✐❛ ✉♠ ♠❛♣❛ r❛✲

  1

  1

  ❝✐♦♥❛❧ φ : C → P ❞❡ ❣r❛✉ ✶✳ ❆ss✐♠✱ φ ◦ ϕ : E → P é ✉♠ ♠❛♣❛ ❞❡ ❣r❛✉ ✶ ❡♥tr❡ ❝✉r✈❛s

  1

  s✉❛✈❡s✱ s❡♥❞♦ ♣♦rt❛♥t♦ ✉♠ ✐s♦♠♦r✜s♠♦✱ ♠❛s ❝♦♠♦ E t❡♠ ❣ê♥❡r♦ ✶ ❡ P t❡♠ ❣ê♥❡r♦ ✵✱ t❡rí❛♠♦s ✉♠❛ ❝♦♥tr❛❞✐çã♦✳ P♦rt❛♥t♦ C é s✉❛✈❡ ❡ ϕ ✉♠ ✐s♦♠♦r✜s♠♦✳ ❘❡❝✐♣r♦❝❛♠❡♥t❡✱ s❡❥❛ C ✉♠❛ ❝✉r✈❛ ❞❛❞❛ ♣❡❧❛ ❡q✉❛çã♦ ❡♠ q✉❡stã♦✳ ❈♦♠♦ ♦

  ♣♦❧✐♥ô♠✐♦ q✉❡ ❞❡✜♥❡ C é ❞❡ ❣r❛✉ ✸✱ ❡♥tã♦ ♣❡❧❛ ❢ór♠✉❧❛ ❞❡ ❣r❛✉✲❣ê♥❡r♦ t❡♠♦s (3

  − 1)(3 − 2) g(C) = = 1.

  2 ❆ss✐♠✱ C é ✉♠❛ ❝✉r✈❛ ❞❡ ❣ê♥❡r♦ ✶✳ ▼❛✐s ❞♦ q✉❡ ✐ss♦✱ ❝♦♠♦ [0, 1, 0] ∈ C(K)✱ ❡♥tã♦ C é ✉♠❛ ❝✉r✈❛ ❡❧í♣t✐❝❛✳

  ❉❡ ❛❣♦r❛ ❡♠ ❞✐❛♥t❡ ✐r❡♠♦s ❝♦♥s✐❞❡r❛r ❝✉r✈❛s ❡❧í♣t✐❝❛s E/K ❞❛❞❛s ♣♦r ❡q✉❛çõ❡s

  2

  3

  2 E : y + a xy + a y = x + a x + a x + a ,

  1

  3

  2

  4

  6 i

  ❝♦♠ a ∈ K ❡ [0, 1, 0] ❝♦♠♦ ♣♦♥t♦ ❜❛s❡✳ ❊ss❛s ❡q✉❛çõ❡s sã♦ ❝❤❛♠❛❞❛s ❡q✉❛çõ❡s ❞❡ ❲❡✐❡rstr❛ss✳

  ❆❣♦r❛ s✉♣♦♥❞♦ char(K) 6= 2✱ ♣♦❞❡♠♦s ❢❛③❡r ❛ s❡❣✉✐♥t❡ s✉❜st✐t✉✐çã♦ y x

  1

  3

  − a − a y , 7→

  ✸✹ ♦❜t❡♥❞♦

  2

  x x x

  1

  3

  1

  3

  1

  3

  y − a − a y − a − a y − a − a

  3

  2 + a x + a = x + a x + a x + a .

  1

  3

  2

  4

  6

  2

  2

  2 ❋❛③❡♥❞♦ ♦ ❞❡s❡♥✈♦❧✈✐♠❡♥t♦ ❞❡ss❛ ❡①♣r❡ssã♦ ♦❜t❡♠♦s

  2

  3

  

2

  y = 4x + b x + 2b x + b ,

  2

  4

  6

  2

  2

  = a + 4a , b = 2a + a a , b = a + 4a

  2

  2

  4

  4

  1

  3

  6

  6

  ♦♥❞❡ b

  1 3 ✳ ❆❣♦r❛ s❡ char(K) 6= 2, 3✱ ❢❛③❡♥❞♦ ♠❛✐s

  x y

  2

  − 3b ❡ss❛s s✉❜st✐t✉✐çõ❡s x 7→ ❡ y 7→ ♥❡ss❛ ú❧t✐♠❛ ❡q✉❛çã♦ ♦❜t❡♠♦s ✉♠❛ ❡q✉❛çã♦ 36 216 ❞❛ ❢♦r♠❛

  2

  3

  y = x x ,

  4

  6

  − 27c − 54c

  2

  3

  = b = b + 36b b

  4

  4

  6

  2

  4

  

6

  ♦♥❞❡ c

  2 − 24b ❡ c 2 − 216b ✳

  ◆❡ss❡s ❝❛s♦s✱ ❞❡✜♥✐♠♦s ♦ ♥ú♠❡r♦ ∆✱ ❝❤❛♠❛❞♦ ♦ ❞✐s❝r✐♠✐♥❛♥t❡ ❞❛ ❝✉r✈❛✱ ❝♦♠♦

  2

  

3

  2

  ∆ = b + 9b b b ,

  8

  2

  4

  6

  −b

  2 − 8b 4 − 27b

  6

  2

  2

  

2

= a a + 4a a a a + a a .

  ♦♥❞❡ b

  8

  6

  2

  6

  1

  3

  4

  2 1 − a 3 − a

  

4

  2

  3

  = x + ax + b ◗✉❛♥❞♦ ✉♠❛ ❡q✉❛çã♦ ❞❡ ❲❡✐❡rstr❛ss s❡ ❛♣r❡s❡♥t❛ ♥❛ ❢♦r♠❛ E : y ✱

  3

  • 2

  ❞✐r❡♠♦s q✉❡ ❡st❛ é ✉♠❛ ❡q✉❛çã♦ ❞❡ ❲❡✐❡rstr❛ss r❡❞✉③✐❞❛✳ ❊ t❡r❡♠♦s ∆ = −16(4a

  27b ) ✳ P♦rt❛♥t♦✱ q✉❛♥❞♦ char(K) 6= 2 ❡ 3 t♦❞❛ ❝✉r✈❛ ❡❧í♣t✐❝❛ s❡ ❡s❝r❡✈❡ ♥❛ ❢♦r♠❛

  2

  3 y = x + ax + b.

  ❊①❡♠♣❧♦ ✷✳✹✳ ◆❛ ✜❣✉r❛ ❛❝✐♠❛✱ t❡♠♦s ❞♦✐s ❡①❡♠♣❧♦s ❞❡ ❝✉r✈❛s ❡❧í♣t✐❝❛s✱ ♦♥❞❡ sã♦ ♠♦str❛❞♦s s❡✉s

  ♣♦♥t♦s ❝♦♠ ❝♦♦r❞❡♥❛❞❛s r❡❛✐s✳ ◆♦ ❡①❡♠♣❧♦ a) t❡♠♦s ✉♠ ❡①❡♠♣❧♦ ❞❡ ❝✉r✈❛ ♥ã♦ s✐♥❣✉❧❛r✱

  ✸✺ ❖ s❡❣✉✐♥t❡ r❡s✉❧t❛❞♦ ❝❛r❛❝t❡r✐③❛✱ ❡♠ t❡r♠♦s ❞♦ ❞✐s❝r✐♠✐♥❛♥t❡✱ q✉❛♥❞♦ ✉♠❛ ❝✉r✈❛

  ❡❧í♣t✐❝❛ E é ♥ã♦ s✐♥❣✉❧❛r✳ Pr♦♣♦s✐çã♦ ✷✳✺✳ ❆ ❝✉r✈❛ E é ♥ã♦ s✐♥❣✉❧❛r s❡✱ ❡ s♦♠❡♥t❡ s❡✱ ∆ 6= 0✳ ❉❡♠♦♥str❛çã♦✳ ❱❡r ✭❈❛♣ít✉❧♦ ✸✱ Pr♦♣♦s✐çã♦ ✶✳✹✱ ❬❙✐❧✈❡r♠❛♥❪✮✳

  ✷✳✶ ▲❡✐ ❞❡ ❣r✉♣♦ ♣❛r❛ ✉♠❛ ❝✉r✈❛ ❡❧í♣t✐❝❛

  ❈♦♥s✐❞❡r❡♠♦s E ✉♠❛ ❝✉r✈❛ ❡❧í♣t✐❝❛ ❡ L ✉♠❛ r❡t❛ ♣r♦❥❡t✐✈❛✳ ❉❛❞♦ q✉❡ E é ❞❡✜♥✐❞❛ ♣♦r ✉♠ ♣♦❧✐♥ô♠✐♦ ❞❡ ❣r❛✉ ✸✱ ♣❡❧♦ ❚❡♦r❡♠❛ ❞❡ ❇é③♦✉t✱ ❛ ✐♥t❡rs❡çã♦ ❡♥tr❡ E ❡ L ❞á ❡♠ ✸ ♣♦♥t♦s ❝♦♥t❛❞❛s ❛s ♠✉❧t✐♣❧✐❝✐❞❛❞❡s✱ ♦✉ s❡❥❛✱ ♦s ✸ ♣♦♥t♦s ♥ã♦ ♣r❡❝✐s❛♠ s❡r ♥❡❝❡ss❛r✐❛♠❡♥t❡ ❞✐st✐♥t♦s✳ ❆ ♣❛rt✐r ❞❡ss❡ ❢❛t♦✱ ♣♦❞❡♠♦s ❞❡✜♥✐r ❣❡♦♠❡tr✐❝❛♠❡♥t❡ ✉♠❛ ♦♣❡r❛çã♦ ❡♠ E q✉❡ ❢❛rá ❞❡ E ✉♠ ❣r✉♣♦ ❛❜❡❧✐❛♥♦✳ ❉❡✜♥✐çã♦ ✷✳✻ ✭▲❡✐ ❞❡ ●r✉♣♦ ♣❛r❛ E✮✳ ❙❡❥❛♠ P, Q ∈ E ❡ L ✉♠❛ r❡t❛ q✉❡ ♣❛ss❛ ♣♦r P ❡ Q ✭❝❛s♦ P s❡❥❛ ✐❣✉❛❧ à Q✱ L s❡rá ❛ r❡t❛ t❛♥❣❡♥t❡ à E ♣❛ss❛♥❞♦ ♣♦r P = Q✮✳ ❙❡❥❛ R

  ′

  ♦ t❡r❝❡✐r♦ ♣♦♥t♦ ❞❡ ✐♥t❡rs❡çã♦ ❡♥tr❡ L ❡ E✱ ❞❡♥♦t❛r❡♠♦s ♣♦r P Q := R✳ ❆❣♦r❛ s❡❥❛ L ❛

  ′

  r❡t❛ q✉❡ ♣❛ss❛ ♣♦r P Q ❡ O✱ ❞❡♥♦t❛r❡♠♦s ♦ t❡r❝❡✐r♦ ♣♦♥t♦ ❞❡ ✐♥t❡rs❡çã♦ ❡♥tr❡ L ❡ E ♣♦r P

  ⊕ Q✱ ♦✉ s❡❥❛✱ P ⊕ Q = (P Q)O✳ ❊①❡♠♣❧♦ ✷✳✼✳

  ◆❡ss❡ ❡①❡♠♣❧♦ ♣♦❞❡♠♦s ✈❡r ❝♦♠♦ é ❞❡✜♥✐❞♦ ❛ s♦♠❛ ❡♥tr❡ ♦s ♣♦♥t♦s P ❡ Q ❡♠ E

  ✳ ❊♠ b) ♦❜s❡r✈❛♠♦s q✉❡ ❛ r❡t❛ L é ❛ r❡t❛ t❛♥❣❡♥t❡ à E ❡♠ P = Q✳ ▼♦str❛r❡♠♦s q✉❡ ❡ss❛ ♦♣❡r❛çã♦ ❞❡✜♥❡ ❡♠ E ✉♠❛ ❡str✉t✉r❛ ❞❡ ❣r✉♣♦ ❛❜❡❧✐❛♥♦✱

  ❝♦♠ E(K) s❡♥❞♦ ✉♠ s✉❜❣r✉♣♦ ❞❡ E✳ ❆♥t❡s ❞✐ss♦✱ t❡♠♦s ♦ s❡❣✉✐♥t❡ r❡s✉❧t❛❞♦✳ ▲❡♠❛ ✷✳✽✳ ❙❡ C é ✉♠❛ ❝✉r✈❛ ❞❡ ❣ê♥❡r♦ ✶ ❡ P, Q ∈ C✱ ❡♥tã♦ (P ) ❡ (Q) sã♦ ❧✐♥❡❛r♠❡♥t❡ ❡q✉✐✈❛❧❡♥t❡s s❡✱ ❡ s♦♠❡♥t❡ s❡✱ P = Q✳

  ✸✻ ❉❡♠♦♥str❛çã♦✳ ❙❡ (P ) ❡ (Q) sã♦ ❧✐♥❡❛r♠❡♥t❡ ❡q✉✐✈❛❧❡♥t❡s✱ s❡❥❛ f ∈ K(C) t❛❧ q✉❡ (P ) − (Q) = div(f )

  ✳ ❆ss✐♠✱ div(f) + (Q) = (P ) > 0✱ ❞❡ ♦♥❞❡ f ∈ L((Q))✳ ❈♦♠♦ deg((Q)) = 1 > 0 = 2g − 2✱ ♦♥❞❡ g = 1 é ♦ ❣ê♥❡r♦ ❞❡ C✱ ❡♥tã♦ ♣❡❧❛s ❝♦♥s❡q✉ê♥❝✐❛s ❞♦ ❚❡♦r❡♠❛ ❞❡

  ❘✐❡♠❛♥♥✲❘♦❝❤✱ ℓ((Q)) = 1✱ ❧♦❣♦ f ∈ L((Q)) = K ❡ ❛ss✐♠✱ (P ) − (Q) = div(f) = 0✱ ♦✉ s❡❥❛✱ P = Q✳ ▼❛✐s ✉♠ r❡s✉❧t❛❞♦ ❛✉①✐❧✐❛r✳

  1

  2

  ▲❡♠❛ ✷✳✾✳ ❙❡ C, C ❡ C sã♦ ❝✉r✈❛s ❝ú❜✐❝❛s s✉❛✈❡s ✭❞❡✜♥✐❞❛s ♣♦r ♣♦❧✐♥ô♠✐♦s ❞❡ ❣r❛✉ ✸✮ ❡ ♦✐t♦ ❞♦s ♣♦♥t♦s ❞❡ ✐♥t❡rs❡çã♦ ❡♥tr❡ C ❡ C

  1 ❡stã♦ ♥❛ ✐♥t❡rs❡çã♦ ❡♥tr❡ C ❡ C 2 ❡♥tã♦ ♦

  ♥♦♥♦ ♣♦♥t♦ ❞❡ss❛s ❞✉❛s ✐♥t❡rs❡çõ❡s ❝♦✐♥❝✐❞❡♠✳

  1

  2

  1

  2 1 , . . . , P

  8

  ❉❡♠♦♥str❛çã♦✳ ❙❡❥❛♠ f ❡ f ♣♦♥t♦s q✉❡ ❞❡✜♥❡♠ C ❡ C ✱ r❡s♣❡❝t✐✈❛♠❡♥t❡✱ ❡ P ♦s ♦✐t♦ ♣r✐♠❡✐r♦s ♣♦♥t♦s ❞❛ ✐♥t❡rs❡çã♦ ❡ P ❡ Q ♦ ♥♦♥♦ ♣♦♥t♦ ❞❡ ❝❛❞❛ ✉♠❛ ❞❛s ✐♥t❡rs❡çõ❡s f

  1

  ✭s❡❣✉✐♥❞♦ ❛ ♦r❞❡♠ ❡♥✉♥❝✐❛❞❛ ♥♦ ❧❡♠❛✮✳ ❚❡♠♦s q✉❡ ∈ K(C) ❡ f

  2

  !

  8

  8 X

  1

  X f

  div( ) = (P ) + (P ) (P ) + (Q) = (P )

  i i − − (Q).

  f

  2 i=1 i=1

  ▼❛s ♣❡❧♦ ❧❡♠❛ ❛♥t❡r✐♦r✱ t❡♠♦s q✉❡ P = Q✱ ♣♦✐s ❛s ❝✉r✈❛s sã♦ ❝ú❜✐❝❛s t❡♥❞♦ ♣♦rt❛♥t♦ ❣ê♥❡r♦ ✶✳ Pr♦♣♦s✐çã♦ ✷✳✶✵✳ ❆ ♦♣❡r❛çã♦ q✉❡ ❞❡✜♥✐♠♦s s♦❜r❡ E é ✉♠❛ ♦♣❡r❛çã♦ ❞❡ ❣r✉♣♦✱ q✉❡ t❡♠ O ❝♦♠♦ ❡❧❡♠❡♥t♦ ♥❡✉tr♦✳ ▼❛✐s ❞♦ q✉❡ ✐ss♦✱ ❡ss❛ ♦♣❡r❛çã♦ é ❝♦♠✉t❛t✐✈❛ ❡ E(K) é ✉♠ s✉❜❣r✉♣♦ ❞❡ E✳ ❉❡♠♦♥str❛çã♦✳ ❆ ❝♦♠✉t❛t✐✈✐❞❛❞❡ ❞❡st❛ ♦♣❡r❛çã♦ é ❞❡❝♦rr❡♥t❡ ❞❛ ♣ró♣r✐❛ ❝♦♥str✉çã♦✱ ♣♦✐s

  ′

  ❛♦ t♦♠❛r ❛s r❡t❛s L ❡ L ❛ ♦r❞❡♠ ❞❛ ❡s❝♦❧❤❛ ❞❡ P ❡ Q ♥ã♦ ✐♠♣♦rt❛✱ ♦✉ s❡❥❛✱ P Q = QP ❡ ♣♦rt❛♥t♦ P ⊕ Q = (P Q)O = (QP )O = Q ⊕ P ✳

  ❆❣♦r❛ ❝♦♠♦ O = [0, 1, 0]✱ t♦♠❛♥❞♦ ❛ r❡t❛ L q✉❡ ✉♥❡ P à O✱ ❞❡♣♦✐s t♦♠❛♥❞♦ ❛

  ′ ′

  r❡t❛ L q✉❡ ✉♥❡ P O à O✱ ✈❡♠♦s q✉❡ L = L ✱ ❛ss✐♠ P ⊕ O = P ✱ ♦✉ s❡❥❛✱ O é ♦ ❡❧❡♠❡♥t♦ ♥❡✉tr♦ ❞❡ss❛ ♦♣❡r❛çã♦✳

  ❉❛❞♦ P ∈ E✱ s❡✉ ❡❧❡♠❡♥t♦ ✐♥✈❡rs♦ é P O✳ ❈♦♠ ❡❢❡✐t♦✱ s❡♥❞♦ L ❛ r❡t❛ q✉❡ ✉♥❡ P

  ′ ′

  ❡ O ❡♥tã♦ P O ∈ L ❡ ❛❣♦r❛ s❡ t♦♠❛r♠♦s L ❛ r❡t❛ q✉❡ ✉♥❡ P O ❡ O t❡♠♦s q✉❡ L = L ❡

  

  ♣♦rt❛♥t♦ ♦ t❡r❝❡✐r♦ ♣♦♥t♦ ❞❡ ✐♥t❡rs❡çã♦ ❞❡ L ❝♦♠ E é P ✳ ❆ss✐♠✱ ❛ r❡t❛ q✉❡ ✉♥❡ P ❡ P O é L✱ ♣♦rt❛♥t♦ P (P O) = O ❡ ❛ss✐♠✱ P ⊕ P O = (P (P O))O = OO = O.

  P❛r❛ ♠♦str❛r♠♦s q✉❡ ❛ ♦♣❡r❛çã♦ é ❛ss♦❝✐❛t✐✈❛✱ ❜❛st❛ ♠♦str❛r♠♦s q✉❡ s❡ P, Q, R ∈ E

  ✸✼

  ′

  ❙❡❥❛ L ❛ r❡t❛ q✉❡ ♣❛ss❛ ♣♦r P ✱ Q ❡ P Q✱ L ❛ r❡t❛ q✉❡ ♣❛ss❛ ♣♦r P ⊕ Q✱ R ❡

  ′′

  (P

  1

  ⊕ Q)R✱ ❡ L ❛ r❡t❛ q✉❡ ♣❛ss❛ ♣♦r O✱ QR ❡ Q ⊕ R✳ ❊♥tã♦ ❛ ❝✉r✈❛ ❝ú❜✐❝❛ C ❞❛❞❛ ♣♦r

  ′ ′′

  L = 0 ✐♥t❡r❝❡❝t❛ E ❡♠ t♦❞♦s ❡ss❡s ♣♦♥t♦s O, P, Q, R, P Q, QR, P ⊕ Q, Q ⊕ R ❡

  · L · L (P

  ⊕ Q)R✳ ❋❛③❡♥❞♦ ♦ ❛♥á❧♦❣♦ ♣❛r❛ ♦ ❝♦♥❥✉♥t♦ ❞❡ ♣♦♥t♦s P, Q ⊕ R, P (Q ⊕ R); Q, R, QR;

  2

  ❡ O, P Q, P ⊕ Q✱ ❝♦♥s❡❣✉✐r❡♠♦s ✉♠❛ ❝ú❜✐❝❛ C q✉❡ ✐♥t❡rs❡❝t❛ E ♥❡ss❡s ♠❡s♠♦s ♣♦♥t♦s✳

  2

  1

  ❱❡♠♦s q✉❡ ♦✐t♦ ❞♦s ♣♦♥t♦s ❞❛ ✐♥t❡rs❡çã♦ ❡♥tr❡ C ❡ E ❛♣❛r❡❝❡♠ ♥❛ ✐♥t❡rs❡çã♦ ❡♥tr❡ C ❡ E

  ✱ ❞❡ ♦♥❞❡ ❝♦♥❝❧✉í♠♦s q✉❡ ♦ ♥♦♥♦ ♣♦♥t♦ ❡♠ ❛♠❜❛s ❛s ✐♥t❡rs❡çõ❡s ❞❡✈❡♠ s❡r ✐❣✉❛✐s✱ s❡♥❞♦ ❛ss✐♠ P (Q ⊕ R) = (P ⊕ Q)R✳ ❊ ❛ ♦♣❡r❛çã♦ é ❛ss♦❝✐❛t✐✈❛✳

  ❆❣♦r❛ s❡ t♦♠❛r♠♦s ❞♦✐s ♣♦♥t♦s K✲r❛❝✐♦♥❛✐s t❡r❡♠♦s q✉❡ ❛ r❡t❛ L q✉❡ ♣❛ss❛ ♣♦r P

  ❡ Q ❡st❛rá ❞❡✜♥✐❞❛ s♦❜r❡ K✱ ❡ ❛ss✐♠ ♦ t❡r❝❡✐r♦ ♣♦♥t♦ ❞❡ ✐♥t❡rs❡çã♦ ❞❡ L ❝♦♠ E s❡rá t❛♠❜é♠ K✲r❛❝✐♦♥❛❧✱ ♣♦✐s ❛ ❡q✉❛çã♦ q✉❡ ♥♦s ♣❡r♠✐t❡ ❝❛❧❝✉❧❛r t❛❧ ✐♥t❡rs❡çã♦ é ✉♠❛ ❡q✉❛çã♦ ❝ú❜✐❝❛ q✉❡ t❡♠ ❞✉❛s r❛í③❡s ❞❡ ❝♦♦r❞❡♥❛❞❛s K✲r❛❝✐♦♥❛✐s ❝♦rr❡s♣♦♥❞❡♥t❡s ❛♦ ♣♦♥t♦ P ❡ Q✳ ❉❛í P Q s❡rá K−r❛❝✐♦♥❛❧✱ ❞❛ ♠❡s♠❛ ❢♦r♠❛ q✉❡ P ⊕ Q = (P Q)O✳

  ❆ ♣❛rt✐r ❞❡ ❛❣♦r❛ ✉s❛r❡♠♦s P + Q ♣❛r❛ ❞❡♥♦t❛r ❛ s♦♠❛ ❡♥tr❡ P ❡ Q✳ ❊ ❞❡✜♥✐♠♦s ♣❛r❛ ❝❛❞❛ m ∈ Z ♦ s❡❣✉✐♥t❡ ❤♦♠♦♠♦r✜s♠♦ ❞❡ ❣r✉♣♦s

  [m] : E → E

  P 7→ [m]P,

  ❞❡✜♥✐❞♦ ♣♦r✿ ❙❡ m = 0✱ ❡♥tã♦ [0]P = O ♣❛r❛ t♦❞♦ P ∈ E✱ ❙❡ m > 0 ❡♥tã♦ [m]P =

  m

  z }| { P +

  · · · + P ✱ ❡ s❡ m < 0 ❡♥tã♦ [m]P = [−m](−P )✱ ♣❛r❛ t♦❞♦ P ∈ E✳ ❊st❛s ❛♣❧✐❝❛çõ❡s q✉❡ ♣r❡s❡r✈❛♠ ❛ ❡str✉t✉r❛ ❞❡ ❣r✉♣♦ ❝♦♠♦ t❛♠❜é♠ ❛ ❡str✉t✉r❛ ❞❡ ✈❛r✐❡❞❛❞❡ é ❝❤❛♠❛❞❛ ❞❡ ✐s♦❣ê♥✐❛✳

  ❆❣♦r❛ ✐r❡♠♦s ❡♥❝♦♥tr❛r ❝♦♦r❞❡♥❛❞❛s ♣❛r❛ P +Q ❡ −P ❡♠ ❢✉♥çã♦ ❞❛s ❝♦♦r❞❡♥❛❞❛s ❞❡ P ❡ Q ♣❛r❛ ❛ ♦♣❡r❛çã♦ ❡♠ E✳ ❙❡❥❛♠

  2

  3

  2 E : y + a xy + a y = x + a x + a x + a ,

  1

  3

  2

  4

  6

  ✉♠❛ ❝✉r✈❛ ❞❡✜♥✐❞❛ ♣♦r ✉♠❛ ❡q✉❛çã♦ ❞❡ ❲❡✐❡rstr❛ss✱ ♦✉ ♠❡s♠♦ ❞❡✜♥✐❞❛ ♣♦r ✉♠❛ ❡q✉❛çã♦ r❡❞✉③✐❞❛

  2

  

3

E : y = x + ax + b.

  , y ) ■♥✐❝✐❛❧♠❡♥t❡ ❝❛❧❝✉❧❛r❡♠♦s ❛s ❝♦♦r❞❡♥❛❞❛s ❞♦ ✐♥✈❡rs♦ ❞❡ P ✳ ❙❡♥❞♦ P = (x ✱ t❡♠♦s q✉❡ −P é ♦ t❡r❝❡✐r♦ ♣♦♥t♦ ❞❡ ✐♥t❡rs❡çã♦ ❡♥tr❡ E ❡ ❛ r❡t❛ q✉❡ ✉♥❡ P ❡ O✱ q✉❡ é ❛

  , y

  1 )

  1

  r❡t❛ ❞❛❞❛ ♣❡❧❛ ❡q✉❛çã♦ x = x ✳ ❙❡♥❞♦ ❛ss✐♠✱ s❡ −P = (x t❡♠♦s q✉❡ y é ❛ ♦✉tr❛

  ✸✽ s♦❧✉çã♦ ✭❛❧é♠ ❞❡ y ✮ ❞❛ ❡q✉❛çã♦ q✉❛❞rát✐❝❛

  2

  3

  2 y + (a x + a )y x x = 0.

  1

  3

  2

  4

  6

  − x − a − a − a ❈♦♠♦

  2

  2

  3

  2

  y + y )y + y y = y + (a x + a )y x x ,

  1

  1

  1

  3

  2

  4

  6

  − (y − x − a − a − a = x

  1

  1

  3

  ❡♥tã♦ y −y − a − a ✳ P♦rt❛♥t♦✱ s❡ P = (x, y) ❡♥tã♦ x )

  1

  3

  −P = (x, −y − a − a ❢♦r♠❛ ❣❡r❛❧ −P = (x, −y) ❢♦r♠❛ r❡❞✉③✐❞❛.

  ❊st❛s sã♦ ❛s ❝❤❛♠❛❞❛s ❢ór♠✉❧❛s ❞❡ ✐♥✈❡rsã♦✳ = (x , y ) = (x , y )

  P❛r❛ ❡♥❝♦♥tr❛r ❛s ❝♦♦r❞❡♥❛❞❛s ❞❛ s♦♠❛ ❡♥tr❡ P

  1

  1 1 ❡ P

  2

  2 2 ✱ s❡

  1 2 ❡ y

  2

  1

  1

  1 3 ❡♥tã♦ t❡r❡♠♦s q✉❡ P

  1

  2

  −y − a − a O✳ ❈❛s♦ ✐ss♦ ♥ã♦ ♦❝♦rr❛✱ ♣♦❞❡♠♦s ❝♦♥s✐❞❡r❛r ❛ r❡t❛ L ❞❡ ❡q✉❛çã♦

  = x = x + P = t✐✈❡r♠♦s x

  L : y = λx + v,

  1

  2

  q✉❡ ♣❛ss❛ ♣♦r P ❡ P ✱ ♦♥❞❡ ❡ss❡ λ ❡ v sã♦ ❞❛❞♦s ♣♦r✿ = x

  1

  2

  • ❙❡ x ✱ ❡♥tã♦✿

  1

  ◆❡ss❡ ❝❛s♦ ❛ r❡t❛ L é t❛♥❣❡♥t❡ à E ❡♠ P ✱ ❡ ❛ss✐♠ s✉❛ ❡q✉❛çã♦ é ❞❛❞❛ ♣♦r dy y

  

1 = (P

1 )(x 1 ).

  − y − x dx ❆❣♦r❛ ❞❡r✐✈❛♥❞♦ ❛ ❡q✉❛çã♦ q✉❡ ❞❡✜♥❡ ❞❡ ❲❡✐❡rstr❛ss ❡♠ r❡❧❛çã♦ à x✱ ♦❜t❡♠♦s dy dy dy

  2 2y + a x + a y + a = 3x + 2a x + a .

  1

  1

  3

  2

  4

  dx dx dx

  2

  dy 3x + 2a x + a y

  2

  1

  4

  

1

  1 1 − a

  (P ) =

  1

  ❆ss✐♠✱ ❡ ♣♦rt❛♥t♦ dx 2y

  1 + a 1 x 1 + a

  3

  2

  dy 3x + 2a x + a y

  2

  1

  4

  1

  1 1 − a

  λ = (P ) =

  1

  dx 2y

  1 + a 1 x 1 + a

  3

  3

  dy + a x + 2a y

  4

  1

  6

  3

  1

  −x

  1 − a v = y (P ) = .

  1

  1

  1

  − x dx 2y + a x + a

  1

  1

  1

  3

  1

  2

  • ❙❡ x 6= x ✱ ❡♥tã♦✿
  • a

  • a
  • a
  • a
  • v
  • P
  • P
  • P

  • a
  • ax + b
  • ax + b)
  • a)
  • 4(x
  • ax + b)(7x
  • ax
  • ax + b)
  • P
  • x
  • P
  • y
  • x

  

4

  − 2ax

  2

  − 8bx + a

  2

  4(x

  3

  , y([2]P ) = (3x

  2

  3

  3

  3

  − 2b) 8(x

  3

  3 .

  ✭✐✐✐✮ ❙❡ P

  1 6= ±P

  2

  x(P

  ✭✐✮ −P = (x, −y) ✭✐✐✮ ❋ór♠✉❧❛ ❞❡ ❞✉♣❧✐❝❛çã♦ x([2]P ) = x

  2

  ) ✳ ❊♥tã♦

  3

  2 ,

  −(λ + a

  1 )x

  3

  − v − a 3 ). ❈♦♠♦ ❝♦♥s❡q✉ê♥❝✐❛ t❡♠♦s ✉♠ r❡s✉♠♦ ❞❡ss❛s ❢ór♠✉❧❛s ♣❛r❛ ♦ ❝❛s♦ ❞❛ ❡q✉❛çã♦ ❞❡

  ❲❡✐❡rstr❛ss r❡❞✉③✐❞❛ E : y

  2

  = x

  ✳ ❙❡ P = (x, y), P

  , y

  1

  = (x

  1

  , y

  1

  ) ❡ P

  2

  = (x

  2

  1

  ) = y

  2

  2

  − y

  1

  x

  2

  − x

  1

  · (x

  1

  ) − y

  3

  1 x

  2

  − y

  2 x

  1

  x

  2

  − x

  1 .

  2

  1

  2

  1

  − y

  1

  x

  2

  − x

  1

  2

  − (x

  2

  − x

  ) y(P

  1

  2

  ) = − y

  2

  − y

  1

  x

  2

  − x

  − x

  

1

  6

  3

  (λx + v) = x

  3

  2

  x

  2

  4

  x + a

  , q✉❡ s❡ ✐❣✉❛❧❛❞❛ à (x − x

  1

  1 )(x

  − x

  2 )(x

  − x

  3 )

  ♥♦s ❞❛rá x

  3

  = λ

  2

  x(λx + v) + a

  3

  λ − a

  1

  ✸✾ ◆❡ss❡ ❝❛s♦ ❛ ✐♥❝❧✐♥❛çã♦ ❞❛ r❡t❛ L é ❞❛❞❛ ♣♦r

  λ = y

  2

  − y

  1

  x

  2

  − x

  ❡ ♦ t❡r♠♦ ✐♥❞❡♣❡♥❞❡♥t❡ v é v = y

  ❆❣♦r❛ s✉❜st✐t✉✐♥❞♦ y ♣♦r λx + v ♥❛ ❡q✉❛çã♦ ❞❡ ❲❡✐❡rstr❛ss✱ ♦❜t❡♠♦s (λx + v)

  1 x

  2

  − y

  2 x

  1

  x

  2

  − x

  1 .

  1

  2

  2

  −P

  = (P

  1 P

  

2

  ) O ❡♥tã♦ P

  1

  2

  = P

  3 O✱ ❞❡ ♦♥❞❡ s❡❣✉❡ q✉❡

  3

  1

  = P

  3 O = P

  1

  2 ✳ P❡❧❛s ❢ór♠✉❧❛s ❞❡ ✐♥✈❡rsã♦✱ ♦❜t❡♠♦s

  P

  1 + P 2 = (λ

  2

  1 λ

  − a

  2

  ) ❡ P

  − x

  3

  1

  − x

  2

  , s❡♥❞♦ ❡st❛ ❛ ♣r✐♠❡✐r❛ ❝♦♦r❞❡♥❛❞❛ ❞♦ ♣♦♥t♦ (x

  3

  , y

  3

  ) = P

  q✉❡ é ♦ t❡r❝❡✐r♦ ♣♦♥t♦ ❞❛ ✐♥t❡rs❡✲ çã♦ ❞❡ L ❝♦♠ E✳ ❆❣♦r❛ ♣❡❧♦ ❢❛t♦ ❞❡ q✉❡ P

  2

  3

  t❛♠❜é♠ ♣❡rt❡♥❝❡ à L✱ ♦❜t❡♠♦s y

  3

  = λx

  3

  ✳ ❈♦♠♦ P

  3

  é (P

  1 P

  ❆❣♦r❛ ✐r❡♠♦s ❢❛③❡r ✉♠ ❡st✉❞♦ s♦❜r❡ ♦s s✉❜❣r✉♣♦s ❞❡ t♦rçã♦ ❞❡ E✱ q✉❡ sã♦ ♦s

  ✹✵ ❉❡✜♥✐çã♦ ✷✳✶✶✳ ❙❡❥❛ E ✉♠❛ ❝✉r✈❛ ❡❧í♣t✐❝❛ ❡ m ✐♥t❡✐r♦ ♣♦s✐t✐✈♦✳ ❉❡✜♥✐♠♦s ❛ m✲t♦rçã♦ ❞❡ E ❝♦♠♦ s❡♥❞♦

  E[m] = {P ∈ E : [m]P = O}. ❉❡ ❢♦r♠❛ ❛♥á❧♦❣❛ ❞❡✜♥✐♠♦s ❛ m✲t♦rçã♦ r❛❝✐♦♥❛❧ ❞❡ E

  E[m](K) = {E ∈ E(K) : [m]P = O}.

  tor

  ❖ s✉❜❣r✉♣♦ ❞❡ t♦rçã♦ ❞❡ E✱ ❞❡♥♦t❛❞♦ ♣♦r E ✱ é ♦ ❝♦♥❥✉♥t♦ ❞♦s ♣♦♥t♦s ❞❡ E ❞❡ ♦r❞❡♠ ✜♥✐t❛✱ ♥❡ss❡ ❝❛s♦ E = E[m].

  tor m>1

  ∪

  tor

  ❊ ❞❡ ✐❣✉❛❧ ❢♦r♠❛ ❞❡✜♥✐♠♦s E(K) ✱ ❝❛s♦ E s❡❥❛ ❞❡✜♥✐❞❛ s♦❜r❡ K✳

  2

  = (x )(x )(x )

  1

  2

  3

  ❊①❡♠♣❧♦ ✷✳✶✷✳ ❈♦♥s✐❞❡r❡ ❛ ❝✉r✈❛ ❡❧í♣t✐❝❛ E : y − e − e − e ✳ ❖s ♣♦♥t♦s ❞❡ 2✲t♦rçã♦ sã♦ ♦s ♣♦♥t♦s t❛✐s q✉❡ P + P = O✱ ♦✉ s❡❥❛✱ P = −P ✳ ❙❡♥❞♦ P = (x, y)✱ x ) = a = 0

  1

  3

  1

  3

  t❡♠♦s q✉❡ −P = (x, y − a − a ❡♠ q✉❡ ♥❡ss❡ ❝❛s♦ a ✳ P♦rt❛♥t♦✱ ♣❛r❛ q✉❡ P s❡❥❛ ✉♠ ♣♦♥t♦ ❞❡ 2✲t♦rçã♦ ❞❡✈❡♠♦s t❡r (x, y) = P =

  −P = (x, −y), , e

  1

  2

  3

  ❞❡ ♦♥❞❡ s❡❣✉❡ q✉❡ y = 0 ❡ x = e ♦✉ e ✳ ❆ss✐♠✱ E[2] = , 0), (e , 0), (e , 0),

  1

  2

  3 {(e O}.

  ❈❧❛r❛♠❡♥t❡ t❡♠♦s q✉❡ Ker([m]) = E[m]✱ ❡ s♦❜r❡ ❡ss❡s s✉❜❣r✉♣♦s t❡♠♦s ♦ s❡❣✉✐♥t❡ r❡s✉❧t❛❞♦✳ Pr♦♣♦s✐çã♦ ✷✳✶✸✳ ❙❡❥❛ E ✉♠❛ ❝✉r✈❛ ❡❧í♣t✐❝❛ ❡ m ✉♠ ✐♥t❡✐r♦ ♣♦s✐t✐✈♦✳ ❊♥tã♦

  2

  ✭❛✮ deg([m]) = m ✳ ✭❜✮ ❙❡ m 6= 0 ❡♠ K✱ ✐st♦ é✱ s❡ char(K) = 0 ♦✉ char(K) = P > 0 ❡ p ♥ã♦ ❞✐✈✐❞❡ m✱

  ❡♥tã♦ Z Z E[m] ∼ . =

  × mZ mZ

  c

  ✭❝✮ ❙❡ char(K) = p > 0 ❡ m = p ✱ ❡♥tã♦ ✈❛❧❡ ✉♠❛ ❞❛s s❡❣✉✐♥t❡s ❛✜r♠❛çõ❡s

  c

  (i) E[p ] = {O}, ∀c = 1, 2, 3, . . .

  Z

  c

  (ii) E[p ] ∼ = , ∀c = 1, 2, 3, . . . .

  

c

  p Z ❉❡♠♦♥str❛çã♦✳ ❱❡r ✭❈❛♣ít✉❧♦ ✸✱ ❈♦r♦❧ár✐♦ ✻✳✹✱ ❬❙✐❧✈❡r♠❛♥❪✮✳

  ✹✶ ❆ ♣❛rt✐r ❞❡ ❛❣♦r❛ ♥♦s ❝♦♥❝❡♥tr❛r❡♠♦s ❡♠ ❡st✉❞❛r ❝♦r♣♦s ❞❡ ♥ú♠❡r♦s✱ ♦✉ s❡❥❛✱ K s❡rá ✉♠❛ ❡①t❡♥sã♦ ✜♥✐t❛ ❞❡ Q✳

  ✷✳✷ ❘❡❞✉çã♦ ❞❡ ✉♠❛ ❝✉r✈❛ ❡❧í♣t✐❝❛

  ◆❡st❛ s❡çã♦ ✈❡r❡♠♦s ❛❧❣✉♠❛s ❞❡✜♥✐çõ❡s ❡ r❡s✉❧t❛❞♦s ❡♠ r❡❧❛çã♦ ❛♦s ♣♦♥t♦s r❛❝✐♦✲ ♥❛✐s ❞❡ ✉♠❛ ❝✉r✈❛ ✈✐st❛ s♦❜r❡ ♦ ❝♦♠♣❧❡t❛♠❡♥t♦ ❞♦ ❝♦r♣♦ K ♣♦r ✉♠❛ ✈❛❧♦r✐③❛çã♦ ❞✐s❝r❡t❛✳ ■ss♦ s❡rá ✐♠♣♦rt❛♥t❡ q✉❛♥❞♦ ♣r♦✈❛r♠♦s ❛ ✈❡rsã♦ ❢r❛❝❛ ❞♦ ❚❡♦r❡♠❛ ❞❡ ▼♦r❞❡❧❧✲❲❡✐❧ ♥♦ ❈❛♣ít✉❧♦ ✸✳ ❉❡✜♥✐çã♦ ✷✳✶✹✳ ❯♠ ✈❛❧♦r ❛❜s♦❧✉t♦ ❞❡ ✉♠ ❝♦r♣♦ K é ✉♠❛ ❛♣❧✐❝❛çã♦

  | · | : K → R q✉❡ s❛t✐s❢❛③ ✭✐✮ |x| > 0 ❡ |x| = 0 s❡✱ ❡ s♦♠❡♥t❡ s❡✱ x = 0✳

  ✭✐✐✮ |xy| = |x| · |y|, ∀x, y ∈ K✳ ✭✐✐✐✮ |x + y| 6 |x| + |y|, ∀x, y ∈ K ✭❞❡s✐❣✉❛❧❞❛❞❡ tr✐❛♥❣✉❧❛r✮✳

  ❉❛❞♦ ✉♠ ✈❛❧♦r ❛❜s♦❧✉t♦✱ ♣♦❞❡♠♦s ❞❡✜♥✐r ✉♠❛ ♠étr✐❝❛ ❡♠ K✱ ❞❛❞❛ ♣♦r d(x, y) = |x − y|✱ q✉❡ ❢❛③ ❝♦♠ q✉❡ K ✜q✉❡ ♠✉♥✐❞♦ ❞❡ ✉♠❛ t♦♣♦❧♦❣✐❛ ✈✐♥❞❛ ❞❡ss❛ ♠étr✐❝❛✳ ❆ss✐♠✱ ❞✐r❡♠♦s q✉❡ ❞♦✐s ✈❛❧♦r❡s ❛❜s♦❧✉t♦s sã♦ ❡q✉✐✈❛❧❡♥t❡s s❡ ❣❡r❛♠ ❛ ♠❡s♠❛ t♦♣♦❧♦❣✐❛✳

  ❆ ♣❛rt✐r ❞❡ ✉♠❛ ✈❛❧♦r✐③❛çã♦ ❞✐s❝r❡t❛ v ♣♦❞❡♠♦s ♠✉♥✐r ♦ ❝♦r♣♦ K ❝♦♠ ✉♠ ✈❛❧♦r ❛❜s♦❧✉t♦ ♣r♦✈❡♥✐❡♥t❡ ❞❡ss❛ ✈❛❧♦r✐③❛çã♦✱ ❞❛❞❛ ♣❡❧❛ ❡①♣r❡ssã♦

  −v(x) v = e .

  |x| ❖♥❞❡ e é ❛ ❜❛s❡ ❞♦ ❧♦❣❛r✐t♠♦ ♥❛t✉r❛❧✳

  n n∈N , a n

  ❉❡✜♥✐çã♦ ✷✳✶✺✳ ❯♠❛ s❡q✉ê♥❝✐❛ {a } ∈ K é ✉♠❛ s❡q✉ê♥❝✐❛ ❞❡ ❈❛✉❝❤② ❝♦♠ r❡❧❛çã♦ à | · | s❡ ♣❛r❛ t♦❞♦ ε > 0✱ ❡①✐st❡ N ∈ N t❛❧ q✉❡

  n m |a − a | 6 ε, ∀n, m > N.

  ❉✐r❡♠♦s q✉❡ ✉♠ ❝♦r♣♦ ♠✉♥✐❞♦ ❞❡ ✉♠ ✈❛❧♦r ❛❜s♦❧✉t♦ (K, | · |) é ❝♦♠♣❧❡t♦ s❡ t♦❞❛ s❡q✉ê♥❝✐❛ ❞❡ ❈❛✉❝❤② t❡♠ ✉♠ ❧✐♠✐t❡ ❡♠ K✳ ❉❛❞♦ ✉♠ ❝♦r♣♦ K ❡ ✉♠ ✈❛❧♦r ❛❜s♦❧✉t♦ | · | ❡♠ K✱ é ♣♦ssí✈❡❧ ♦❜t❡r ✉♠ ❝♦♠♣❧❡t❛✲

  ′

  ♠❡♥t♦ ❞❡ K ❝♦♠ r❡s♣❡✐t♦ à | · |✱ ✐st♦ é✱ ❡①✐st❡ ✉♠❛ ❡①t❡♥sã♦ K ❞❡ K✱ q✉❡ t❡♠ ✉♠ ✈❛❧♦r

  ′

  ✹✷ || · ||✳ ❆ss✐♠✱ ♣♦❞❡♠♦s ❢❛❧❛r ❞♦ ❝♦♠♣❧❡t❛♠❡♥t♦ ❞❡ K ❝♦♠ r❡s♣❡✐t♦ ❛ ✉♠❛ ✈❛❧♦r✐③❛çã♦ ❞✐s✲ ❝r❡t❛ v✱ ♦♥❞❡ q✉❡r❡♠♦s ❞✐③❡r q✉❡ é ♦ ❝♦♠♣❧❡t❛♠❡♥t♦ ❞❡ K ❝♦♠ r❡s♣❡✐t♦ ❛♦ ✈❛❧♦r ❛❜s♦❧✉t♦ ♣r♦✈❡♥✐❡♥t❡ ❞❡ v✳ ❉❡✜♥✐çã♦ ✷✳✶✻✳ ❯♠ ✈❛❧♦r ❛❜s♦❧✉t♦ | · | é ❞✐t♦ ❛rq✉✐♠❡❞✐❛♥♦ s❡ {|n · 1 K

  | : n ∈ N} é Pr♦♣♦s✐çã♦ ✷✳✶✼✳ ❯♠ ✈❛❧♦r ❛❜s♦❧✉t♦ é ♥ã♦ ❛rq✉✐♠❡❞✐❛♥♦ s❡✱ ❡ s♦♠❡♥t❡ s❡ s❛t✐s❢❛③ ❛ ❞❡s✐❣✉❛❧❞❛❞❡ tr✐❛♥❣✉❧❛r ❢♦rt❡ |x + y| 6 max{|x|, |y|}, ∀x, y ∈ K.

  ❉❡♠♦♥str❛çã♦✳ ❱❡r ✭❈❛♣ít✉❧♦ ✷✱ Pr♦♣♦s✐çã♦ ✸✳✻✱ ❬◆❡✉❦✐r❝❤❪✮✳ ❊①❡♠♣❧♦ ✷✳✶✽✳ ✭✐✮ ❙♦❜r❡ Q✱ t❡♠♦s ♦ ✈❛❧♦r ❛❜s♦❧✉t♦ ✉s✉❛❧

  = max

  ∞

  |q| {q, −q}, q✉❡ é ❛rq✉✐♠❡❞✐❛♥♦ ❡ ❝✉❥♦ ❝♦♠♣❧❡t❛♠❡♥t♦ ❞❡ Q ❝♦♠ r❡s♣❡✐t♦ à ❡ss❡ ✈❛❧♦r ❛❜s♦❧✉t♦ é R✳

  p

  ✭✐✐✮ ❈♦♥s✐❞❡r❛♥❞♦ p ♣r✐♠♦ ❡ v ❛ ✈❛❧♦r✐③❛çã♦ p✲á❞✐❝❛✱ ♦ ✈❛❧♦r ❛❜s♦❧✉t♦ p✲á❞✐❝♦ é ❞❛❞♦ ♣♦r

  −v p (q) = p .

p

  |q| ❊ss❡ ✈❛❧♦r ❛❜s♦❧✉t♦ é ♥ã♦ ❛rq✉✐♠❡❞✐❛♥♦✳ ❖ ❝♦♠♣❧❡t❛♠❡♥t♦ ❞❡ Q ❝♦♠ r❡s♣❡✐t♦ à ❡ss❡ ✈❛❧♦r ❛❜s♦❧✉t♦ é ❝♦♥❤❡❝✐❞♦ ❝♦♠♦ ♦ ❝♦r♣♦ ❞♦s ♥ú♠❡r♦s p✲á❞✐❝♦s✱ ❞❡♥♦t❛❞♦ ♣♦r Q p

  ✳ ➱ ❝♦♥❤❡❝✐❞♦ q✉❡ ❡ss❡s ❞♦✐s ✈❛❧♦r❡s ❛❜s♦❧✉t♦s sã♦ ♦s ú♥✐❝♦s ✈❛❧♦r❡s ❛❜s♦❧✉t♦s s♦❜r❡

  Q ✱ ❛ ♠❡♥♦s ❞❡ ❡q✉✐✈❛❧ê♥❝✐❛ ✭❈❛♣ít✉❧♦ ✷✱ Pr♦♣♦s✐çã♦ ✸✳✼✱ ❬◆❡✉❦✐r❝❤❪✮✳ ❉✐r❡♠♦s ❡♥tã♦ q✉❡

  ∞ p

  | · | ❡ | · | sã♦ ♦s ✈❛❧♦r❡s ❛❜s♦❧✉t♦s st❛♥❞❛r❞ ❞❡ Q✳

  v

  P❛r❛ ♦ ❝♦r♣♦ K ♠✉♥✐❞♦ ❞❡ ✉♠❛ ✈❛❧♦r✐③❛çã♦ ❞✐s❝r❡t❛✱ ❞❡♥♦t❛r❡♠♦s ♣♦r K ❛♦

  v

  ❝♦♠♣❧❡t❛♠❡♥t♦ ❞❡ K ❝♦♠ r❡s♣❡✐t♦ ❛ v✳ ❚❡♠♦s q✉❡ K t❡♠ ❝♦♠♦ ❛♥❡❧ ❞❡ ✐♥t❡✐r♦s R = : v(x) > 0

  v v

  {x ∈ K }, q✉❡ é ✉♠ ❛♥❡❧ ❧♦❝❛❧ ❝✉❥♦ ✐❞❡❛❧ ♠á①✐♠❛❧ é m

  

v = v : v(x) > 0 v R v ,

  {x ∈ K } = π = R /m

  v v v v v

  ♦♥❞❡ π é ✉♠ ✉♥✐❢♦r♠✐③❛♥t❡ ❡♠ R ✳ ❉❡♥♦t❛r❡♠♦s t❛♠❜é♠ ♣♦r k ♦ ❝♦r♣♦ r❡s✐✲

  v

  ❞✉❛❧ ❞❡ R ✳

  ✹✸

  K

  ❉❡♥♦t❛r❡♠♦s ♣♦r M ♦ ❝♦♥❥✉♥t♦ ❞❡ ✈❛❧♦r❡s ❛❜s♦❧✉t♦s ❞❡ K ❝✉❥❛ r❡str✐çã♦ à Q

  ∞

  ❝♦✐♥❝✐❞❡ ❝♦♠ ✉♠ ❞♦s ✈❛❧♦r❡s ❛❜s♦❧✉t♦s st❛♥❞❛r❞ ❞❡ Q✱ ♣♦r M K ♦ ❝♦♥❥✉♥t♦ ❞♦s ✈❛❧♦r❡s ❛❜s♦❧✉t♦s ❛rq✉✐♠❡❞✐❛♥♦s ❡ ♣♦r M ♦ ❝♦♥❥✉♥t♦ ❞♦s ✈❛❧♦r❡s ❛❜s♦❧✉t♦s ♥ã♦ ❛rq✉✐♠❡❞✐❛♥♦s✳

  K

  ❙❡❥❛ E/K ✉♠❛ ❝✉r✈❛ ❡❧í♣t✐❝❛ E : y + a

  1 xy + a 3 y = x + a 2 x + a 4 x + a 6 .

  −2 −3

  x, u y) ❋❛③❡♥❞♦ ❛s s✉❜st✐t✉✐çõ❡s (x, y) 7→ (u ♦❜t❡♠♦s ✉♠❛ ♥♦✈❛ ❡q✉❛çã♦

  2

  3

  3

  2

  2

  4

  6

  y + (a u)xy + (a u )y = x + (a u )x + (a u )x + (a u ),

  1

  3

  2

  4

  6 i

  a ❡♠ q✉❡ ♦s ❛♥t✐❣♦s ❝♦❡✜❝✐❡♥t❡s a i sã♦ s✉❜st✐t✉✐❞♦s ♣♦r u i ✳ ❆ss✐♠✱ ♣❡❧❛ ❡s❝♦❧❤❛ ❞❡ u ❞❡ ❢♦r♠❛ ❛❞❡q✉❛❞❛✱ ♣♦❞❡♠♦s ♦❜t❡r ❡q✉❛çõ❡s ❞❡ ❲❡✐❡rstr❛ss ❝♦♠ ❝♦❡✜❝✐❡♥t❡s ❡♠ R v ✱ ❡ ❛ss✐♠ ♦ ❞✐s❝r✐♠✐♥❛♥t❡ ∆ ❡st❛rá ❡♠ R v ❡ s❡rá t❛❧ q✉❡ v(∆) > 0✳ ❉❡✜♥✐çã♦ ✷✳✶✾✳ ❙❡❥❛ E/K ✉♠❛ ❝✉r✈❛ ❡❧í♣t✐❝❛✳ ❉✐r❡♠♦s q✉❡ ✉♠❛ ❡q✉❛çã♦ ❞❡ ❲❡✐❡rstr❛ss ♣❛r❛ E é ♠✐♥✐♠❛❧ ❡♠ r❡❧❛çã♦ à v s❡ v(∆) ❛t✐♥❣❡ ♦ ✈❛❧♦r ♠í♥✐♠♦ ♥❛s ❝♦♥❞✐çõ❡s ❞❡ q✉❡ v(a ) > 0

  i

  ✳ ❆❣♦r❛ ❡st❛r❡♠♦s ✐♥t❡r❡ss❛❞♦s ❡♠ ❛♥❛❧✐s❛r ♦s ♣♦♥t♦s ❞❡ E q✉❛♥❞♦ ❝♦♥s✐❞❡r❛♠♦s

  v

  ❛s s♦❧✉çõ❡s ♠ó❞✉❧♦ π ✱ ♦♥❞❡ v ∈ M K ✭❡st❛♠♦s ❝♦♥s✐❞❡r❛♥❞♦ ❛ ✈❛❧♦r✐③❛çã♦ ❞✐s❝r❡t❛ ❞❛ q✉❛❧ ♣r♦✈é♠ ♦ ✈❛❧♦r ❛❜s♦❧✉t♦✮✱ ♦✉ s❡❥❛✱ ❝♦♥s✐❞❡r❛♥❞♦ ♦ ❤♦♠♦♠♦r✜s♠♦ ❞❡ r❡❞✉çã♦ ♠ó❞✉❧♦ π

  v

  R = R /m , t

  v v v v

  → k 7→ t,

  v v

  ❡s❝♦❧❤❡♠♦s ✉♠❛ ❡q✉❛çã♦ ♠✐♥✐♠❛❧ ♣❛r❛ E/K ❡ ♦❧❤❛♠♦s s❡✉s ❝♦❡✜❝✐❡♥t❡s ♠ó❞✉❧♦ π ✳ ❈♦♠ E

  v v

  ✐ss♦ ♦❜t❡r❡♠♦s ✉♠❛ ❝✉r✈❛ s♦❜r❡ k ✱ q✉❡ s❡rá ❞❡♥♦t❛❞❛ ♣♦r ˜

  2

  3

  2

  ˜ E v : y + ˜ a

  1 xy + ˜ a 3 y = x + ˜ a 2 x + ˜ a 4 x + ˜ a 6 .

  , y , z ] ) , y , z

  v v

  ❉❛❞♦ ✉♠ ♣♦♥t♦ P = [x ∈ E(K ❝♦♠ ❝♦♦r❞❡♥❛❞❛s x ∈ R ✱ ❡♥tã♦ P = [˜ x , ˜ y , ˜ z ] E(k )

  v

  ♦ ♣♦♥t♦ ˜ ♣❡rt❡♥❝❡ ❛ ˜ ✳ ❊ ✐ss♦ ♣❡r♠✐t❡ ❞❡✜♥✐r ❛ ❛♣❧✐❝❛çã♦ ❞❡ r❡❞✉çã♦ E(K ) E(k ), P P .

  v v

  → ˜ 7→ ˜ ❖ ❝♦♥❥✉♥t♦ ❞♦s ♣♦♥t♦s ♥ã♦ s✐♥❣✉❧❛r❡s ❞❛ ❝✉r✈❛ r❡❞✉③✐❞❛ ❢♦r♠❛♠ ✉♠ ❣r✉♣♦ ❞❡♥♦✲

  E (k ) t❛❞♦ ♣♦r ˜ ns v ✳ ❆❧é♠ ❞✐ss♦✱ ♣♦❞❡♠♦s ❞❡✜♥✐r t❛♠❜é♠ ♦s s❡❣✉✐♥t❡s ❝♦♥❥✉♥t♦s E (K v ) = v ) : ˜ P E ns (k v )

  {P ∈ E(K ∈ ˜ }, E

  1 (K v ) = v ) : ˜ P = ˜ {P ∈ E(K O}.

  (K ) (K )

  v 1 v

  ❉✐r❡♠♦s q✉❡ E é ♦ ❝♦♥❥✉♥t♦ ❞♦s ♣♦♥t♦s ❞❡ r❡❞✉çã♦ ♥ã♦ s✐♥❣✉❧❛r ❡ q✉❡ E é ♦

  ✹✹ E /k

  v v

  ❚♦♠❛❞♦ v ∈ M K ✱ ❞✐r❡♠♦s q✉❡ E/K t❡♠ ✉♠❛ ❜♦❛ r❡❞✉çã♦ ❡♠ v✱ s❡ ˜ é ✉♠❛ ❝✉r✈❛ s✉❛✈❡✱ ❝❛s♦ ❝♦♥trár✐♦ ❞✐r❡♠♦s q✉❡ t❡♠ ✉♠❛ ♠á r❡❞✉çã♦✳ ❆❣♦r❛ ❡♥✉♥❝✐❛r❡♠♦s ✉♠ r❡s✉❧t❛❞♦ q✉❡ ❣❛r❛♥t❡ q✉❛♥❞♦ ❛ ❝✉r✈❛ E t❡♠ ✉♠❛ ❜♦❛ r❡❞✉çã♦ ❝♦♠ r❡s♣❡✐t♦ à ✉♠❛ ✈❛❧♦r✐③❛çã♦✳ Pr♦♣♦s✐çã♦ ✷✳✷✵✳ ❙❡❥❛ E/K ✉♠❛ ❝✉r✈❛ ❡❧í♣t✐❝❛ ❞❛❞❛ ♣♦r ✉♠❛ ❡q✉❛çã♦ ❞❡ ❲❡✐❡rstr❛ss ♠✐♥✐♠❛❧

  2

  3

  2 E : y + a 1 xy + a 3 y = x + a 2 x + a 4 x + a 6 ,

  ❝♦♠ ❞✐s❝r✐♠✐♥❛♥t❡ ∆✳ ❊♥tã♦✱ E t❡♠ ✉♠❛ ❜♦❛ r❡❞✉çã♦ ❝♦♠ r❡s♣❡✐t♦ ❛ v ∈ M s❡✱ ❡

  K ×

  E /k

  v v

  s♦♠❡♥t❡ s❡✱ v(∆) = 0✱ ✐st♦ é✱ ∆ ∈ R v ✳ ❊ ♥❡st❡ ❝❛s♦ ˜ é ✉♠❛ ❝✉r✈❛ ❡❧í♣t✐❝❛✳ ❉❡♠♦♥str❛çã♦✳ ❱❡r ✭❈❛♣ít✉❧♦ ✶✱ Pr♦♣♦s✐çã♦ ✺✳✶✱ ❬❙✐❧✈❡r♠❛♥❪✮✳

  ❆❣♦r❛ ❡♥✉♥❝✐❛♠♦s ✉♠ r❡s✉❧t❛❞♦ q✉❡ t❡♠ s✉❛ ✐♠♣♦rtâ♥❝✐❛ ♥❛ ♣r♦✈❛ ❞❛ ✈❡rsã♦ ❢r❛❝❛ ❞♦ ❚❡♦r❡♠❛ ❞❡ ▼♦r❞❡❧❧✲❲❡✐❧ ✭❡♥✉♥❝✐❛❞♦ ❡ ❞✐s❝✉t✐❞♦ ♥♦ ♣ró①✐♠♦ ❝❛♣ít✉❧♦✮✳ Pr♦♣♦s✐çã♦ ✷✳✷✶✳ ❙❡❥❛ E/K ✉♠❛ ❝✉r✈❛ ❡❧í♣t✐❝❛✱ m ✉♠ ✐♥t❡✐r♦ ♣♦s✐t✐✈♦ ❡ v ∈ M K t❛❧ q✉❡ v(m) = 0✳ ❙❡ E t❡♠ ✉♠❛ ❜♦❛ r❡❞✉çã♦ ❡♠ r❡❧❛çã♦ à v✱ ❡♥tã♦ ❛ ❛♣❧✐❝❛çã♦ ❞❡ r❡❞✉çã♦

  E(K)[m] E (k ) → ˜ v v P P .

  7→ ˜ é ✐♥❥❡t✐✈❛✳ ❉❡♠♦♥str❛çã♦✳ ❱❡r ✭❈❛♣ít✉❧♦ ✼✱ Pr♦♣♦s✐çã♦ ✸✳✶✱ ❬❙✐❧✈❡r♠❛♥❪✮✳

  ❈❛♣ít✉❧♦ ✸ ❖ ❚❡♦r❡♠❛ ❞❡ ▼♦r❞❡❧❧✲❲❡✐❧

  ❊st❡ ❝❛♣ít✉❧♦ é ❞❡❞✐❝❛❞♦ à ❞❡♠♦♥str❛çã♦ ❞♦ ❚❡♦r❡♠❛ ❞❡ ▼♦r❞❡❧❧✲❲❡✐❧✱ r❡s✉❧t❛❞♦ ❡st❡ q✉❡ ❞✐③ q✉❡ ♦ ❣r✉♣♦ ❞❡ ▼♦r❞❡❧❧✲❲❡✐❧ ✭❝♦♥❥✉♥t♦ ❞♦s ♣♦♥t♦s K✲r❛❝✐♦♥❛✐s✮ ❞❡ q✉❛❧q✉❡r ❝✉r✈❛ ❡❧í♣t✐❝❛ s♦❜r❡ ✉♠ ❝♦r♣♦ ❞❡ ♥ú♠❡r♦s K✱ é ✜♥✐t❛♠❡♥t❡ ❣❡r❛❞♦✳ ❊st❡ ❚❡♦r❡♠❛ ♣r♦✈❛❞♦ ♣♦r ❆♥❞ré ❲❡✐❧ ❡♠ ✶✾✷✽ é ✉♠❛ ❡①t❡♥sã♦ ❞❛ ✈❡rsã♦ ♣❛r❛ ♦ ❝♦r♣♦ ❞♦s r❛❝✐♦♥❛✐s✱ ♣r♦✈❛❞❛ ♣♦r ▲♦✉✐s ▼♦r❞❡❧❧ ❡♠ ✶✾✷✷✳

  ❊ss❛ ♣r♦✈❛ é ❞✐✈✐❞✐❞❛ ❡♠ ❞✉❛s ❣r❛♥❞❡s ❡t❛♣❛s✱ ✉♠❛ ❞❡❧❛s é ♦ ❝❤❛♠❛❞♦ ❚❡♦r❡♠❛ ❋r❛❝♦ ❞❡ ▼♦r❞❡❧❧✲❲❡✐❧✱ ♠❛s ❛♣❡♥❛s ❛ ✈❡rsã♦ ❢r❛❝❛ ♥ã♦ é s✉✜❝✐❡♥t❡✳ ❆ ♦✉tr❛ ❡t❛♣❛ é ♦ ❝❤❛♠❛❞♦ ❚❡♦r❡♠❛ ❞❡ ❉❡s❝✐❞❛✱ q✉❡ ✐♥❞✐❝❛rá q✉❡ t✐♣♦ ❞❡ ❢✉♥çã♦✱ q✉❡ s❡rá ❝❤❛♠❛❞❛ ❢✉♥çã♦ ❛❧t✉r❛✱ ♣♦❞❡ s❡r ✉s❛❞❛ ♣❛r❛ ♣r♦✈❛r q✉❡ ✉♠ ❣r✉♣♦ ❛❜❡❧✐❛♥♦ é ✜♥✐t❛♠❡♥t❡ ❣❡r❛❞♦✳

  ✸✳✶ ❆ ✈❡rsã♦ ❢r❛❝❛ ❞♦ ❚❡♦r❡♠❛

  ◆❡st❛ s❡çã♦ s❡rã♦ ❛♣r❡s❡♥t❛❞♦s ❛❧❣✉♥s r❡s✉❧t❛❞♦s té❝♥✐❝♦s q✉❡ ♥ã♦ s❡rã♦ ❞❡♠♦♥s✲ tr❛❞♦s✱ ♠❛s q✉❡ t❡rã♦ ✐♥❞✐❝❛❞❛s ❛s ❞❡✈✐❞❛s r❡❢❡rê♥❝✐❛s ♣❛r❛ s✉❛s ❞❡♠♦♥str❛çõ❡s✳ ❚❡♦r❡♠❛ ✸✳✶ ✭▼♦r❞❡❧❧✲❲❡✐❧ ❋r❛❝♦✮✳ ❙❡❥❛ K ✉♠ ❝♦r♣♦ ❞❡ ♥ú♠❡r♦s✱ E/K ✉♠❛ ❝✉r✈❛

E(K)

  ❡❧í♣t✐❝❛ ❞❡✜♥✐❞❛ s♦❜r❡ K ❡ m ✉♠ ✐♥t❡✐r♦ ♠❛✐♦r ♦✉ ✐❣✉❛❧ à ✷✳ ❊♥tã♦ é ✜♥✐t♦✳ mE(K) ❆♥t❡s ❞❡ ❛♣r❡s❡♥t❛r ❛ ♣r♦✈❛ ❞❡st❡ r❡s✉❧t❛❞♦✱ ♣r❡❝✐s❛♠♦s ❞❡ ♠❛✐s ❛❧❣✉♠❛s ❢❡rr❛✲

  ♠❡♥t❛s✳ Pr♦✈❛r❡♠♦s ♦ s❡❣✉✐♥t❡ ❧❡♠❛✱ q✉❡ ♥♦s ♣❡r♠✐t✐rá s✉♣♦r q✉❡ E[m] ⊆ E(K)✳ ▲❡♠❛ ✸✳✷✳ ❙❡❥❛ K ✉♠ ❝♦r♣♦ ❞❡ ♥ú♠❡r♦s ❡ L/K ✉♠❛ ❡①t❡♥sã♦ ❞❡ ●❛❧♦✐s ✜♥✐t❛✳ ❙❡

E(L) E(K)

  é ✜♥✐t♦✱ ❡♥tã♦ t❛♠❜é♠ é ✜♥✐t♦✳ mE(L) mE(K) ❉❡♠♦♥str❛çã♦✳ ❈♦♥s✐❞❡r❡♠♦s ❛ ✐♥❝❧✉sã♦ E(K) ֒→ E(L)✳ ❊st❛ ✐♥❞✉③ ✉♠❛ ❛♣❧✐❝❛çã♦

E(K) E(L)

  , → mE(K) mE(L)

  ✹✻ ❝✉❥♦ ♥ú❝❧❡♦ é ❞❛❞♦ ♣♦r

E(K)

  ∩ mE(L) Φ = .

  {P + mE(K) : P ∈ E(K) ∩ mE(L)} = mE(K) = P

  P P

  ❉❛❞♦ P + mE(K) ∈ Φ✱ ❝♦♥s✐❞❡r❡♠♦s Q ∈ E(L) s❛t✐s❢❛③❡♥❞♦ [m]Q ✭❡st❡ Q

  P

  ♥ã♦ ♣r❡❝✐s❛ s❡r ú♥✐❝♦✮✳ ❆❣♦r❛ ❞❡✜♥✐♠♦s ❛ s❡❣✉✐♥t❡ ❢✉♥çã♦ λ : G(L/K)

  P

  → E[m]

  τ τ .

  P

  7→ Q P − Q ❊st❛ ❛♣❧✐❝❛çã♦ ❡stá ❜❡♠ ❞❡✜♥✐❞❛✱ ✈✐st♦ q✉❡

  τ τ τ

  [m](Q ) = ([m]Q ) = P

  P P P P − Q − [m]Q − P = O, (P ∈ E(K)), τ P

  ❡ ❛ss✐♠ Q P − Q ∈ E[m]✳

  ′

  = λ

  P P

  ❆❣♦r❛ ✈❡❥❛♠♦s ♦ s❡❣✉✐♥t❡✿ ❙❡❥❛♠ P, P ∈ E(K) ∩ mE(L) t❛✐s q✉❡ λ ✳ ❚❡♠♦s q✉❡ ′ ′ τ

  (Q ) = Q ,

  P P P P

  − Q − Q ∀τ ∈ G(L/K),

  τ τ ′ ′ ′

  = λ (τ ) = λ (τ ) = Q ✈✐st♦ q✉❡ Q P P P P ✳ P♦rt❛♥t♦✱ Q P P

  P − Q P − Q − Q ∈ E(K)✳ ❉❡

  ♦♥❞❡ s❡❣✉❡ q✉❡

  ′

  P = [m]Q − P P − [m]Q P ∈ mE(K),

  ′

  ( ❡ ❛ss✐♠ P ≡ P ♠♦❞ mE(K))✳ ■ss♦ ♣r♦✈❛ q✉❡ ❛ ❛ss♦❝✐❛çã♦

  Φ ,

  P

  → F(G(L/K), E[m]), P 7→ λ é ✐♥❥❡t♦r❛✳ ▼❛s t❛♥t♦ E[m] q✉❛♥t♦ G(L/K) sã♦ ✜♥✐t♦s✱ ❞❡ ♦♥❞❡ s❡❣✉❡ q✉❡ Φ é ✜♥✐t♦✳

  ❋✐♥❛❧♠❡♥t❡✱ t❡♠♦s q✉❡ ❛ s❡q✉ê♥❝✐❛ ❡①❛t❛

E(K) E(L)

  , → Φ → → mE(K) mE(L)

E(K) E(K)

  ❝♦❧♦❝❛ ❡♥tr❡ ❞♦✐s ❣r✉♣♦s ✜♥✐t♦s✱ ❡ ❛ss✐♠✱ é ✜♥✐t♦✳ mE(K) mE(K) ❊♠ ❝♦♥s❡q✉ê♥❝✐❛ ❞❡ss❡ ❧❡♠❛✱ ♣♦❞❡♠♦s ❛ss✉♠✐r q✉❡ E[m] ⊆ E(K)✱ ♣♦✐s s❡ ❛ss✐♠

  ♥ã♦ ♦ ❢♦ss❡✱ ♣♦❞❡rí❛♠♦s ❝♦♥str✉✐r ✉♠❛ ❡①t❡♥sã♦ ❞❡ ●❛❧♦✐s ✜♥✐t❛ L/K✱ q✉❡ ❝♦♥té♠ t♦❞❛s ❛s ❝♦♦r❞❡♥❛❞❛s ❞♦s ♣♦♥t♦s ❞❡ m✲t♦rçã♦✱ q✉❡ ❥á s❛❜❡♠♦s q✉❡ sã♦ ✜♥✐t♦s ✭❱❡r ❈♦r♦❧ár✐♦ ✺✳✹✱ ❬❇✉tt❪✮✳

  ❉❛q✉✐ ❡♠ ❞✐❛♥t❡ ✐r❡♠♦s s✉♣♦r q✉❡ E[m] ⊆ E(K)✳ ❊ ❛ ♣❛rt✐r ❞❡ ❛❣♦r❛✱ ♠♦str❛✲

  ✹✼ E(K) r❡♠♦s q✉❡ ❛ ✜♥✐t✉❞❡ ❞♦ ❣r✉♣♦ ♣♦❞❡ s❡r s✉❜st✐t✉✐❞❛ ♣❡❧❛ ✜♥✐t✉❞❡ ❞❡ ✉♠❛ ❝❡rt❛ mE(K)

  ❡①t❡♥sã♦ ❞❡ ❝♦r♣♦s ❞❡ K✳

  ✸✳✶✳✶ ❖ ❡♠♣❛r❡❧❤❛♠❡♥t♦ ❞❡ ❑✉♠♠❡r

E(K)

  P❛r❛ ♠♦str❛r♠♦s ❛ ❡q✉✐✈❛❧ê♥❝✐❛ ❡♥tr❡ ❛ ✜♥✐t✉❞❡ ❞♦ ❣r✉♣♦ ❝♦♠ ❛ ✜♥✐t✉❞❡ mE(K) ❞❡ ✉♠❛ ❝❡rt❛ ❡①t❡♥sã♦ ❞❡ K✱ ♣r❡❝✐s❛r❡♠♦s ❞❡✜♥✐r ♦ ❡♠♣❛r❡❧❤❛♠❡♥t♦ ❞❡ ❑✉♠♠❡r✱ q✉❡ é ✈✐st♦ ❛ s❡❣✉✐r✳

  ■r❡♠♦s ❝♦♥s✐❞❡r❛r ❛ s❡❣✉✐♥t❡ ❛♣❧✐❝❛çã♦ κ : E(K)

  × G(K/K) → E[m], = P

  P P

  ❞❡✜♥✐❞❛ ❞❛ s❡❣✉✐♥t❡ ❢♦r♠❛✳ ❙❡♥❞♦ P ∈ E(K)✱ ❡s❝♦❧❤❡♠♦s Q ∈ E(K) t❛❧ q✉❡ [m]Q ✱ ❡ ✐ss♦ é ♣♦ssí✈❡❧ ✈✐st♦ q✉❡ ❛ ❛♣❧✐❝❛çã♦ [m] : E → E é s♦❜r❡❥❡t✐✈❛✳ ❊♥tã♦ ❞❡✜♥✐♠♦s

  τ κ(P, τ ) = Q .

  P P − Q

  ❊st❛ ❛♣❧✐❝❛çã♦ é ❝❤❛♠❛❞❛ ❊♠♣❛r❡❧❤❛♠❡♥t♦ ❞❡ ❑✉♠♠❡r ❡ s❛t✐s❢❛③ ❛s s❡❣✉✐♥✲ t❡s ♣r♦♣r✐❡❞❛❞❡s✱ ❡✈✐❞❡♥❝✐❛❞❛s ♣❡❧❛ ♣r♦♣♦s✐çã♦ ❛ s❡❣✉✐r✳ Pr♦♣♦s✐çã♦ ✸✳✸✳ ❆ ❛♣❧✐❝❛çã♦ κ s❛t✐s❢❛③✿

  ✭✐✮ ❊stá ❜❡♠ ❞❡✜♥✐❞♦✱ ✐st♦ é✱ ✐♥❞❡♣❡♥❞❡ ❞❛ ❡s❝♦❧❤❛ ❞❡ Q P ❡ κ(P, τ) ∈ E[m] ❡❢❡t✐✈❛♠❡♥t❡✳ ✭✐✐✮ ➱ ❜✐❧✐♥❡❛r✱ ✐st♦ é✱ κ(P + Q, τ) = κ(P, τ) + κ(Q, τ) ❡ κ(P, τσ) = κ(P, τ) + κ(P, σ)✱ ♣❛r❛ t♦❞♦ P, Q ∈ E(K), τ, σ ∈ G(K/K).

  ✭✐✐✐✮ ❖ ♥ú❝❧❡♦ ❞❡ κ à ❡sq✉❡r❞❛ é mE(K)✱ ✐st♦ é✱ κ(P, τ) = O ♣❛r❛ t♦❞♦ τ ∈ G(K/K) s❡✱ ❡ s♦♠❡♥t❡ s❡✱ P ∈ mE(K)✳

  −1

E(K))

  ✭✐✈✮ ❖ ♥ú❝❧❡♦ ❞❡ κ à ❞✐r❡✐t❛ é G(K/L)✱ ♦♥❞❡ L = K([m] q✉❡ é ❛ ♠❡♥♦r ❡①t❡♥sã♦

  −1

E(K)

  ❞❡ ❝♦r♣♦s ❞❡ K q✉❡ ❝♦♥té♠ t♦❞❛s ❛s ❝♦♦r❞❡♥❛❞❛s ❞♦s ♣♦♥t♦s ❡♠ [m] ✳ ✭✈✮ ❖ ❊♠♣❛r❡❧❤❛♠❡♥t♦ ❞❡ ❑✉♠♠❡r ✐♥❞✉③ ✉♠❛ ❢♦r♠❛ ❜✐❧✐♥❡❛r ♣❡r❢❡✐t❛

  E(K)/mE(K) × G(L/K) → E[m],

  ✐st♦ é✱ ✉♠❛ ❛♣❧✐❝❛çã♦ ❜✐❧✐♥❡❛r ✭❝♦♠♦ ✈✐st♦ ♥♦ ✐t❡♠ ✭✐✐✮✮ ❡ q✉❡ s❡✉s ♥ú❝❧❡♦s à ❡sq✉❡r❞❛ ❡ ❞✐r❡✐t❛ sã♦ tr✐✈✐❛✐s✳

  ❉❡♠♦♥str❛çã♦✳ ✭✐✮ ▼♦str❡♠♦s q✉❡ κ(P, τ) ∈ E[m]✳ ❉❛❞♦ q✉❡✱ s❡♥❞♦ P = (x, y) ❡♥tã♦

  τ τ τ

  P = (x , y ) ❡ q✉❡ G(K/K) ✜①❛ ♦s ♣♦♥t♦s K✲r❛❝✐♦♥❛✐s✱ ❡♥tã♦ t❡♠♦s q✉❡

  τ τ

  [m]κ(P, τ ) = [m]Q = P

  P

  ✹✽ ❊ ♣♦rt❛♥t♦✱ κ(P, τ) ∈ E[m]✳

  ′

  , Q

  P P

  ❆❣♦r❛ ♠♦str❡♠♦s q✉❡ ♥ã♦ ❞❡♣❡♥❞❡ ❞❛ ❡s❝♦❧❤❛ ❞❡ Q ✳ ❙❡ t✐✈❡r♠♦s Q P ❞✐st✐♥t♦s

  ′ ′

  P P

  = [m]Q = P = Q + S t❛✐s q✉❡ [m]Q P ✱ ❡♥tã♦ Q P ♣❛r❛ ❛❧❣✉♠ S ∈ E[m] ⊆ E(K)✳

  ❆ss✐♠✱

  

′ τ ′ τ τ τ τ

  (Q ) = (Q P + S) P + S) = Q P + S P ,

  P − Q P − (Q P − Q − S = Q P − Q τ

  ✈✐st♦ q✉❡ S ∈ E(K) ❡ ♣♦rt❛♥t♦ S − S = O. P♦rt❛♥t♦ κ ❡stá ❜❡♠ ❞❡✜♥✐❞❛✳ ✭✐✐✮ P❛r❛ ❛ ❧✐♥❡❛r✐❞❛❞❡ ❡♠ G(K/K) t❡♠♦s✳ ❉❛❞♦s τ, σ ∈ G(K/K)✱

  τ σ τ σ σ σ

  κ(P, τ σ) = Q = (Q ) + (Q ) = κ(P, τ ) + κ(P, σ),

  

P P P

P − Q P − Q P − Q

  ❡ ❝♦♠♦ κ(P, τ) ∈ E[m] ⊆ E(K)✱ ❡♥tã♦ κ(P, τσ) = κ(P, τ) + κ(P, σ). P❛r❛ ❛ ❧✐♥❡❛r✐❞❛❞❡ ❡♠ E(K)✱ ❥á s❛❜❡♠♦s q✉❡ ❛ ✐♠❛❣❡♠ ❞❡ κ(P, τ) ♥ã♦ ❞❡♣❡♥❞❡ ❞❛

  P

  ❡s❝♦❧❤❛ ❞♦ r❡♣r❡s❡♥t❛♥t❡ Q ✱ s❡♥❞♦ ❛ss✐♠✱ ♣❛r❛ P + R✱ ❡s❝♦❧❤❡♠♦s ♦ r❡♣r❡s❡♥t❛♥t❡ Q = Q + Q

  P +R P R

  ❞❡ ♦♥❞❡ t❡r❡♠♦s

  τ τ τ κ(P + R, τ ) = Q = (Q ) + (Q ) = κ(P, τ ) + κ(R, τ ).

  P +R P R

P +R − Q P − Q R − Q

  = P

  P P

  ✭✐✐✐✮ Pr✐♠❡✐r♦✱ s✉♣♦♥❞♦ P ∈ mE(K)✱ t♦♠❡♠♦s Q ∈ E(K) t❛❧ q✉❡ [m]Q ✳

  τ

  κ(P, τ ) = Q = Q =

  

P P P

P − Q − Q O, ∀τ ∈ G(K/K), P ✈✐st♦ q✉❡ Q é ✜①❛❞♦ ♣♦r G(K/K).

  P♦r ♦✉tr♦ ❧❛❞♦✱ s❡ κ(P, τ) = O, ♣❛r❛ t♦❞♦ τ ∈ G(K/K)✱ ❡♥tã♦

  τ

  Q = Q ,

  P P ♣❛r❛ t♦❞♦ τ ∈ G(K/K). P

  ❉❛í✱ Q ∈ E(K) ❡ ❛ss✐♠ P ∈ mE(K)✳ ✭✐✈✮ ❙❡❥❛ τ ∈ G(K/L)✱ ❡♥tã♦

  τ

  κ(P, τ ) = Q P = − Q O, ∀P ∈ E(K),

  P P

  ✈✐st♦ q✉❡ Q ∈ E(L) ♣❡❧❛ ❞❡✜♥✐çã♦ ❞❡ L✳ P♦r ♦✉tr♦ ❧❛❞♦✱ s❡ τ ∈ G(K/K) é t❛❧ q✉❡ κ(P, τ) = O ♣❛r❛ t♦❞♦ P ∈ E(K)✱ ❡♥tã♦ ♣❛r❛ t♦❞♦ Q ∈ E(K) ❝♦♠ [m]Q ∈ E(K) t❡♠✲s❡

  τ O = κ([m]Q, τ) = Q − Q.

  ✹✾ P♦rt❛♥t♦✱ τ ✜①❛ Q✱ ❡ s❡ Q = (x(Q), y(Q)) ❡♥tã♦ τ ✜①❛ t❛♠❜é♠ ♦ ❝♦r♣♦ ❞❡ ❞❡✜♥✐çã♦ ❞❡ Q✱ K(x(Q), y(Q))✳ ▼❛s ❜❛st❛ ♦❜s❡r✈❛r q✉❡ L é ♣r❡❝✐s❛♠❡♥t❡ ❛ ✉♥✐ã♦ ❞❡ t♦❞♦s ❡ss❡s ❝♦r♣♦s ❞❡ ❞❡✜♥✐çã♦✱ ❞❡ ♠♦❞♦ q✉❡ τ t❛♠❜é♠ ✜①❛ L✳ ❉❡ ♦♥❞❡ s❡q✉❡ q✉❡ τ ∈

G(K/L)

G(K/K)

G(K/K)

  ❘❡❝✐♣r♦❝❛♠❡♥t❡✱ ✉s❛♥❞♦ ✉♠ ❛r❣✉♠❡♥t♦ ❛♥á❧♦❣♦✱ ♠♦str❛✲s❡ q✉❡ ❛ ✜♥✐t✉❞❡ ❞❡

  P❛r❛ ♣r♦✈❛r ❛ ✈❡rsã♦ ❢r❛❝❛ ❞♦ ❚❡♦r❡♠❛ ❞❡ ▼♦r❞❡❧❧✲❲❡✐❧✱ ♣r❡❝✐s❛♠♦s ❞❡ ♠❛✐s ❛❧❣✉♠❛s ❞❡✜♥✐çõ❡s ❞❛ ❚❡♦r✐❛ ❆❧❣é❜r✐❝❛ ❞♦s ◆ú♠❡r♦s✳

  ✸✳✶✳✷ ❆ ❞❡♠♦♥str❛çã♦ ❞❛ ✈❡rsã♦ ❢r❛❝❛

  ❣❛r❛♥t❡ ❛ ✜♥✐t✉❞❡ ❞❡ E(K)/mE(K)✳ ❆ss✐♠✱ ❞❛q✉✐ ❡♠ ❞✐❛♥t❡ ♥♦s ❞❡❞✐❝❛r❡♠♦s ❛ ♠♦str❛r ❛ ✜♥✐t✉❞❡ ❞❛ ❡①t❡♥sã♦ L/K✳

  κ( ·, τ) : E(K)/mE(K) → E[m]. ❈♦♠♦ E[m] ❡ E(K)/mE(K) sã♦ ✜♥✐t♦s✱ ❡♥tã♦ ♦ ❣r✉♣♦ Hom(E(K)/mE(K), E[m]) t❛♠✲ ❜é♠ é ✜♥✐t♦✳ ❆❣♦r❛ ❝♦♠♦ ❛ ❢♦r♠❛ ❜✐❧✐♥❡❛r ✐♥❞✉③✐❞❛ é ♣❡r❢❡✐t❛✱ t❡♠♦s q✉❡ κ(·, τ) = κ(·, σ) s❡✱ ❡ s♦♠❡♥t❡ s❡✱ τ = σ✱ ❞❡ ♦♥❞❡ s❡❣✉❡ q✉❡ G(L/K) é ✜♥✐t♦✳

  ✳ ✭✈✮ ❈♦♥s✐❞❡r❛♥❞♦ ❛ t♦rr❡ ❞❡ ❝♦r♣♦s K ⊆ L ⊆ K✱ ❥á s❛❜❡♠♦s q✉❡ G(K/L) é ✉♠ s✉❜❣r✉♣♦

  ❆❣♦r❛✱ ♣♦❞❡r❡♠♦s tr♦❝❛r ♦ ♣r♦❜❧❡♠❛ ❞❡ ♠♦str❛r ❛ ✜♥✐t✉❞❡ ❞♦ ❣r✉♣♦ E(K)/mE(K) ♣❡❧♦ ❞❡ ♠♦str❛r ❛ ✜♥✐t✉❞❡ ❞❛ ❡①t❡♥sã♦ G(L/K)✳ ❈♦♠ ❡❢❡✐t♦✱ s❡ E(K)/mE(K) é ✜♥✐t♦✱ ❡♥tã♦ ♣❛r❛ ❝❛❞❛ τ ∈ G(L/K) ♣♦❞❡♠♦s ✐♥❞✉③✐r ✉♠ ❤♦♠♦♠♦r✜s♠♦

  ♥❛t✉r❛❧✱ ❞❡ ♠♦❞♦ q✉❡ ❛ ❜✐❧✐♥❡❛r✐❞❛❞❡ s❡ ♣r❡s❡r✈❛ ❡ ♦s ♥ú❝❧❡♦s à ❡sq✉❡r❞❛ ❡ à ❞✐r❡✐t❛ ♣❛ss❛♠ ❛ s❡r❡♠ tr✐✈✐❛✐s✳

  E(K)/mE(K) × G(L/K) → E[m], q✉❡ é ❞❡✜♥✐❞❛ ❞❡ ♠♦❞♦ ♥❛t✉r❛❧ ❛ ♣❛rt✐r ❞❡ κ ♣❛ss❛♥❞♦ ❛♦ q✉♦❝✐❡♥t❡ ❞❡ ♠❛♥❡✐r❛

  G(K/L) . ❚♦♠❛♠♦s ❛ ❛♣❧✐❝❛çã♦

  P♦rt❛♥t♦✱ ❛ ❡①t❡♥sã♦ L/K é ❞❡ ●❛❧♦✐s ❡ ♠❛✐s ❞♦ q✉❡ ✐ss♦✱ G(L/K) ∼ =

  → Hom(E(K), E[m]) τ 7→ κ(·, τ).

  ♥♦r♠❛❧ ❞❡ G(K/K)✱ ✈✐st♦ q✉❡ é ♥ú❝❧❡♦ ❞♦ ❤♦♠♦♠♦r✜s♠♦

G(L/K)

  ✺✵ ❉❡✜♥✐çã♦ ✸✳✹✳ ❙❡❥❛ L/K ✉♠❛ ❡①t❡♥sã♦ ✜♥✐t❛ ❞❡ ❝♦r♣♦s ❝♦♠♣❧❡t♦s ❡ v ✉♠❛ ✈❛❧♦r✐③❛çã♦

  , k ]

  v v

  ❡♠ L✳ ❉✐③❡♠♦s q✉❡ ❛ ❡①t❡♥sã♦ L/K é ♥ã♦ r❛♠✐✜❝❛❞❛ ❡♠ v s❡ [L : K] = [l ♦♥❞❡ l

  v ❡ k v sã♦ ♦s ❝♦r♣♦s r❡s✐❞✉❛✐s ❝♦♠ r❡s♣❡✐t♦ à v✳

  ❉❡✜♥✐çã♦ ✸✳✺✳ ❈❤❛♠❛♠♦s ❞❡ ❣r✉♣♦ ❞❡ ✐♥ér❝✐❛ ❛♦ s❡❣✉✐♥t❡ s✉❜❣r✉♣♦ ❞❡ G(K/K)✱ I v = v < 1, v {τ ∈ G(K/K) : |τ(x) − x| ♣❛r❛ t♦❞♦ x ∈ R }.

  −1

E(K))

  Pr♦♣♦s✐çã♦ ✸✳✻✳ ❙❡❥❛ L = K([m] ♦ ❝♦r♣♦ q✉❡ ❞❡✜♥✐♠♦s ❛♥t❡r✐♦r♠❡♥t❡✳ ❊♥tã♦ ✭❛✮ ❆ ❡①t❡♥sã♦ L/K é ❛❜❡❧✐❛♥❛ ❡ t❡♠ ❡①♣♦❡♥t❡ m✱ ✐st♦ é✱ ♦ ❣r✉♣♦ ❞❡ ●❛❧♦✐s G(L/K) é

  ❛❜❡❧✐❛♥♦ ❡ ❡ t♦❞♦ ❡❧❡♠❡♥t♦ ❞❡ G(L/K) t❡♠ ♦r❞❡♠ q✉❡ ❞✐✈✐❞❡ m✳

  ∞ : ˜ E (k ) : v(m) . v v

  ✭❜✮ ❙❡❥❛ S = {v ∈ M K é s✐♥❣✉❧❛r} ∪ {v ∈ M K 6= 0} ∪ M K ❆ ❡①t❡♥sã♦

L/K

  K

  é ♥ã♦ r❛♠✐✜❝❛❞❛ ❢♦r❛ ❞❡ S✱ ♦✉ s❡❥❛✱ s❡ v ∈ M ❡ v 6∈ S✱ ❡♥tã♦ ❛ ❡①t❡♥sã♦

L/K

  ♥ã♦ s❡ r❛♠✐✜❝❛ ❡♠ v✳ ❉❡♠♦♥str❛çã♦✳ ✭❛✮ P❡❧♦ q✉❡ ✈✐♠♦s ❛♥t❡r✐♦r♠❡♥t❡✱ ♦ ❡♠♣❛r❡❧❤❛♠❡♥t♦ ❞❡ ❑✉♠♠❡r ❞❡✜♥❡

  ✉♠ ❤♦♠♦♠♦r✜s♠♦ ❞❡ ❣r✉♣♦s ✐♥❥❡t✐✈♦ φ : G(L/K)

  → Hom(E(K), E[m]), τ 7→ κ(·, τ), q✉❡ ♠♦str❛ q✉❡ G(L/K) é ❛❜❡❧✐❛♥♦✳ ❆❣♦r❛✱ s❡ τ ∈ G(L/K)✱ t❡♠♦s

  m m

  φ(τ ) = κ( ) = [m]κ( ·, τ ·, τ) = O.

  m

  = id ❉❡ ♠♦❞♦ q✉❡ τ ∈ G(L/K)✱ ♦ q✉❡ ✐♠♣❧✐❝❛ ❡♠ ord(τ)|m✳

  K

  ✭❜✮ ❈♦♥s✐❞❡r❡ v ∈ M ❝♦♠ v 6∈ S✳ ❉❛❞♦ q✉❡ L é ❛ ✉♥✐ã♦ ❞❡ t♦❞❛s ❛s ❡①t❡♥sõ❡s

  ′

  K = K(Q) ♣❛r❛ Q ∈ E(K) ❝♦♠ [m]Q ∈ E(K)✱ ❡♥tã♦ ♣r❡❝✐s❛♠♦s ♠♦str❛r q✉❡ ❝❛❞❛

  ✉♠❛ ❞❛s ❡①t❡♥sõ❡s K(Q)/K é ♥ã♦ r❛♠✐✜❝❛❞❛ ❡♠ v✳

  ′ ′ K

  ❙❡❥❛ v ∈ M t❛❧ q✉❡ v r❡str✐t♦ ❛ K ❝♦✐♥❝✐❞❡ ❝♦♠ v✳ ❈♦♥s✐❞❡r❡♠♦s ❛ ❝♦rr❡s♣♦♥❞❡♥t❡

  ′

  /k

  v

  ❡①t❡♥sã♦ k v ❞❡ ❝♦r♣♦s r❡s✐❞✉❛✐s✳ ❈♦♠♦ v 6∈ S✱ E t❡♠ ✉♠❛ ❜♦❛ r❡❞✉çã♦ ❝♦♠ r❡❧❛çã♦ ❛ v✱ ❡ ♣♦rt❛♥t♦ t❛♠❜é♠ t❡♠

  ′

  ✉♠❛ ❜♦❛ r❡❞✉çã♦ ❝♦♠ r❡❧❛çã♦ ❛ v ✱ ❥á q✉❡ ♣♦❞❡♠♦s ❝♦♥s✐❞❡r❛r ❛ ♠❡s♠❛ ❡q✉❛çã♦ ❞❡

  ′

  (∆) = 0 ❲❡✐❡rstr❛ss ♦♥❞❡ t❡r❡♠♦s v(∆) = v ✳ ❈♦♥s✐❞❡r❡♠♦s t❛♠❜é♠ ❛ ❛♣❧✐❝❛çã♦ ❞❡ r❡❞✉çã♦

  ′ ′ E(K ) E (k ).

  → ˜ v

  v v

  P❡❧❛ ❞❡✜♥✐çã♦ ❞❡ I ✱ s✉❛ ❛t✉❛çã♦ s♦❜r❡ k é tr✐✈✐❛❧✱ ❞❡ ♠♦❞♦ q✉❡ t❛♠❜é♠ ❛t✉❛

  v

  ✺✶ ′ ′ E (k )

  v v

  tr✐✈✐❛❧♠❡♥t❡ s♦❜r❡ ˜ v ✳ P♦rt❛♥t♦✱ ❞❛❞♦ τ ∈ I ✱ t❡♠♦s

  τ τ

  Q − Q = Q − Q = O. P♦r ♦✉tr♦ ❧❛❞♦✱ ❝♦♠♦ [m]Q ∈ E(K)✱ ❡♥tã♦

  τ τ

  [m](Q − Q) = ([m]Q) − [m]Q = O.

  τ

  ❆ss✐♠✱ Q − Q é ✉♠ ♣♦♥t♦ ❞❡ m−t♦rçã♦ q✉❡ ♣❡rt❡♥❝❡ ❛♦ ♥ú❝❧❡♦ ❞❛ ❛♣❧✐❝❛çã♦ ❞❡ r❡❞✉çã♦✳ ▼❛s ♣❡❧♦ q✉❡ ✈✐♠♦s ♥♦ ✜♥❛❧ ❞♦ ❈❛♣ít✉❧♦ ✷✱ ❛ ❛♣❧✐❝❛çã♦ ❞❡ r❡❞✉çã♦

  ′ ′

  E(K )[m] E (k ) → ˜ v

  v τ v

  é ✐♥❥❡t✐✈❛✱ ♣♦rt❛♥t♦ Q − Q = O✳ ❉❡ ♠♦❞♦ q✉❡ t♦❞♦ ❡❧❡♠❡♥t♦ ❞❡ I ✜①❛ Q✱ ❡

  ′ ′

  = K(Q) ♣♦rt❛♥t♦ t❡♠♦s q✉❡ K ♥ã♦ s❡ r❛♠✐✜❝❛ s♦❜r❡ K ❡♠ v ✳ ▼♦str❛♥❞♦ q✉❡

  ′

  K /K ♥ã♦ s❡ r❛♠✐✜❝❛ ❢♦r❛ ❞❡ S✱ ✜♥❛❧✐③❛♥❞♦ ❛ ♣r♦✈❛✳

  ❆❣♦r❛ ❝♦♠♦ ú❧t✐♠♦ ♣❛ss♦ ♣❛r❛ ❛ ♣r♦✈❛ ❞❛ ✈❡rsã♦ ❢r❛❝❛✱ t❡♠♦s ♦ s❡❣✉✐♥t❡ r❡s✉❧t❛❞♦✳

  K

  ❚❡♦r❡♠❛ ✸✳✼ ✭❍❡r♠✐t❡✮✳ ❙❡❥❛ S ⊆ M ✜♥✐t♦✳ ❊♥tã♦ ❡①✐st❡ ❛♣❡♥❛s ✉♠ ♥ú♠❡r♦ ✜♥✐t♦ ❞❡

  ′

  /K ❡①t❡♥sõ❡s K ♥ã♦ r❛♠✐✜❝❛❞❛s ❢♦r❛ ❞❡ S ❝♦♠ ❣r❛✉ ❧✐♠✐t❛❞♦✱ ♦✉ s❡❥❛✱

  ′

  : K |K | < n

  ♣❛r❛ ❛❧❣✉♠ n✳ ❉❡♠♦♥str❛çã♦✳ ❱❡r ✭❈❛♣ít✉❧♦ ✺✱ ♣á❣✳ ✶✷✷✱ ❍❡r♠✐t❡✬s ❚❤❡♦r❡♠✱ ❬▲❛♥❣❪✮✳

  ❆❣♦r❛ ✉s❡♠♦s ❡ss❡s ❞♦✐s ú❧t✐♠♦s r❡s✉❧t❛❞♦s ♣❛r❛ ♣r♦✈❛r ❛ ✈❡rsã♦ ❢r❛❝❛ ❞♦ ❚❡♦r❡♠❛ ❞❡ ▼♦r❞❡❧❧✲❲❡✐❧✳ ❚❡♦r❡♠❛ ✸✳✽ ✭▼♦r❞❡❧❧✲❲❡✐❧ ❋r❛❝♦✮✳ ❙❡❥❛ K ✉♠ ❝♦r♣♦ ❞❡ ♥ú♠❡r♦s✱ E/K ✉♠❛ ❝✉r✈❛

E(K)

  ❡❧í♣t✐❝❛ ❞❡✜♥✐❞❛ s♦❜r❡ K ❡ m ✉♠ ✐♥t❡✐r♦ ♠❛✐♦r ♦✉ ✐❣✉❛❧ à ✷✳ ❊♥tã♦ é ✜♥✐t♦✳ mE(K) ❉❡♠♦♥str❛çã♦✳ ❈♦♠♦ ❥á ✈✐♠♦s ❛♥t❡r✐♦r♠❡♥t❡✱ ❞❛❞♦ m > 2✱ é s✉✜❝✐❡♥t❡ ♣r♦✈❛r q✉❡ ❛

  

−1

E(K))

  ❡①t❡♥sã♦ L/K é ✜♥✐t❛✱ ♦♥❞❡ L = K([m] ✱ ❥á q✉❡ ❛ss✐♠ t❡r❡♠♦s q✉❡ ♦ ❣r✉♣♦ E(K)/mE(K) t❛♠❜é♠ s❡rá ✜♥✐t♦✳ ❈♦♥s✐❞❡r❛♥❞♦ ♦ ❝♦♥❥✉♥t♦

  ∞

  S = : ˜ E (k ) : v(m) ,

  v v

  {v ∈ M K é s✐♥❣✉❧❛r} ∪ {v ∈ M K 6= 0} ∪ M K t❡♠♦s q✉❡ ♣❡❧❛ Pr♦♣♦s✐çã♦ ✸✳✻✱ q✉❡ ♣❛r❛ ❝❛❞❛ Q ∈ E(K) t❛❧ q✉❡ [m]Q ∈ E(K)✱ ❛ ❡①t❡♥sã♦

  ′

  K = K(Q) é ♥ã♦ r❛♠✐✜❝❛❞❛ ❢♦r❛ ❞❡ S✳ ▼❛s ❝♦♠♦ ❝❛❞❛ ✉♠ ❞♦s ❝♦♥❥✉❣❛❞♦s ❞❡ ●❛❧♦✐s ❞❡

  ✺✷

  ′

  Q /K sã♦ ❞❛ ❢♦r♠❛ Q + R ♣❛r❛ ❛❧❣✉♠ R ∈ E[m] ⊆ E(K)✱ ❡♥tã♦ ♦ ❣r❛✉ ❞❛ ❡①t❡♥sã♦ K é

  2

  = ♥♦ ♠á①✐♠♦ m |E[m]|✱ ♦✉ s❡❥❛✱

  2 [K(Q) : K] 6 m .

  ❆❣♦r❛✱ ♣❡❧♦ ❚❡♦r❡♠❛ ❞❡ ❍❡r♠✐t❡✱ ❢❛③❡♥❞♦ Q ✈❛r✐❛r ❡♥tr❡ t♦❞♦s ♦s ♣♦♥t♦s ❡♠

  −1 ′

  [m] E(K) = K(Q)

  ✱ ❡①✐st✐rã♦ ❛♣❡♥❛s ✉♠ ♥ú♠❡r♦ ✜♥✐t♦ ❞❡ ❡①t❡♥sõ❡s ❞❛ ❢♦r♠❛ K ✳ ▼❛s

  ′

  ❝♦♠♦ L é ❛ ✉♥✐ã♦ ❞❡ t♦❞♦s ♦s K ✱ t❡♠♦s q✉❡ ❛ ❡①t❡♥sã♦ L/K é ✜♥✐t❛✳ ❋✐♥❛❧✐③❛♥❞♦ ❛ ♣r♦✈❛ ❞❡ss❡ t❡♦r❡♠❛✳

  ✸✳✷ ❖ ❚❡♦r❡♠❛ ❞❛ ❉❡s❝✐❞❛

  Pr♦✈❛r❡♠♦s ❛❣♦r❛ ✉♠ r❡s✉❧t❛❞♦ q✉❡ ❞á ❣❛r❛♥t✐❛s ❞❡ q✉❛♥❞♦ ✉♠ ❣r✉♣♦ ❛❜❡❧✐❛♥♦ é ✜♥✐t❛♠❡♥t❡ ❣❡r❛❞♦✱ ❛tr❛✈és ❞❛ ❡①✐stê♥❝✐❛ ❞❡ ❝❡rt❛s ❢✉♥çõ❡s q✉❡ sã♦ ❝❤❛♠❛❞❛s ❞❡ ❢✉♥çõ❡s ❛❧t✉r❛✳ ▼❛✐s ♣r❡❝✐s❛♠❡♥t❡✱ ♦ ♣ró①✐♠♦ r❡s✉❧t❛❞♦ ❝♦♥❤❡❝✐❞♦ ❝♦♠♦ ❚❡♦r❡♠❛ ❞❛ ❉❡s❝✐❞❛ ❣❛r❛♥t❡ q✉❡ ✉♠ ❣r✉♣♦ ❛❜❡❧✐❛♥♦ é ✜♥✐t❛♠❡♥t❡ ❣❡r❛❞♦ ❞❡s❞❡ q✉❡ ❡①✐st❛ ✉♠❛ ❢✉♥çã♦ ❝♦♠ ❝❡rt❛s ♣r♦♣r✐❡❞❛❞❡s ❞❡ ❧✐♠✐t❛çã♦✳ ❉❡♣♦✐s ❞❡ ♣r♦✈❛❞♦ ❡ss❡ r❡s✉❧t❛❞♦✱ ❢❛r❡♠♦s ✉♠ ❡st✉❞♦ s♦❜r❡ ❢✉♥çõ❡s ❛❧t✉r❛s ♥♦ ❡s♣❛ç♦ ♣r♦❥❡t✐✈♦✱ ♣❛r❛ ❡♥✜♠ ❝♦♥str✉✐r♠♦s ❢✉♥çõ❡s ❛❧t✉r❛ ❡♠ ❝✉r✈❛s ❡❧í♣t✐❝❛s ❛ ✜♠ ❞❡ ♣r♦✈❛r ♦ ❚❡♦r❡♠❛ ❞❡ ▼♦r❞❡❧❧✲❲❡✐❧✳ ❚❡♦r❡♠❛ ✸✳✾ ✭❚❡♦r❡♠❛ ❞❛ ❉❡s❝✐❞❛✮✳ ❙❡❥❛ G ✉♠ ❣r✉♣♦ ❛❜❡❧✐❛♥♦✳ ❙✉♣♦♥❤❛ q✉❡ ❡①✐st❛ ✉♠❛ ❢✉♥çã♦ h : G

  → R q✉❡ s❛t✐s❢❛③ ❛s s❡❣✉✐♥t❡s ♣r♦♣r✐❡❞❛❞❡s✿ ✭✐✮ ❉❛❞♦ Q ∈ G✳ ❊①✐st❡ ✉♠❛ ❝♦♥st❛♥t❡ c

  1 ✱ ❞❡♣❡♥❞❡♥❞♦ ❞❡ G ❡ Q✱ t❛❧ q✉❡

  h(P + Q) 6 2h(P ) + c

  1 , ♣❛r❛ t♦❞♦ P ∈ G.

  2

  ✭✐✐✮ ❊①✐st❡ ✉♠ ✐♥t❡✐r♦ m > 2 ❡ ✉♠❛ ❝♦♥st❛♥t❡ c ✱ ❞❡♣❡♥❞❡♥❞♦ ❞❡ G✱ t❛❧ q✉❡

  2

  h(mP ) > m h(P ) ,

  2 − c ♣❛r❛ t♦❞♦ P ∈ G.

  ✭✐✐✐✮ P❛r❛ q✉❛❧q✉❡r ❝♦♥st❛♥t❡ c

  3 ✱ ♦ ❝♦♥❥✉♥t♦

  3

  {P ∈ G : h(P ) 6 c } é ✜♥✐t♦✳ ❙✉♣♦♥❤❛ ❛✐♥❞❛ q✉❡✱ ♣❛r❛ ♦ ✐♥t❡✐r♦ m ❡♠ (ii)✱ ♦ q✉♦❝✐❡♥t❡ G/mG é ✜♥✐t♦✳ ❊♥tã♦ G

  ✺✸ , . . . , Q

  1 n

  ❉❡♠♦♥str❛çã♦✳ ❙❡❥❛ {Q } ✉♠ ❝♦♥❥✉♥t♦ ❞❡ r❡♣r❡s❡♥t❛♥t❡s ❞♦s ❡❧❡♠❡♥t♦s ❡♠ G/mG✱ ❡ P ∈ G q✉❛❧q✉❡r✳ ❊♥tã♦

  1 i 1 ♣❛r❛ ❛❧❣✉♠ 1 6 i

  1 i

  6 P = mP + Q , n,

  ♦✉ s❡❥❛✱ P ♣❡rt❡♥❝❡ à ❝❧❛ss❡ ❞❡ Q

  1 ❡♠ G/mG✳ , P , . . .

  1

  2

  ❉❡ ❢♦r♠❛ r❡❝✉rs✐✈❛ ❝♦♥str✉✐♠♦s ✉♠❛ s❡q✉ê♥❝✐❛ P ❞❡ ❢♦r♠❛ q✉❡ P = mP + Q .

  k k+1 i k

  • 1

  ❉♦ ✐t❡♠ ✭✐✐✮ t❡♠♦s q✉❡

  

2

  h(mP ) > m h(P )

  

k k

  2

  − c

  1

  1

  k ) 6 (h(mP j ) + c 2 ) = (h(P j−1 i j ) + c 2 ).

  ⇒ h(P − Q

  2

  2

  m m

  1j i j

  ❈♦♥s✐❞❡r❡ c ❛ ❝♦♥st❛♥t❡ ❞❛❞❛ ♥♦ ✐t❡♠ ✭✐✮✱ r❡❧❛t✐✈❛ à −Q ✳ ❚❡♠♦s ❡♥tã♦

  1 h(P ) 6 (2h(P ) + c + c ).

  j j−1 1j

  2

  2

  m

  ′ ′

  = max : 1 6 j 6 n + c

  1j

  2

  ❈♦♥s✐❞❡r❛♥❞♦ c

  1 {c } ❡ c = c 1 ✱ ♦❜t❡♠♦s

  1 h(P ) 6 (2h(P ) + c).

  j j−1

  2

  m ❆♣❧✐❝❛♥❞♦ ✐ss♦ ❞❡ ❢♦r♠❛ r❡❝✉rs✐✈❛ ♦❜t❡♠♦s

  !

  k k−1

  2

  2 2 c h(P ) 6 h(P ) + 1 +

  • k

  · · · + ·

  2

  2

  2

  2

  m m m m

  

k

k

  2

  2

  2 c

  − 1

  m

  = h(P ) + ·

  2

  2

  2

  m

  2 m

  − 1

  m

  1 c

  6 h(P ) +

  k

  2

  2 m − 2 1 c

  6 h(P ) + ,

  k

  2

  2

  2

  1

  6 ♣♦✐s m > 2 ❡ ❛ss✐♠ ✳

  2

  m

  2 P❛r❛ k s✉✜❝✐❡♥t❡♠❡♥t❡ ❣r❛♥❞❡ ♦❜t❡♠♦s c h(P ) 6 1 + .

  k

  2

  ✺✹ , . . . , Q

  k

1 n

  ❈♦♠♦ P é ❝♦♠❜✐♥❛çã♦ ❧✐♥❡❛r ❞❡ P ❡ Q ✱ ♣♦r ✐♥❞✉çã♦ ♠♦str❛♠♦s q✉❡

  k

  X

  k j−1 + P = m P m Q . k i j j=1

  c P❡❧❛ ❝♦♥❞✐çã♦ ✭✐✐✐✮✱ ♦ ❝♦♥❥✉♥t♦ {Q ∈ G : h(Q) 6 1 + } é ✜♥✐t♦✱ s✉♣♦♥❤❛♠♦s q✉❡

  2 , . . . , Q

  n+1 r i

  s❡❥❛ {Q }. ❉❛í✱ t♦❞♦ ❡❧❡♠❡♥t♦ P ∈ G s❡rá ❝♦♠❜✐♥❛çã♦ ❧✐♥❡❛r ❞❡ss❡s Q ✱ ♣❛r❛ i = 1, . . . , r ✳ ▼♦str❛♥❞♦ q✉❡ G é ✜♥✐t❛♠❡♥t❡ ❣❡r❛❞♦✳

  ✸✳✸ ❆❧t✉r❛s ♥♦ ❊s♣❛ç♦ Pr♦❥❡t✐✈♦

  ❊st❛r❡♠♦s ✐♥t❡r❡ss❛❞♦s ❡♠ ✉s❛r ♦ ❚❡♦r❡♠❛ ❞❛ ❉❡s❝✐❞❛ ♣❛r❛ ♣r♦✈❛r q✉❡ ♦ ❣r✉♣♦

E(K)

  é ✜♥✐t❛♠❡♥t❡ ❣❡r❛❞♦✳ P❛r❛ ✐ss♦ ❢❛r❡♠♦s ✉♠ ❡st✉❞♦ ❞❡ ❢✉♥çõ❡s ❛❧t✉r❛ ♥♦ ❡s♣❛ç♦ ♣r♦❥❡t✐✈♦✱ ♣❛r❛ ❡♥t❡♥❞❡r ❝♦♠♦ ♣♦❞❡♠♦s r❡str✐♥❣✐r t❛✐s ❢✉♥çõ❡s ❛♦s ♣♦♥t♦s ❞❡ ✉♠❛ ❝✉r✈❛ ❡❧í♣t✐❝❛✳

  ❆♣r❡s❡♥t❛r❡♠♦s ✉♠ ❡①❡♠♣❧♦ ❞❡ ❢✉♥çã♦ ❛❧t✉r❛ ♥♦ ❡s♣❛ç♦ ♣r♦❥❡t✐✈♦ s♦❜r❡ ♦s r❛❝✐✲ ♦♥❛✐s✱ ❡ ♥❡ss❡ ♣r♦❝❡ss♦ ✈❡r❡♠♦s ❛ ✐♠♣♦rtâ♥❝✐❛ ❞♦ s❡✉ ❛♥❡❧ ❞❡ ✐♥t❡✐r♦s✱ ♥♦ ❝❛s♦ Z✱ s❡r ✉♠ ❞♦♠í♥✐♦ ❞❡ ✐❞❡❛✐s ♣r✐♥❝✐♣❛✐s✱ ♦ q✉❡ ♥ã♦ ♥❡❝❡ss❛r✐❛♠❡♥t❡ ♦❝♦rr❡ ♣❛r❛ ❝♦r♣♦s ❞❡ ♥ú♠❡r♦s ❛r❜✐trár✐♦s✳

  n

  (Q) ❊①❡♠♣❧♦ ✸✳✶✵✳ ❙❡♥❞♦ P ∈ P ✉♠ ♣♦♥t♦ ❝♦♠ ❝♦♦r❞❡♥❛❞❛s r❛❝✐♦♥❛✐s✳ ❱✐st♦ q✉❡ Z é ✉♠ ❞♦♠í♥✐♦ ❞❡ ✐❞❡❛✐s ♣r✐♥❝✐♣❛✐s✱ ♣♦❞❡♠♦s ❡♥❝♦♥tr❛r ❝♦♦r❞❡♥❛❞❛s ❤♦♠♦❣ê♥❡❛s ♣❛r❛ P ❞❡ t❛❧ ❢♦r♠❛ q✉❡ ❛♠❜❛s ❡st❡❥❛♠ ❡♠ Z ❡ s❡❥❛♠ ♣r✐♠❛s ❡♥tr❡ s✐✱ ♦✉ s❡❥❛✱

  P = [x , . . . , x ],

  n

  , . . . , x , . . . , x ) = 1

  n n

  ❝♦♠ x ∈ Z ❡ mdc(x ✳ ❉❡st❡ ♠♦❞♦ ❞❡✜♥✐♠♦s ❛ ❛❧t✉r❛ ♥❛t✉r❛❧ ❞❡ P ❝♦♠♦ s❡♥❞♦

  H(P ) = max

  n {|x |, . . . , |x |}. n

  (Q) : P❛r❛ ❡st❛ ❞❡✜♥✐çã♦✱ ❝❧❛r❛♠❡♥t❡ t❡r❡♠♦s q✉❡ ❞❛❞♦ c ❝♦♥st❛♥t❡✱ ♦ ❝♦♥❥✉♥t♦ {P ∈ P

  n

  H(P ) 6 c ❡❧❡♠❡♥t♦s✳

  } é ✜♥✐t♦✱ ❡ t❡♠ ♥♦ ♠á①✐♠♦ (2c + 1) ❆❣♦r❛ ❝♦♠❡ç❛♠♦s ♦ ♣r♦❝❡❞✐♠❡♥t♦ ♣❛r❛ ♦❜t❡r ❢✉♥çõ❡s ❛❧t✉r❛ ♥♦ ❡s♣❛ç♦ ♣r♦❥❡t✐✈♦ s♦❜r❡ ❝♦r♣♦s ❞❡ ♥ú♠❡r♦s ❛r❜✐trár✐♦s✱ ♣❛r❛ ✐ss♦ r❡❧❡♠❜r❡♠♦s ❛ s❡❣✉✐♥t❡ ❞❡✜♥✐çã♦✳

  ❉❡✜♥✐çã♦ ✸✳✶✶✳ ❉❛❞♦ K ✉♠ ❝♦r♣♦ ❞❡ ♥ú♠❡r♦s✱ s❡✉ ❝♦♥❥✉♥t♦ ❞❡ ✈❛❧♦r❡s ❛❜s♦❧✉t♦s

  K

  st❛♥❞❛r❞✱ ❞❡♥♦t❛❞♦ ♣♦r M ✱ é ♦ ❝♦♥❥✉♥t♦ ❞❡ s❡✉s ✈❛❧♦r❡s ❛❜s♦❧✉t♦s q✉❡ q✉❛♥❞♦ r❡str✐t♦ à Q ❝♦✐♥❝✐❞❡ ❝♦♠ ✉♠ ❞♦s ✈❛❧♦r❡s ❛❜s♦❧✉t♦s st❛♥❞❛r❞ ❞❡ Q✱ ✐st♦ é✱ ♦ ✈❛❧♦r ❛❜s♦❧✉t♦ r❡❛❧ ♦✉ ♦ ✈❛❧♦r ❛❜s♦❧✉t♦ p✲á❞✐❝♦ ♣❛r❛ ❛❧❣✉♠ p ♣r✐♠♦✳

  ✺✺

  K v

  ❉❡✜♥✐çã♦ ✸✳✶✷✳ ❙❡❥❛ v ∈ M ✳ ❖ ❣r❛✉ ❧♦❝❛❧ ❡♠ v✱ ❞❡♥♦t❛❞♦ ♣♦r n é n = [K : Q ],

  

v v v

v v

  ♦♥❞❡ K ❡ Q sã♦ ♦s r❡s♣❡❝t✐✈♦s ❝♦♠♣❧❡t❛♠❡♥t♦s ❞❡ K ❡ Q ❝♦♠ r❡s♣❡✐t♦ ❛ v✳ ❆ s❡❣✉✐r ❡♥✉♥❝✐❛♠♦s ❞♦✐s r❡s✉❧t❛❞♦s ❞❛ ❚❡♦r✐❛ ❆❧❣é❜r✐❝❛ ❞♦s ◆ú♠❡r♦s ❝♦♠ ❛s

  ❞❡✈✐❞❛s r❡❢❡rê♥❝✐❛s✳ ❚❡♦r❡♠❛ ✸✳✶✸ ✭❋ór♠✉❧❛ ❞❡ ❊①t❡♥sã♦✮✳ ❙❡❥❛ Q ⊆ K ⊆ L ✉♠❛ t♦rr❡ ❞❡ ❝♦r♣♦s ❞❡ ♥ú♠❡✲

  K

  r♦s✱ ❡ s❡❥❛ v ∈ M ✳ ❊♥tã♦

  X n = [L : K] .

  w · n v w∈M L , w|v

  ❖♥❞❡ w|v ❞❡♥♦t❛ q✉❡ ❛ r❡str✐çã♦ ❞❡ w ❛ K ❝♦✐♥❝✐❞❡ ❝♦♠ v ❉❡♠♦♥str❛çã♦✳ ❱❡r ✭❈❛♣ít✉❧♦ ✷✱ ❈♦r♦❧ár✐♦ ✶✱ ❬▲❛♥❣❪✮✳

  ∗

  ❚❡♦r❡♠❛ ✸✳✶✹ ✭❋ór♠✉❧❛ ❞♦ Pr♦❞✉t♦✮✳ ❙❡❥❛ x ∈ K ✳ ❊♥tã♦ Y

  v n = 1.

  |x|

  v v∈M K

  ❉❡♠♦♥str❛çã♦✳ ❱❡r ✭❈❛♣ít✉❧♦ ✷✱ ❈♦r♦❧ár✐♦ ✷✱ ❬▲❛♥❣❪✮✳

  n

  , . . . , x ] (K)

  n

  ❉❡✜♥✐çã♦ ✸✳✶✺✳ ❙❡❥❛ P = [x ∈ P ✱ ❛ ❛❧t✉r❛ ❞❡ P ✭❡♠ r❡❧❛çã♦ à K✮ é ❞❡✜♥✐❞❛ ❝♦♠♦

  Y

  n v H (P ) = max , . . . , .

K v n v

  {|x | |x | }

  v∈M K n

  (K) Pr♦♣♦s✐çã♦ ✸✳✶✻✳ ❙❡❥❛ P ∈ P ✳

  (P )

  K

  ✭❛✮ ❆ ❛❧t✉r❛ H ❡stá ❜❡♠ ❞❡✜♥✐❞❛✱ ♦✉ s❡❥❛✱ ♥ã♦ ❞❡♣❡♥❞❡ ❞❛ ❡s❝♦❧❤❛ ❞❛s ❝♦♦r❞❡♥❛❞❛s ❤♦♠♦❣ê♥❡❛s ❡s❝♦❧❤✐❞❛s✳

  n K (P ) > 1 (K)

  ✭❜✮ ❆ ❛❧t✉r❛ s❛t✐s❢❛③ H ✱ ♣❛r❛ t♦❞♦ P ∈ P ✳ ✭❝✮ ❙❡❥❛ L ✉♠❛ ❡①t❡♥sã♦ ✜♥✐t❛ ❞❡ K✳ ❊♥tã♦

  [L:K] H (P ) = H (P ) .

L K

  ❉❡♠♦♥str❛çã♦✳ ✭❛✮ ❚❡♠♦s q✉❡ ❛s ♦✉tr❛s ❝♦♦r❞❡♥❛❞❛s ❤♦♠♦❣ê♥❡❛s ♣❛r❛ ♦ ♣♦♥t♦ P sã♦ ❞❛ ❢♦r♠❛

  ∗ [λx , . . . , λx ], . n

  ✺✻ ❆ss✐♠✱ ♣❡❧❛ ❢ór♠✉❧❛ ❞♦ ♣r♦❞✉t♦ ♦❜t❡♠♦s

  Y Y

  n v n v n v

  max , . . . , = max , . . . ,

  v n v v n v

  {|λx | |λx | } |λ| v {|x | |x | }

  v∈M K v∈M K

  Y Y

  n v n v

  = max , . . . ,

  v n v

  |λ| v {|x | |x | }

  v∈M K v∈M K

  Y

  n v = max , . . . , . v n v

  {|x | |x | }

  

v∈M K

  ✭❜✮ ❉❛❞♦ P ♥♦ ❡s♣❛ç♦ ♣r♦❥❡t✐✈♦✱ s❡♠♣r❡ ♣♦❞❡♠♦s ❡♥❝♦♥tr❛r ❝♦♦r❞❡♥❛❞❛s ❤♦♠♦❣ê♥❡❛s ❡♠ = 1

  i i v

  q✉❡ ❛♦ ♠❡♥♦s ✉♠❛ ❞❡ss❛s ❝♦♦r❞❡♥❛❞❛s s❡❥❛ ✶✱ s❡♥❞♦ ❡ss❛ ❝♦♦r❞❡♥❛❞❛ x t❡♠♦s |x |

  K

  ♣❛r❛ t♦❞♦ ✈❛❧♦r ❛❜s♦❧✉t♦ ♥♦r♠❛❧✐③❛❞♦ v ∈ M ✳ ❆ss✐♠✱ t♦❞♦s ♦s ❢❛t♦r❡s ❞♦ ♣r♦❞✉t♦ ❡♠ H (P ) (P ) > 1

  K K

  sã♦ ♣❡❧♦ ♠❡♥♦s ✶✱ ❞❡ ♠♦❞♦ q✉❡ H ✳ ✭❝✮ P❡❧❛ ❢ór♠✉❧❛ ❞❡ ❡①t❡♥sã♦ t❡♠♦s ♦ s❡❣✉✐♥t❡

  Y

  n w

  H (P ) = max , . . . ,

  L w n w

  {|x | |x | }

  w∈M L

  Y Y

  n v

  = max , . . . ,

  v n v ✈✐st♦ q✉❡x i

  {|x | |x | } ∈ K

  v∈M K w∈M , w|v L

  Y

  [L:K]n v

  = max , . . . ,

  v n v

  {|x | |x | }

  v∈M K [L:K]

  = H (P ) .

  K

  ❖❜s❡r✈❛çã♦ ✸✳✶✼✳ ❖❜s❡r✈❡ q✉❡ s❡ K = Q✱ ❡♥tã♦ H Q ❝♦✐♥❝✐❞❡ ❝♦♠ ❛ ❛❧t✉r❛ ❞❡✜♥✐❞❛ ♥♦

  n

  , . . . , x ] (Q) , . . . , x ❡①❡♠♣❧♦ ❛♥t❡r✐♦r✳ ❈♦♠ ❡❢❡✐t♦✱ s❡ t✐✈❡r♠♦s P = [x

  1 ❝♦♠ x n

  ∈ P ∈ Z , . . . , x n ) = 1

  p (x i ) > 0

  ❡ mdc(x ✱ ❡♥tã♦ ♣❛r❛ q✉❛❧q✉❡r ♣r✐♠♦ p t❡r❡♠♦s ord ❡ ❛♦ ♠❡♥♦s ✉♠

  6

  i p (x i ) = 0 i i v

  1 ❞♦s x t❡rá ord ✭♣♦✐s ♦s x sã♦ ♣r✐♠♦s ❡♥tr❡ s✐✮✳ ❆ss✐♠✱ |x | ♣❛r❛ t♦❞♦ i✱ ❡

  i v = 1 Q (P )

  |x | ♣❛r❛ ❛❧❣✉♠ i✱ ✐ss♦ ♣❛r❛ t♦❞♦ ✈❛❧♦r ❛❜s♦❧✉t♦ ♥ã♦ ❛rq✉✐♠❡❞✐❛♥♦✳ ❊ ❛ss✐♠✱ H só ❞❡♣❡♥❞❡ ❞♦ ✈❛❧♦r ❛❜s♦❧✉t♦ r❡❛❧✱ q✉❡ é ♦ q✉❡ ❞❡✜♥❡ ❛ ❢✉♥çã♦ ❛❧t✉r❛ ❞♦ ❡①❡♠♣❧♦ ❛♥t❡r✐♦r✳

  n

  (Q) : H (P ) 6 c

  Q

  ❊♠ ♣❛rt✐❝✉❧❛r✱ ♦ ❝♦♥❥✉♥t♦ {P ∈ P } é ✜♥✐t♦✱ ♣❛r❛ t♦❞♦ c ∈ R✳

  K

  ❖ ♦❜❥❡t✐✈♦ ❛❣♦r❛ é t❡♥t❛r ❡st❡♥❞❡r ❡ss❛ ♣r♦♣r✐❡❞❛❞❡ ♣❛r❛ t♦❞❛ H ✳

  n

  (Q) ❉❡✜♥✐çã♦ ✸✳✶✽✳ P❛r❛ P ∈ P ✱ ❛ ❛❧t✉r❛ ❛❜s♦❧✉t❛ ❞❡ P ✱ ❞❡♥♦t❛❞❛ ♣♦r H(P )✱ é

  n

  (K) ❞❡✜♥✐❞❛ ❝♦♠♦ s❡❣✉❡✿ ❙❡❥❛ K ✉♠ ❝♦r♣♦ ❞❡ ♥ú♠❡r♦s ♣❛r❛ ♦ q✉❛❧ P ∈ P ✱ ❡♥tã♦

  1/[K:Q]

  H(P ) = H (P ) ,

  

K

  ♦♥❞❡ ❝♦♥s✐❞❡r❛♠♦s ❛ r❛✐③ r❡❛❧ ♣♦s✐t✐✈❛✳

  ✺✼ P♦❞❡♠♦s ✈❡r q✉❡ ♣❡❧♦ ✐t❡♠ (c) ❞❛ ♣r♦♣♦s✐çã♦ ❛♥t❡r✐♦r✱ ❡ss❛ ❢✉♥çã♦ ❡stá ❜❡♠ ❞❡✲

  1 [L:Q]

  (P ) =

  L

  ✜♥✐❞❛✳ ❈♦♠ ❡❢❡✐t♦✱ s❡ ❡s❝♦❧❤❡r♠♦s ♦✉tr♦ ❝♦r♣♦ ❞❡ ♥ú♠❡r♦s L/K✱ ❡♥tã♦ H

  [L:K]

  1 [L:Q] [K:Q]

  (H (P )) = H (P )

  K K ✳ ❆❣♦r❛✱ ♣❡❧♦ ✐t❡♠ (b) ❞❛ ♠❡s♠❛ ♣r♦♣♦s✐çã♦✱ t❡♠♦s q✉❡ n

  H(P ) > 1 (Q).

  ✱ ♣❛r❛ t♦❞♦ P ∈ P ❉❡✜♥✐çã♦ ✸✳✶✾✳ ❯♠ ♠♦r✜s♠♦ ❞❡ ❣r❛✉ d ❡♥tr❡ ❡s♣❛ç♦s ♣r♦❥❡t✐✈♦s é ✉♠❛ ❛♣❧✐❝❛çã♦

  n m

  F : P → P

  P (P ), . . . , f (P )],

  m

  7→ [f , . . . , f , . . . , x ]

  m n

  ♦♥❞❡ f ∈ Q[x sã♦ ♣♦❧✐♥ô♠✐♦s ❤♦♠♦❣ê♥❡♦s ❞❡ ❣r❛✉ d s❡♠ ♥❡♥❤✉♠ ③❡r♦ ♥ã♦ tr✐✈✐❛❧ ❡♠ ❝♦♠✉♠ ❡♠ Q✳ ❉✐r❡♠♦s q✉❡ F ❡stá ❞❡✜♥✐❞♦ s♦❜r❡ K s❡ ♣✉❞❡r♠♦s ❡♥❝♦♥tr❛r ♣♦❧✐♥ô♠✐♦s q✉❡ ❞❡✜♥❛♠ F ❝✉❥♦s ❝♦❡✜❝✐❡♥t❡s ❡st❡❥❛♠ ❡♠ K✳

  n m

  ❚❡♦r❡♠❛ ✸✳✷✵✳ ❙❡❥❛ F : P ✉♠ ♠♦r✜s♠♦ ❞❡ ❣r❛✉ d✳ ❊♥tã♦ ❡①✐st❡♠ ❝♦♥st❛♥t❡s → P

  ♣♦s✐t✐✈❛s c

  1 ❡ c 2 q✉❡ ❞❡♣❡♥❞❡♠ ❞❡ F ✱ t❛✐s q✉❡ d d n

1 H(P ) H(F (P )) 6 c 2 H(P ) , (Q).

  6 c

  ∀P ∈ P , . . . , f ] , . . . , f

  m m

  ❉❡♠♦♥str❛çã♦✳ ❙❡❥❛ F = [f ♠♦r✜s♠♦ ❞❡ ❣r❛✉ d✱ ❡♠ q✉❡ f ♥ã♦ t❡♠

  n , . . . , x ] (Q). n

  ③❡r♦s ♥ã♦ tr✐✈✐❛✐s ❡♠ ❝♦♠✉♠✱ ❡ s❡❥❛ P = [x ∈ P ❊s❝♦❧❤❛ K ✉♠ ❝♦r♣♦ ❞❡ , . . . , x

  n

  ♥ú♠❡r♦s q✉❡ ❝♦♥t❡♥❤❛ ❛s ❝♦♦r❞❡♥❛❞❛s x ❡ t❛♠❜é♠ ♦s ❝♦❡✜❝✐❡♥t❡s ❞♦s ♣♦❧✐♥ô♠✐♦s f

i K

  ✳ P❛r❛ ❝❛❞❛ ✈❛❧♦r ❛❜s♦❧✉t♦ v ∈ M ❞❡✜♥❛ = max , . . . , = max (P ) , . . . , (P )

  

v v n v v v m v

  |P | {|x | |x | } ❡ |F (P )| {|f | |f | }, ❡

  

v = max v : a i

|F | {|a| é ❝♦❡✜❝✐❡♥t❡ ❞❡ ❛❧❣✉♠ f }.

  ❆ss✐♠✱ ♣❡❧❛ ❞❡✜♥✐çã♦ ❞❡ ❛❧t✉r❛ t❡r❡♠♦s Y Y

  n v n v

  H (P ) = (F (P )) = ,

  K K

  |P | v ❡ H |F (P )| v

  

v∈M v∈M

K K

  Q

  n v

  (F ) = ❡ ❞❡✜♥✐♠♦s H K |F | ✳

  v∈M K v ∞

  ❉❡✜♥❛ t❛♠❜é♠ ❛ ❢✉♥çã♦ ǫ(v) q✉❡ é 1 s❡ v ∈ M ❡ é 0 ❝❛s♦ v ∈ M ✳ ❉❡ ♠♦❞♦

  K K v

  q✉❡ ❞❛ ❞❡s✐❣✉❛❧❞❛❞❡ tr✐❛♥❣✉❧❛r ♣❛r❛ ♦ ✈❛❧♦r ❛❜s♦❧✉t♦ | · | t❡♠♦s

  ǫ(v)

  • max

  1 n v 1 v n v

  |t · · · + t | {|t | |t | },

  6 n , . . . ,

  K

  ♣❛r❛ q✉❛❧q✉❡r q✉❡ s❡❥❛ v ∈ M ✳

  ✺✽

  K

  ❙❡❥❛ v ∈ M ✱ ♣❡❧❛ ❞❡s✐❣✉❛❧❞❛❞❡ ❛❝✐♠❛ t❡♠♦s

  ǫ(v) d

  6 (P ) c ,

  i v v

  |f | |F | · |P | v ♣❛r❛ ❛❧❣✉♠❛ ❝♦♥st❛♥t❡ c ♣♦s✐t✐✈❛ ✭♣♦❞❡♠♦s ❡s❝♦❧❤❡r c ❝♦♠♦ ♦ ♥ú♠❡r♦ ❞❡ t❡r♠♦s ❞♦ ♣♦✲

  i i

  ❧✐♥ô♠✐♦ f ✮✱ ✈✐st♦ q✉❡ ♦s ♣♦❧✐♥ô♠✐♦s f sã♦ ❤♦♠♦❣ê♥❡♦s ❞❡ ❣r❛✉ d✱ ♣♦❞❡♥❞♦ s❡r ❡s❝r✐t♦s ♥❛ ❢♦r♠❛

  n n

  Y Y

  d d j

  1 jc

  f = a x + x .

  i

1 c

j · · · + a j j=0 j=0

  P♦❞❡♠♦s t♦♠❛r ✉♠ c ❛❞❡q✉❛❞♦ ♣❛r❛ q✉❡ ✈❛❧❤❛ ♣❛r❛ t♦❞♦s ♦s i✳

  ǫ(v) d

  6 c . ❆ss✐♠✱ |F (P )| v |F | v · |P | P♦rt❛♥t♦

  v

  Y Y Y Y

  v v v v n ǫ(v)n n dn

  6 c . |F (P )| v · |F | v · |P | v

  v∈M K v∈M K v∈M K v∈M K

  ▼❛s ❝♦♠♦ ǫ(v) = 0 ♣❛r❛ t♦❞♦ ✈❛❧♦r ❛❜s♦❧✉t♦ ♥ã♦ ❛rq✉✐♠❡❞✐❛♥♦✱ ❡♥tã♦ Y Y

  ǫ(v)n v n v [K:Q] c = c = c . v∈M K v∈M

  K

  ❊ t♦♠❛♥❞♦ ❛ r❛í③ [K : Q]✲és✐♠❛✱ ♦❜t❡♠♦s

  1

  !

  [K:Q]

  Y

  

n v

  H(F (P )) = |F (P )| v

  v∈M K

  1

  1

  ! !

  [K:Q] [K:Q]

  Y Y

  v v n dn

  6 c · |F | v · |P | v

  v∈M K v∈M K d

  = c .

  · H(F ) · H(P ) = c

  2 ❇❛st❛ t♦♠❛r c · H(F ).

  ❆❣♦r❛ ✐r❡♠♦s ♠♦str❛r ❛ ❝♦t❛ ✐♥❢❡r✐♦r✳

  n+1

  (Q) : f (Q) = (Q) = 0

  m

  ❚❡♠♦s q✉❡ {Q ∈ A · · · = f } = {(0, . . . , 0)}✳ P❡❧♦ ❚❡♦r❡♠❛ ❞♦s ❩❡r♦s ❞❡ ❍✐❧❜❡rt✱ ♦ ✐❞❡❛❧

  I = (f , . . . , f ) , . . . , x ]

  m n

  ⊆ Q[x

  d i

  ❝♦♥té♠ ❛❧❣✉♠❛ ♣♦tê♥❝✐❛ x ♣❛r❛ i = 0, . . . , n✱ ♣♦✐s

  i

  √ x I = I(V (I)).

  i

  ∈

  ✺✾ , . . . , x ]

  ij n

  ❆ss✐♠✱ ❡①✐st❡♠ ♣♦❧✐♥ô♠✐♦s g ∈ Q[x ❡ ✉♠ ✐♥t❡✐r♦ e > 1 t❛✐s q✉❡

  

m

  X

  e

  x = g f ,

  ij j i

j=0

  , . . . , x ] ♣❛r❛ ❝❛❞❛ i = 0, . . . , n✳ ❊ ♣♦❞❡♠♦s ❛ss✉♠✐r q✉❡ g ∈ K[x sã♦ t♦❞♦s ❞❡ ❣r❛✉

  e

  e

  j

  − d✱ ❥á q✉❡ x i é ❤♦♠♦❣ê♥❡♦ ❞❡ ❣r❛✉ e ❡ t♦❞♦s ♦s f sã♦ ❤♦♠♦❣ê♥❡♦s ❞❡ ❣r❛✉ d✳ ❆❞♦t❛♥❞♦ ❛s s❡❣✉✐♥t❡s ♥♦t❛çõ❡s

  Y

  n v

  = max : b (G) = ,

  v v é ❝♦❡✜❝✐❡♥t❡ ❞❡ ❛❧❣✉♠ g ij K

  |G| {|b| } ❡ H |G| v

  v∈M K

  (G)

  K

  t❡♠♦s q✉❡ t❛♥t♦ e q✉❛♥t♦ H ♥ã♦ ❞❡♣❡♥❞❡♠ ❞❡ P ✳ , . . . , x ]

  n

  ❙❡♥❞♦ P = [x ✱ t❡♠♦s q✉❡

  m

  X

  e ǫ(v)

  6 = g (P )f (P ) D max (P )f (P ) ; 0 6 j 6 m

  i ij j v ij j v

  |x | v | | {|g | }

  j=0 ǫ(v)

6 D max ij (P ) v j (P ) v ; 0 6 j 6 m

  {|g | |f | }, f

  ij j

  ♦♥❞❡ D só ❞❡♣❡♥❞❡ ❞♦s ♣♦❧✐♥ô♠✐♦s g ✳ ❆❣♦r❛ ❝♦♥s✐❞❡r❛♥❞♦ ♦ ♠á①✐♠♦ s♦❜r❡ t♦❞♦s ♦s i✱ t❡♠♦s

  e ǫ(v)

6 D max (P ) .

  ij v v

  |P | v |g | |F (P )|

  i,j ij

  ❈♦♠♦ ❝❛❞❛ ✉♠ ❞♦s g é ❤♦♠♦❣ê♥❡♦ ❞❡ ❣r❛✉ e − d✱ ❡♥tã♦ ❝♦♠♦ ❝♦♥s❡q✉ê♥❝✐❛ ❞❛ ❞❡s✐❣✉❛❧✲ ❞❛❞❡ tr✐❛♥❣✉❧❛r

  

ǫ(v)

e−d

  ij v v

6 D ,

  |g |

  1 |G| |P | v

  1

  ❡♠ q✉❡ ❛ ❝♦♥st❛♥t❡ D t❛♠❜é♠ ♥ã♦ ❞❡♣❡♥❞❡ ❞❡ P ✳ P♦rt❛♥t♦✱

  d ǫ(v)

  6 (DD ) . |P |

  1 |G| v |F (P )| v v K

  ◆♦✈❛♠❡♥t❡ t♦♠❛♥❞♦ ♦ ♣r♦❞✉t♦ s♦❜r❡ t♦❞♦s v ∈ M ❡ t♦♠❛♥❞♦ ❛ r❛í③ [K : Q]✲és✐♠❛✱ ♦❜t❡♠♦s

  d

  

2

6 H(P ) D

  · H(F (P )), = D = 1/D

  2

  1

  1

  2

  ♦♥❞❡ D · D · H(G). ❈♦♥s✐❞❡r❡ c ✳ (Q)

  n+1

  ❈♦r♦❧ár✐♦ ✸✳✷✶✳ ❙❡❥❛ A ∈ GL ✳ ❊♥tã♦ ❛ ♠✉❧t✐♣❧✐❝❛çã♦ ♣♦r A ✐♥❞✉③ ✉♠ ❛✉t♦♠♦r✲

  n n

  , c

  1

  2

  ✜s♠♦ A : P → P ✳ ❊♥tã♦ ❡①✐st❡♠ ❝♦♥st❛♥t❡s c q✉❡ ❞❡♣❡♥❞❡♠ ❞❛s ❡♥tr❛❞❛s ❞❡ A t❛✐s q✉❡ c H(P ) 6 H(A(P )) 6 c H(P ),

  1

  2

  ✻✵

  n (Q).

  ♣❛r❛ t♦❞♦ P ∈ P ❉❡♠♦♥str❛çã♦✳ ❈♦♠♦ A é ❧✐♥❡❛r✱ ❡❧❛ ✐♥❞✉③ ✉♠ ♠♦r✜s♠♦ ❞❡ ❣r❛✉ ✶✱ ❜❛st❛♥❞♦ ❛♣❧✐❝❛r ♦ r❡s✉❧t❛❞♦ ❞♦ t❡♦r❡♠❛ ❛♥t❡r✐♦r✳

  ❆❣♦r❛ ✈❛♠♦s tr❛t❛r ❞❡ ❡♥t❡♥❞❡r ❛ r❡❧❛çã♦ ❡♥tr❡ ♦s ❝♦❡✜❝✐❡♥t❡s ❞❡ ✉♠ ♣♦❧✐♥ô♠✐♦ ❡ ❛ ❛❧t✉r❛ ❞❡ s✉❛s r❛í③❡s✳ ❙♦❜r❡ ✐ss♦✱ t❡♠♦s ♦ s❡❣✉✐♥t❡ r❡s✉❧t❛❞♦✳

  d

  • t d = a (t

  1 ) d )

  ❚❡♦r❡♠❛ ✸✳✷✷✳ ❙❡❥❛ f(t) = a · · · + a − α · · · (t − α ∈ Q[t] ✉♠ ♣♦❧✐♥ô♠✐♦ ❞❡ ❣r❛✉ d✳ ❊♥tã♦

  d d

  Y Y

  −d d−1 2 H(α ) 6 H([a , . . . , a ]) 6 2 H(α ). j d j j=1 j=1

  ❉❡♠♦♥str❛çã♦✳ ■♥✐❝✐❛❧♠❡♥t❡ ♦❜s❡r✈❡♠♦s q✉❡ ❛ ❞❡s✐❣✉❛❧❞❛❞❡ ♥ã♦ s❡ ❛❧t❡r❛ ❝❛s♦ ♠✉❧t✐♣❧✐✲ q✉❡♠♦s f ♣♦r ✉♠❛ ❝♦♥st❛♥t❡ ♥ã♦ ♥✉❧❛✳ ❆ss✐♠✱ s❡♠ ♣❡r❞❛ ❞❡ ❣❡♥❡r❛❧✐❞❛❞❡ ♣♦❞❡♠♦s = 1

  ❛ss✉♠✐r q✉❡ a ✱ ❢❛③❡♥❞♦ ❞❡ f ✉♠ ♣♦❧✐♥ô♠✐♦ ♠ô♥✐❝♦✳ , . . . , α )

  ❙❡❥❛ K = Q(α

  1 d ✱ ❡ ♣❛r❛ v ∈ M K ✱ ❞❡♥♦t❡♠♦s ♣♦r ǫ(v) = 2 s❡ v ❢♦r ❛rq✉✐✲

  ♠❡❞✐❛♥♦ ❡ ǫ(v) = 1 ❝❛s♦ v s❡❥❛ ♥ã♦ ❛rq✉✐♠❡❞✐❛♥♦✳ ❖❜s❡r✈❛♠♦s q✉❡ ❡ss❛ ❢✉♥çã♦ ǫ ♥ã♦ é ❛ ♠❡s♠❛ q✉❡ ❞❡✜♥✐♠♦s ❛♥t❡r✐♦r♠❡♥t❡✱ q✉❡

  ❛ss♦❝✐❛✈❛ ✶ ❛ ✈❛❧♦r❡s ❛❜s♦❧✉t♦s ❛rq✉✐♠❡❞✐❛♥♦ ❡ ✵ ❛ ♥ã♦ ❛rq✉✐♠❡❞✐❛♥♦s✳ ❈♦♠ ❡st❛ ♥♦t❛çã♦✱ t❡♠♦s q✉❡

  6 ǫ(v) max , , x, y

  v v v K |x + y| {|x| |y| }, ♣❛r❛ v ∈ M ∈ K.

  ❆❣♦r❛ ♦❜s❡r✈❡ q✉❡ s❡ v ∈ M ❡ |x| v v ✱ ❡♥tã♦ ❛ ❞❡s✐❣✉❛❧❞❛❞❡ s❡ t♦r♥❛ ✉♠❛ ✐❣✉❛❧❞❛❞❡

  K 6= |y|

  ✭✐ss♦ é ❝♦♥s❡q✉ê♥❝✐❛ ❞❛ ❞❡s✐❣✉❛❧❞❛❞❡ tr✐❛♥❣✉❧❛r ❢♦rt❡✮✳ Pr♦✈❛r❡♠♦s ❛ s❡❣✉✐♥t❡ ❞❡s✐❣✉❛❧❞❛❞❡✱ ✉s❛♥❞♦ ✐♥❞✉çã♦ ♥♦ ❣r❛✉ ❞♦ ♣♦❧✐♥ô♠✐♦ f

  d d

  Y Y

  −d d−1

  ǫ(v) max j v , 1 j v max j v , 1 {|α | } 6 max {|a | } 6 ǫ(v) {|α | }.

  06j6d j=1 j=1

  1 P❛r❛ d = 1 t❡♠♦s q✉❡ f(t) = t − α ✱ ❡ ♣❛r❛ ❡ss❡ ❝❛s♦ ❛ ❞❡s✐❣✉❛❧❞❛❞❡ é ❝❧❛r❛✱ ✈✐st♦ q✉❡ −1 −1

  2

  1 v , 1 1 v , 1

  · max{|α | } 6 1 · max{|α | } = max

  1 v , 1

  {|α | } = 2 max , 1

  1 v

  {|α | } = 1 max , 1

  1 v {|α | }.

  ✻✶ ❆❣♦r❛✱ ♣♦r ❤✐♣ót❡s❡ ❞❡ ✐♥❞✉çã♦✱ s✉♣♦♥❤❛♠♦s q✉❡ ❛ ❞❡s✐❣✉❛❧❞❛❞❡ é ✈á❧✐❞❛ ♣❛r❛ ♣♦❧✐♥ô♠✐♦s ❞❡ ❣r❛✉ d − 1✱ ❝♦♠ r❛í③❡s ❡♠ K✱ ❡ s❡❥❛ f ♥❛s ❤✐♣ót❡s❡s✳ ❙❡❥❛ k ✐♥❞í❝❡ t❛❧ q✉❡

  > , |α k | v |α i | v ∀i = 1, . . . , d.

  ❈♦♥s✐❞❡r❡♠♦s t❛♠❜é♠ ♦ ♣♦❧✐♥ô♠✐♦

  d−1 g(t) = (t ) ) ) ) = b + t . 1 k−1 k+1 d d−1

  − α · · · (t − α · (t − α · · · (t − α · · · + b )g(t) = b =

  k i i k i−1 −1

  ❈❧❛r❛♠❡♥t❡✱ t❡♠♦s q✉❡ f(t) = (t − α ✱ ❡ ❛❧é♠ ❞✐ss♦ a − α · b ✭♦♥❞❡ b b = 0

  d

  ✮✳ P♦rt❛♥t♦✱ max j v j k b j−1 v

  {|a | } = max {|b − α | } (❞❡s✐❣✉❛❧❞❛❞❡ tr✐❛♥❣✉❧❛r)

  06j6d 06j6d

  6 ǫ(v) max , b

  j v k j−1 v

  {|b | |α | }

  06j6d

  6 ǫ(v) max , 1

  j v k v

  {|b | } max{|α | }

  06j6d d

  Y

  d−1

  6 ǫ(v) max , 1

  

j v

{|α | } (♣❡❧❛ ❤✐♣ót❡s❡ ❞❡ ✐♥❞✉çã♦). j=1

  ❆❣♦r❛✱ ♣❛r❛ ♠♦str❛r ❛ ❧✐♠✐t❛çã♦ ✐♥❢❡r✐♦r ✈❛♠♦s s❡♣❛r❛r ❡♠ ❞♦✐s ❝❛s♦s

  6 k v ǫ(v).

  • ❈❛s♦ ✶✿ ❙❡ |α | P❡❧❛ ❡s❝♦❧❤❛ ❞♦ í♥❞✐❝❡ k✱ t❡♠♦s

  d

  Y

  d d

  j v k v

  {|α | } 6 max{|α | }

  6 max , 1 , 1 ǫ(v) ,

  j=1

  = 1 ❞❡ ♦♥❞❡ s❡❣✉❡ ♦ r❡s✉❧t❛❞♦✱ ♣♦✐s ❝♦♠♦ a ✱ ❡♥tã♦

  d

  Y

  −d

  ǫ(v) max j v , 1 j v {|α | } 6 1 6 max {|a | }.

  06j6d j=1

  > ǫ(v)

  k v i−1 v

  • ❈❛s♦ ✷✿ ❙❡ |α | ✳ ❙❡❥❛ i ♦ í♥❞✐❝❡ ♣❛r❛ ♦ q✉❛❧ |b | é ♠á①✐♠♦✳ ❊♥tã♦ max .

  j v j k j−1 v i k i−1 v

  {|a | } = max {|b − α · b | } > |b − α · b |

  06j6d 06j6d

  ❙❡ v ∈ M K ✱ ❡♥tã♦

  −1

  = = ǫ(v) max

  i k i−1 v k v i−1 v k v j v |b − α · b | |α | |b | |α | {|b | }.

  06j6d−1

  ✻✷

  ∞

  ❙❡ v ∈ M K ✱ ❡♥tã♦ > b

  

i k i−1 v k v i−1 v i v

  |b − α | |α | |b | − |b | >

  ( k v j−1 v |α | − 1)|b |

  = ( k v j v |α | − 1) max {|b | }

  −1

  > ǫ(v) max

  k v j v

  |α | {|b | },

  06j6d−1 k v > ǫ(v) = 2 k v k v /ǫ(v)

  ♣♦✐s |α | ✭❡ ❛ss✐♠ |α | − 1 > |α | ✮✳ P♦rt❛♥t♦ ✐♥❞❡♣❡♥❞❡♥t❡ s❡ v é ❛rq✉✐♠❡❞✐❛♥♦ ♦✉ ♥ã♦✱ t❡♠♦s

  −1

  max max , 1

  j v j v k v {|a | } > ǫ(v) {|b | } max{|α | }. 06j6d 06j6d−1

  ❊ ❛❣♦r❛ ❜❛st❛ ❛♣❧✐❝❛r ❛ ❤✐♣ót❡s❡ ❞❡ ✐♥❞✉çã♦ ♣❛r❛ g ❡ ♦❜t❡r❡♠♦s ♦ r❡s✉❧t❛❞♦ ♣❛r❛ f✳ ❚❡♠♦s ♣r♦✈❛❞♦ ❡♥tã♦ ❛s ❞❡s✐❣✉❛❧❞❛❞❡s

  d d

  Y Y

  −d d−1

  ǫ(v) max , 1 max , 1

  j v j v j v {|α | } 6 max {|a | } 6 ǫ(v) {|α | }.

  06j6d j=1 j=1 v

  ❊ ♣❛r❛ ♦❜t❡r ♦ r❡s✉❧t❛❞♦ ❞♦ t❡♦r❡♠❛✱ ❜❛st❛ ❡❧❡✈❛r ❡ss❡s t❡r♠♦s ❛ n ✱ ❞❡♣♦✐s t♦♠❛r ♦

  1 K

  ♣r♦❞✉t♦ s♦❜r❡ t♦❞♦s ♦s v ∈ M ❡ ♣♦r ✜♠ ❡❧❡✈❛r ❛ ✱ ❡ ✈❡r q✉❡ ❝♦♠♦ ǫ(v) 6 2 ❡♥tã♦

  [K:Q] −d −d d−1 d−1

  6

  2 ❡ ǫ(v) ✳

  6 2 ǫ(v)

  ❊①✐❜✐r❡♠♦s ❛❣♦r❛ ✉♠ r❡s✉❧t❛❞♦ q✉❡ ❣❛r❛♥t❡ q✉❡ ❛ ❢✉♥çã♦ ❛❧t✉r❛ é ✐♥✈❛r✐❛♥t❡ q✉❛♥t♦ à ❛çã♦ ❞♦ ❣r✉♣♦ ❞❡ ●❛❧♦✐s G(Q/Q)✳

  n

  (Q) ❚❡♦r❡♠❛ ✸✳✷✸✳ ❙❡❥❛ P ∈ P ❡ τ ∈ G(Q/Q)✳ ❊♥tã♦

  τ H(P ) = H(P ). n

  (K) ❉❡♠♦♥str❛çã♦✳ ❙❡❥❛ K/Q ✉♠❛ ❡①t❡♥sã♦ t❛❧ q✉❡ P ∈ P ✳ ❚❡♠♦s q✉❡ τ ❞❡✜♥❡ ✉♠ ✐s♦♠♦r✜s♠♦

  τ

  τ : K = → K {τ(α) : α ∈ K}. ❡ q✉❡ ✐♥❞✉③ t❛♠❜é♠ ✉♠ ✐❞❡♥t✐✜❝❛çã♦ ❡♥tr❡ ♦s ❝♦♥❥✉♥t♦s ❞❡ ✈❛❧♦r❡s ❛❜s♦❧✉t♦s ❞❡ K ❡ ❞❡

  τ

  K

  τ τ

  τ : M K K , v , → M 7→ v

  τ v = v

  ♦♥❞❡ |τ(x)| |x| ✳ ❆ss✐♠✱ τ t❛♠❜é♠ ✐♥❞✉③ ✉♠ ✐s♦♠♦r✜s♠♦ ❡♥tr❡ ♦s ❝♦♠♣❧❡t❛♠❡♥t♦s

  τ τ τ

  K v , v v ❡ K v ❞❡ ♠♦❞♦ q✉❡ ♦s ❣r❛✉s ❧♦❝❛✐s n ❡ n sã♦ ✐❣✉❛✐s✳

  ✻✸ P♦rt❛♥t♦

  Y

  τ τ n w τ

  H (P ) = max

  

K w

  {|x i | }

  w∈M Kτ

  Y

  n vτ

  = max

  i v

  {|x | }

  v∈M K

  Y

  n v

  = max

  i v

  {|x | }

  v∈M K = H (P ).

  K τ

  : Q] ❈♦♠♦ [K : Q] = [K ✱ ❡♥tã♦ ♣❡❧❛ ❞❡✜♥✐çã♦ ❞❛ ❛❧t✉r❛ ❛❜s♦❧✉t❛ ♦❜t❡♠♦s

  τ H(P ) = H(P ).

  ❋✐♥❛❧♠❡♥t❡✱ ❛♥t❡s ❞❡ ♣r♦ss❡❣✉✐r♠♦s ♣❛r❛ ❞❡✜♥✐r ❢✉♥çõ❡s ❛❧t✉r❛s s♦❜r❡ ❝✉r✈❛s ❡❧í♣✲ t✐❝❛s✱ ♣r♦✈❛r❡♠♦s ♦ r❡s✉❧t❛❞♦ q✉❡ ❣❛r❛♥t❡ q✉❡ ❛ ❢✉♥çã♦ ❛❧t✉r❛ H s❛t✐s❢❛③ ❛ ❝♦♥❞✐çã♦ ❞❡ ✜♥✐t✉❞❡ ❞♦ t❡♦r❡♠❛ ❞❛ ❞❡s❝✐❞❛✳ ❚❡♦r❡♠❛ ✸✳✷✹✳ ❙❡❥❛♠ c ❡ d ❝♦♥st❛♥t❡s ♣♦s✐t✐✈❛s✳ ❊♥tã♦ ❝♦♥❥✉♥t♦

  n

  (Q : H(P ) 6 c {P ∈ P ❡ [Q(P ) : Q] 6 d},

  é ✜♥✐t♦✳ ❖♥❞❡ Q(P ) é ♦ ❝♦r♣♦ ❞❡ ❞❡✜♥✐çã♦ ♠✐♥✐♠❛❧ ❞❡ P ✱ ✐st♦ é✱ ❛ ♠❡♥♦r ❡①t❡♥sã♦ ❞❡ Q q✉❡ ❝♦♥té♠ ❛s ❝♦♦r❞❡♥❛❞❛s ❤♦♠♦❣ê♥❡❛s ♥♦r♠❛❧✐③❛❞❛s ❞❡ P ✳ ❊♠ ♣❛rt✐❝✉❧❛r✱ ♣❛r❛ t♦❞♦ ❝♦r♣♦ ❞❡ ♥ú♠❡r♦s K✱ ♦ ❝♦♥❥✉♥t♦

  n

  (K) : H (P ) 6 c

  K

  {P ∈ P } é ✜♥✐t♦✳

  n

  (Q) ❉❡♠♦♥str❛çã♦✳ ❙❡❥❛ P ∈ P ✳ ❊s❝♦❧❤❛♠♦s ❝♦♦r❞❡♥❛❞❛s ❤♦♠♦❣ê♥❡❛s ♣❛r❛ P ❞❡ ❢♦r♠❛

  , . . . , x ] , . . . , x )

  n n

  q✉❡ ❛❧❣✉♠❛ ❞❡❧❛s s❡❥❛ ✶✱ P = [x ✳ ❊♥tã♦ Q(P ) = Q(x ❡ t❡♠♦s ❛ s❡❣✉✐♥t❡ ❡st✐♠❛t✐✈❛ ♣❛r❛ ❛ ❛❧t✉r❛ ❞❡ P ✱

  Y

  v n

  H Q(P ) (P ) = max i v {|x | }

  

06j6n

v∈M Q (P )

    Y

  v n

  >   max max i v , 1

  {|x | }

  06j6n

v∈M Q

(P ) = max H Q(P ) (x i ).

  06j6n

  ✻✹ ) 6 c ) :

  1 i

  P♦rt❛♥t♦✱ s❡ H(P ) 6 c ❡ [Q(P ) : Q] 6 d✱ ❡♥tã♦ max H(x ❡ max[Q(x Q] 6 d. ❆ss✐♠ ♦ ♣r♦❜❧❡♠❛ s❡ r❡❞✉③ ❛ ♣r♦✈❛r q✉❡ ♦ ❝♦♥❥✉♥t♦

  {x ∈ Q(x) : H(x) 6 c ❡ [Q(x) : Q] 6 d} é ✜♥✐t♦✳

  ❙❡❥❛ x ♥♦ ❝♦♥❥✉♥t♦ ❛❝✐♠❛✱ t❡♠♦s q✉❡ e = [Q(x) : Q] 6 d✳ ❈♦♥s✐❞❡r❡ ♦ ❝♦♥❥✉♥t♦ , . . . , x

  1 e

  ❞♦s ❝♦♥❥✉❣❛❞♦s ❞❡ x✱ {x = x }✳ ❊♥tã♦ ♦ ♣♦❧✐♥ô♠✐♦ ♠✐♥✐♠❛❧ ❞❡ x s♦❜r❡ Q é

  e e−1

  • f (t) = (t ) ) = t + a t t + a

  x 1 e 1 e−1 e − x · · · (t − x · · · + a ∈ Q[t].

  P♦❞❡♠♦s ❡♥tã♦ ♦❜t❡r ❛ s❡❣✉✐♥t❡ ❡st✐♠❛t✐✈❛

  e

  Y

  e−1

  H([1, a

  1 , . . . , a e ]) 6 2 H(x j ) j=1

e−1 e

  = 2 H(x)

  

d

  6 (2c) ♣♦✐s H(x) 6 c ❡ e 6 d.

  , . . . , a

  1 e

  P❡❧♦ ❢❛t♦ ❞❡ q✉❡ a ∈ Q✱ ❡ q✉❡ ❡①✐st❡ ❛♣❡♥❛s ✉♠❛ q✉❛♥t✐❞❛❞❡ ✜♥✐t❛ ❞❡

  e d

  (Q) ♣♦♥t♦s Q ∈ P ❝♦♠ ❛❧t✉r❛ ❧✐♠✐t❛❞❛ ♣♦r (2c) ✱ ❡♥tã♦ ❡①✐st❡ ❛♣❡♥❛s ✉♠❛ q✉❛♥t✐❞❛❞❡

  (t) (t)

  x x

  ✜♥✐t❛ ❞❡ ♣♦❧✐♥ô♠✐♦s f ✳ ❈♦♠♦ ❝❛❞❛ ✉♠ ❞♦s ♣♦❧✐♥ô♠✐♦s f t❡♠ ♥♦ ♠á①✐♠♦ d r❛í③❡s ❡♠ Q✱ ✈❡♠♦s q✉❡ ❞❡ ❢❛t♦ ♦ ❝♦♥❥✉♥t♦

  {x ∈ Q(x) : H(x) 6 c ❡ [Q(x) : Q] 6 d} é ✜♥✐t♦✳ P♦rt❛♥t♦

  n

  (Q : H(P ) 6 c {P ∈ P ❡ [Q(P ) : Q] 6 d},

  é ✜♥✐t♦✳ ❆❣♦r❛✱ s❡ K é ✉♠ ❝♦r♣♦ ❞❡ ♥ú♠❡r♦s ❞❡ ❣r❛✉ e✱ ❡♥tã♦ t❡♠♦s ❛ ✐♥❝❧✉sã♦

  n n 1/e

  (K) : H (P ) 6 c (Q : H(P ) 6 c

  K {P ∈ P } ⊆ {P ∈ P ❡ [Q(P ) : Q] 6 e}. n

  (K) : H (P ) 6 c

  K

  ▼♦str❛♥❞♦ q✉❡ {P ∈ P } é ✜♥✐t♦✳

  ✻✺

  ✸✳✹ ❆❧t✉r❛s ❡♠ ❈✉r✈❛s ❊❧í♣t✐❝❛s

  ◆❡st❛ s❡çã♦ ❞❡♠♦♥str❛r❡♠♦s ♦ ❚❡♦r❡♠❛ ❞❡ ▼♦r❞❡❧❧✲❲❡✐❧✱ ♠❛s ♣❛r❛ ✐ss♦ ♣r❡❝✐s❛✲ ♠♦s ❡♥t❡♥❞❡r ❝♦♠♦ ❛s ❢✉♥çõ❡s ❛❧t✉r❛s ❞❡✜♥✐❞❛s ♥♦ ❡s♣❛ç♦ ♣r♦❥❡t✐✈♦ s❡ ❝♦♠♣♦rt❛♠ ❡♠ r❡❧❛çã♦ à ♦♣❡r❛çã♦ ❞❡ ❣r✉♣♦ ❞❡✜♥✐❞❛ s♦❜r❡ ✉♠❛ ❝✉r✈❛ ❡❧í♣t✐❝❛✳

  ❈♦♠❡ç❛r❡♠♦s ❞❡✜♥✐♥❞♦ ❛ s❡❣✉✐♥t❡ ♥♦t❛çã♦✿ ❙❡❥❛♠ f ❡ g ❢✉♥çõ❡s r❡❛✐s ❞❡✜♥✐❞❛s ❡♠ ✉♠ ❝♦♥❥✉♥t♦ S✳ ❊s❝r❡✈❡♠♦s f = g + O(1)

  , c

  1

  2

  ♣❛r❛ ❞✐③❡r q✉❡ ❡①✐st❡♠ ❝♦♥t❛♥t❡s c t❛✐s q✉❡

  1

  6 c f (P ) ,

  2 − g(P ) 6 c ∀P ∈ S.

  ❙❡ ❛♣❡♥❛s ❛ ❞❡s✐❣✉❛❧❞❛❞❡ ✐♥❢❡r✐♦r é s❛t✐s❢❡✐t❛✱ ❡s❝r❡✈❡♠♦s f > g + O(1)✱ ❝❛s♦ s♦♠❡♥t❡ ❛ ❞❡s✐❣✉❛❧❞❛❞❡ s✉♣❡r✐♦r ❢♦r s❛t✐s❢❡✐t❛✱ ❡♥tã♦ ❞❡♥♦t❛♠♦s ♣♦r f 6 g + O(1)✳

  ❙❡♥❞♦ E/K ✉♠❛ ❝✉r✈❛ ❡❧í♣t✐❝❛ ❡ f ∈ K(E) ♥ã♦ ❝♦♥st❛♥t❡✱ t❡♠♦s q✉❡ f ❞❡✜♥❡ ✉♠ ♠♦r✜s♠♦

  1

  f : E → P

  P 7→ [1, 0] s❡ f t❡♠ ♣ó❧♦ ❡♠ P

  P 7→ [f(P ), 1] ❝❛s♦ ❝♦♥trár✐♦. ◆❛ ❞❡✜♥✐çã♦ ❞❛ ❢✉♥çã♦ ❛❧t✉r❛ H✱ ♦❜s❡r✈❛♠♦s q✉❡ ❡❧❛ ❛t✉❛ ❞❡ ❢♦r♠❛ ♠✉❧t✐♣❧✐❝❛✲ t✐✈❛✱ q✉❡r❡♠♦s ❝♦♥str✉✐r ✉♠❛ ❢✉♥çã♦ ❛❧t✉r❛ s♦❜r❡ ✉♠❛ ❝✉r✈❛ ❡❧í♣t✐❝❛ ❞❡ ♠♦❞♦ ❛ t❡r ❜♦❛s

  ♣r♦♣r✐❡❞❛❞❡s ❛❞✐t✐✈❛s✳ ❉❡✜♥✐çã♦ ✸✳✷✺✳ ❉❡✜♥✐♠♦s ❛ ❛❧t✉r❛ ✭❧♦❣❛rít♠✐❝❛ ❛❜s♦❧✉t❛✮ ♥♦ ❡s♣❛ç♦ ♣r♦❥❡t✐✈♦ ❝♦♠♦ s❡♥❞♦ ❛ ❢✉♥çã♦

  n

  h : P (Q) → R

  P 7→ log H(P ). P❡❧♦ ❢❛t♦ ❞❡ q✉❡ H(P ) > 1✱ t❡♠♦s h(P ) > 0 ♣❛r❛ t♦❞♦ P ✳ ❉❡✜♥✐çã♦ ✸✳✷✻✳ ❙❡❥❛ E/K ✉♠❛ ❝✉r✈❛ ❡❧í♣t✐❝❛✱ f ∈ K(E) ♥ã♦ ❝♦♥st❛♥t❡✳ ❆ ❛❧t✉r❛ ❡♠ E

  ✭r❡❧❛t✐✈❛ ❛ f✮ é ❛ ❢✉♥çã♦ h f : E(K) → R

  P 7→ h(f(P )).

  ✻✻ ❙♦❜r❡ ❡ss❛ ❢✉♥çã♦ t❡♠♦s ❣❛r❛♥t✐❞❛ ❛ ❝♦♥❞✐çã♦ ❞❡ ✜♥✐t✉❞❡ ♣❛r❛ ♦ ❚❡♦r❡♠❛ ❞❛

  ❉❡s❝✐❞❛✱ ❞❛❞❛ ♣❡❧♦ r❡s✉❧t❛❞♦ ❛❜❛✐①♦✳ Pr♦♣♦s✐çã♦ ✸✳✷✼✳ ❙❡❥❛ E/K ✉♠❛ ❝✉r✈❛ ❡❧í♣t✐❝❛✱ f ∈ K(E) ✉♠❛ ❢✉♥çã♦ ♥ã♦ ❝♦♥st❛♥t❡ ❡ c ✉♠❛ ❝♦♥st❛♥t❡✳ ❊♥tã♦ ♦ ❝♦♥❥✉♥t♦

  (P ) 6 c

  f

  {P ∈ E(K) : h } é ✜♥✐t♦✳ ❉❡♠♦♥str❛çã♦✳ ❈♦♠♦ ❛ ❢✉♥çã♦ f ❡stá ❞❡✜♥✐❞❛ s♦❜r❡ K✱ ❡❧❛ ❧❡✈❛ ♣♦♥t♦s P ∈ E(K) ❡♠

  1

  (K) ♣♦♥t♦s ❞❡ P ✳ P♦rt❛♥t♦ f ❞❡✜♥❡ ✉♠❛ ❝♦rr❡s♣♦♥❞ê♥❝✐❛ ❡♥tr❡ ♦s ❝♦♥❥✉♥t♦s

  1 c

  (P ) 6 c (K) : H(Q) 6 e

  f

  {P ∈ E(K) : h } → {Q ∈ P } P → Q = f(P ).

  ❱✐st♦ q✉❡ ♦ ❝♦♥tr❛❞♦♠í♥✐♦ ❞❡ss❛ ❢✉♥çã♦ é ✜♥✐t♦ ❡ ❛ ♣ré✲✐♠❛❣❡♠ ❞❡ ❝❛❞❛ ✉♠ ❞❡ s❡✉s (P ) 6 c

  f

  ❡❧❡♠❡♥t♦s é ✜♥✐t♦✱ ❡♥tã♦ ♦ ❝♦♥❥✉♥t♦ {P ∈ E(K) : h } é ✜♥✐t♦✳ ❖ ♣ró①✐♠♦ t❡♦r❡♠❛ ♥♦s ❞á ✉♠❛ r❡❧❛çã♦ ❡♥tr❡ ❛s ❛❧t✉r❛s ♥❛ ❝✉r✈❛ ❡❧í♣t✐❝❛ ❡ s✉❛

  ❧❡✐ ❞❡ ❣r✉♣♦✳ ❚❡♦r❡♠❛ ✸✳✷✽✳ ❙❡❥❛ E/K ✉♠❛ ❝✉r✈❛ ❡❧í♣t✐❝❛ ❡ f ∈ K(E) ✉♠❛ ❢✉♥çã♦ ♣❛r ♥ã♦ ❝♦♥st❛♥t❡ ✭✐st♦ é✱ f(P ) = f(−P ) ♣❛r❛ t♦❞♦ P ∈ E(K)✮✳ ❊♥tã♦✱ ♣❛r❛ t♦❞♦ P, Q ∈ E(K) t❡♠♦s h (P + Q) + h (P (P ) + 2h (Q) + O(1).

  f f f f

  − Q) = 2h ❉❡♠♦♥str❛çã♦✳ ❈♦♥s✐❞❡r❡♠♦s ❛ ❡q✉❛çã♦ ❞❡ ❲❡✐❡rtr❛ss ❞❛ ❢♦r♠❛

  2

  

3

E : y = x + ax + b.

  Pr♦✈❛r❡♠♦s ♦ t❡♦r❡♠❛ ♣❛r❛ ❛ ❢✉♥çã♦ f = x✱ q✉❡ é ✉♠❛ ❢✉♥çã♦ ♣❛r s♦❜r❡ E(K)✳ ( (P ) = h (

  x x x

  ❈♦♠♦ h O) = 0 ❡ h −P )✱ ♦ r❡s✉❧t❛❞♦ é ✐♠❡❞✐❛t♦ q✉❛♥❞♦ P = O ♦✉ Q =

  O✱ ♣♦✐s ♥❡ss❡ ❝❛s♦ t❡r❡♠♦s h (P + Q) + h (P (P ) + 2h (Q).

  x x x x

  − Q) = 2h ❆ss✐♠✱ ❛ss✉♠✐r❡♠♦s q✉❡ P, Q 6= O, ❡ ❞❡✜♥❛♠♦s x(P ) = [x

  1 , 1], x(Q) = [x 2 , 1] x(P + Q) = [x , 1], x(P , 1].

  3

  4

  − Q) = [x

  3

  4

  ✻✼ , y , 1] , y , 1]

  1

  

1

  2

  2

  ❙❡ t✐✈❡r♠♦s P 6= ±Q✱ P = [x ❡ Q = [x ✱ ❡♥tã♦ ♣❡❧❛s ❡①♣r❡ssõ❡s ❛❧❣é❜r✐❝❛s q✉❡ ❡♥❝♦♥tr❛♠♦s ♣❛r❛ ❞❡♥♦t❛r ❛ s♦♠❛ ❡♥tr❡ ♣♦♥t♦s ❞❡ E✱ ♦❜t❡♠♦s

  2

  y

  2

  

1

  − y x = x(P + Q) = + x )

  3

  1

  2

  − (x x

  2

  

1

  − x

  2

  2

  3

  y y + y + x x (x + x )

  2 − 2y 1 − x

  2

  =

  2

  (x

  2 1 )

  − x (a + x x )(x + x ) + 2b y

  1

  

2

  1

  2

  1

  2

  − 2y = .

  2

  (x + x ) x

  

1

  2

  1

  2

  − 4x

  2

  2

  3

  3

  • ax + b + ax + b

  1

  2

  ◆❛ ú❧t✐♠❛ ✐❣✉❛❧❞❛❞❡ s✉❜st✐t✉✐♠♦s y

  1 ❡ y 2 ♣♦r x 1 ❡ x 2 ✱ r❡s♣❡❝t✐✈❛♠❡♥t❡✳

  , , 1]

  2

  2

  ❆❣♦r❛ ✉s❛♥❞♦ q✉❡ −Q = [x −y ♦❜t❡♠♦s ✉♠ r❡s✉❧t❛❞♦ ❛♥á❧♦❣♦ ♣❛r❛ x(P − Q)✱ (a + x x )(x + x ) + 2b + 2y y

  1

  2

  

1

  2

  1

  2 x = .

  4

  2

  (x + x ) x

  1

  

2

  1

  2

  − 4x P♦rt❛♥t♦✱

  2

  2(x + x )(a + x x ) + 4b (x x + x )

  1

  2

  1

  2

  1 2 − a) − 4b(x

  1

  2 x + x = x = .

  3 4 ❡ x

  3

  4

  2

  2

  (x + x ) x (x + x ) x

  1

  2

  1

  2

  1

  2

  1

  2

  − 4x − 4x

  2

  2

  , ❆❣♦r❛✱ ❞❡✜♥✐♠♦s ♦ ♠❛♣❛ g : P → P ❞❛❞♦ ♣♦r

  2

  2

  2

  g([t, u, v]) = [u , (v − 4tv, 2u(at + v) + 4bt − at) − 4btu]. ❈♦♥str✉✐♠♦s ♦ s❡❣✉✐♥t❡ ❞✐❛❣r❛♠❛

  ♦♥❞❡ G(P, P ) = (P + Q, P − Q) ❡ T é ❛ ❝♦♠♣♦s✐çã♦ ❞❛s s❡❣✉✐♥t❡s ❛♣❧✐❝❛çõ❡s

  1

1 E

  × E → P × P (P, Q) , β ], [α , β ])

  1

  1

  2

  2

  7→ (x(P ), y(P )) = ([α

  ✻✽ ❡

  1

  1

  

2

P

  × P → P ([α

  1 , β 1 ], [α 2 , β 2 ])

1 β

2 , α 1 β 2 + α 2 β 1 , α 1 α 2 ].

  7→ [β + x , x x ].

  1

  2

  1

  

2

  3

  4

  ❊ ❛ss✐♠ ♦❜t❡♠♦s T (P, Q) = [1, x ❆s ❢ór♠✉❧❛s q✉❡ ♦❜t✐✈❡♠♦s ♣❛r❛ x ❡ x ♠♦str❛♠ q✉❡ ❡ss❡ ❞✐❛❣r❛♠❛ é ❝♦♠✉t❛t✐✈♦✱ ✈✐st♦ q✉❡ g(T (P, Q)) = g([1, x + x , x x ])

  1

  2

  1

  2

  2

  2

  = [(x + x ) x , 2(x + x )(a + x x ) + 4b, (x x + x )]

  1

  2

  1

  2

  1

  

2

  1

  2

  1

  2

  1

  2

  − 4x − a) − 4b(x = [1, x + x , x x ]

  3

  4

  3

  4

  = T (P + Q, P − Q) = T (G(P, Q)).

  ❆❣♦r❛ ♠♦str❛r❡♠♦s q✉❡ g é ✉♠ ♠♦r✜s♠♦✱ ❡ ♣❛r❛ ✐ss♦ ♣r❡❝✐s❛♠♦s ♠♦str❛r q✉❡ ♦s ♣♦❧✐♥ô✲ ♠✐♦s q✉❡ ❞❡✜♥❡♠ g ♥ã♦ ♣♦ss✉❡♠ ③❡r♦s ❡♠ ❝♦♠✉♠ ❝♦♠ ❡①❝❡ssã♦ ❞❡ q✉❛♥❞♦ t = u = v = 0✳ ❙✉♣♦♥❤❛ ❡♥tã♦ q✉❡ g([t, u, v]) = 0✳

  2

  2

  ❙❡ t = 0 ❡♥tã♦ u − 4tv = 0 ❡ (v − at) − 4btu = 0 ✐♠♣❧✐❝❛♠ q✉❡ u = v = 0✳ u ❆ss✐♠✱ ♣♦❞❡♠♦s s✉♣♦r q✉❡ t 6= 0✱ ❡ ❞❡✜♥✐♠♦s x = ✳

  2t

  2

  2

  = v/t ❆ ♣❛rt✐r ❞❛ ❡q✉❛çã♦ u − 4tv = 0 ❡ ❞❡ x = u/2t✱ ♦❜t❡♠♦s x ✳ ❆ss✐♠✱ s❡

  2

  2

  2u(at + v) + 4bt = 0 ❡ (v − at) − 4btu = 0,

  2

  ❢❛③❡♥❞♦ ❛s ❞❡✈✐❞❛s s✉❜st✐t✉✐çõ❡s ❡ ❞✐✈✐❞✐♥❞♦ ❛s ❡q✉❛çõ❡s ♣♦r t ✱ ♦❜t❡r❡♠♦s ✉♠ s✐st❡♠❛ ❞❡ ❡q✉❛çõ❡s ❡♠ x✱

  2

  3

  ψ(x) = 4x(a + x ) + 4b = 4(x + ac + b) = 0 ❡

  2

  2

  4

  2

  2 φ(x) = (x = 0.

  − a) − 8bx = x − 2ax − 8bx + a ❆❣♦r❛ ♣❛r❛ ♠♦str❛r q✉❡ ψ ❡ φ ♥ã♦ ♣♦ss✉❡♠ ③❡r♦s ❡♠ ❝♦♠✉♠✱ ❜❛st❛ ✈❡r✐✜❝❛r ❛ s❡❣✉✐♥t❡ ✐❣✉❛❧❞❛❞❡

  2

  3

  3

  2

  (12x + 16a)φ(x) + 27b ) − (3x − 5ax − 27b)ψ(x) = 4(4a 6= 0. ❊ss❛ ú❧t✐♠❛ ❞❡s✐❣✉❛❧❞❛❞❡ s❡ ✈❡r✐✜❝❛ ♣❡❧♦ ❢❛t♦ q✉❡ q✉❡ ✉♠ ❞♦s ♠❡♠❜r♦s é ✉♠ ♠ú❧t✐♣❧♦ ♥ã♦ ♥✉❧♦ ❞♦ ❞✐s❝r✐♠✐♥❛♥t❡ ❞❛ ❡q✉❛çã♦ ❞❡ ❲❡✐❡rstr❛ss✱ q✉❡ é ♥ã♦ ♥✉❧♦ ♣❡❧♦ ❢❛t♦ ❞❡ q✉❡ ❛ ❝✉r✈❛ ❡❧í♣t✐❝❛ é s✉❛✈❡✳ P♦rt❛♥t♦✱ ❛s ❢✉♥çõ❡s q✉❡ ❞❡✜♥❡♠ g ♥ã♦ ♣♦❞❡♠ t❡r ③❡r♦s ❡♠ ❝♦♠✉♠ ♥ã♦ tr✐✈✐❛✐s✱ ♠♦str❛♥❞♦ q✉❡ g é ✉♠ ♠♦r✜s♠♦✳

  ✻✾ ❘❡t♦r♥❛♥❞♦ ❛♦ ❞✐❛❣r❛♠❛ ❝♦♠✉t❛t✐✈♦✱ ♦❜t❡♠♦s h(T (P + Q, P

  − Q)) = h(T ◦ G(P, Q)) = h(g

  ◦ T (P, Q)) = 2h(T (P, Q)) + O(1).

  ❊ss❛ ú❧t✐♠❛ ✐❣✉❛❧❞❛❞❡ s❡ ❞❡✈❡ ♣❡❧♦ s❡❣✉✐♥t❡ ❢❛t♦✿ ❉❛❞♦ q✉❡ g é ✉♠ ♠♦r✜s♠♦ ❞❡ ❣r❛✉ ✷✱

  1

  2

  ♣♦✐s ❡stá ❞❡✜♥✐❞♦ ♣♦r ♣♦❧✐♥ô♠✐♦s ❤♦♠♦❣ê♥❡♦s ❞❡ ❣r❛✉ ✷✱ ❡①✐st❡♠ ❝♦♥st❛♥t❡s c ❡ c t❛✐s

  2

  2

  6

  1

  2

  1

  q✉❡ c ✱ ❡ ✉s❛♥❞♦ ❛ ❞❡✜♥✐çã♦ ❞❡ h ❡♥❝♦♥tr❛r❡♠♦s log c h(g(P ))

  6 H(P ) H(g(P )) 6 c H(P )

  2 − 2h(P ) 6 log c ✱ ♦✉ s❡❥❛✱ h(g(P )) = 2h(P ) + O(1).

  ❆❣♦r❛ ♣❛r❛ ❝♦♠♣❧❡t❛r ❛ ♣r♦✈❛ ❞♦ r❡s✉❧t❛❞♦ ♣❛r❛ f = x ❜❛st❛ ♣r♦✈❛r q✉❡ h(T (R , R )) = h (R ) + h (R ) + O(1), , R

  1 2 x 1 x

  2

  1

  2

  ∀R ∈ E(K), ❥á q✉❡ s❡ ❡ss❛ ❡①♣r❡ssã♦ ✈❛❧❡✱ ❡♥tã♦ h (P + Q) + h (P

  x x

  − Q) + O(1) = h(T (P + Q, P − Q)) = 2h(T (P, Q)) + O(1) = 2h (P ) + 2h (Q) + O(1),

  x x

  ❞❡ ♦♥❞❡ ♦❜t❡r❡♠♦s ♦ r❡s✉❧t❛❞♦ ❞❡s❡❥❛❞♦ ♣❛r❛ f = x✳ = =

  1

  2

  ❖❜s❡r✈❡♠ q✉❡ s❡ R O ♦✉ R O ❡♥tã♦ ♦ r❡s✉❧t❛❞♦ ✈❛❧❡✱ ❞❛❞♦ q✉❡ h(T (R , R )) = h (R ) + h (R ).

  1 2 x 1 x 2 ❙✉♣♦♥❞♦ ❡♥tã♦ q✉❡ ❛♠❜♦s sã♦ ❞✐❢❡r❡♥t❡s ❞❡ O✱ t❡r❡♠♦s

  x(R

  1 ) = [α

1 , 1], x(R

2 ) = [α 2 , 1],

  ❞❡ ♦♥❞❡ s❡❣✉❡ q✉❡ h(T (R , R )) = h([1, α + α , α α ]) (R ) + h (R ) = h(α ) + h(α ).

  1

  2

  1

  2

  1

2 x

1 x

  2

  1

  2

  ❡ h )(t )

  1

  2

  ❯s❛♥❞♦ ♦ ♣♦❧✐♥ô♠✐♦ f(t) = (t − α − α ♦❜t❡♠✲s❡

  −2

2 H(α

  1 )H(α 2 ) 6 H([1, α

1 + α

2 , α 1 α 2 ]) 6 2H(α 1 )H(α 2 ),

  ❡ ♣♦rt❛♥t♦ h(α ) + h(α ) + α , α α ]) 6 h(α ) + h(α ) + log 2,

  1

  2

  1

  2

  1

  2

  1

  2

  − log 4 6 h([1, α ♦✉ s❡❥❛✱ h(T (R , R )) = h (R ) + h (R ) + O(1).

  1 2 x 1 x

  2

  ✼✵ ❆ ♣r♦✈❛ ♣❛r❛ ✉♠ f ❛r❜✐trár✐♦ é ❝♦♥s❡q✉ê♥❝✐❛ ❞♦ ❧❡♠❛ s❡❣✉✐♥t❡ ❡ ❞❡ss❡ r❡s✉❧t❛❞♦ q✉❡ ♦❜t✐✈❡♠♦s ♣❛r❛ x✳ ◆❛ ♣r♦✈❛ ❞♦ ❚❡♦r❡♠❛ ❞❡ ▼♦r❞❡❧❧✲❲❡✐❧ ✐r❡♠♦s ❝♦♥s✐❞❡r❛r ✉♠❛

  ❢✉♥çã♦ ♣❛r ❡ s✉❛ ❢✉♥çã♦ ❛❧t✉r❛ ❝♦rr❡s♣♦♥❞❡♥t❡✱ ❡ ✉♠ r❡s✉❧t❛❞♦ ❛♥t❡r✐♦r ❛ ❡❧❡ ❣❛r❛♥t✐rá q✉❡ ❡st❛ ❢✉♥çã♦ ❛❧t✉r❛ s❛t✐s❢❛③ ❛s ❤✐♣ót❡s❡s ❞♦ ❚❡♦r❡♠❛ ❞❡ ❉❡s❝✐❞❛✳ ❆ss✐♠✱ ♣r❡❝✐s❛♠♦s ❛♣❡♥❛s ❞❛ ❢✉♥çã♦ x ❡ s✉❛ ❛❧t✉r❛ ❛ss♦❝✐❛❞❛ ♣❛r❛ ❛ ♣r♦✈❛ ❞♦ t❡♦r❡♠❛✳ ▲❡♠❛ ✸✳✷✾✳ ❙❡❥❛♠ f, g ∈ K(E) ❢✉♥çõ❡s ♣❛r❡s✳ ❊♥tã♦ (def g)h f = (deg f )h g + O(1).

  ❉❡♠♦♥str❛çã♦✳ ❱❡r ✭❈❛♣ít✉❧♦ ✽✱ ▲❡♠❛ ✻✳✸✱ ❬❙✐❧✈❡r♠❛♥❪✮✳ ❯s❛♥❞♦ ❡ss❡ ❧❡♠❛✱ ♦ r❡st❛♥t❡ ❞♦ r❡s✉❧t❛❞♦ ❞♦ t❡♦r❡♠❛ ❛♥t❡r✐♦r s❛✐ ❝♦♠♦ ❝♦r♦❧ár✐♦✱

  ✈✐st♦ q✉❡ ❜❛st❛ ✉s❛r ❛ ✐❣✉❛❧❞❛❞❡ ❞♦ ❧❡♠❛ ❡ ♦ r❡s✉❧t❛❞♦ ❞♦ t❡♦r❡♠❛ ♣❛r❛ ❛ ❢✉♥çã♦ x✱ ❡ ❛ss✐♠ ♦❜t❡r ♦ r❡s✉❧t❛❞♦ ❞♦ t❡♦r❡♠❛ ♣❛r❛ ✉♠❛ f ♣❛r q✉❛❧q✉❡r✳

  ❆❣♦r❛✱ ❡ss❡ ❝♦r♦❧ár✐♦ ❣❛r❛♥t❡ q✉❡ ❞❡ ❢❛t♦ ❡ss❛s ❢✉♥çõ❡s ❛❧t✉r❛ h f ♣❛r❛ f ♣❛r✱ s❛t✐s❢❛③❡♠ ❛s ❤✐♣ót❡s❡s ❞♦ ❚❡♦r❡♠❛ ❞❡ ❉❡s❝✐❞❛✳ ❈♦r♦❧ár✐♦ ✸✳✸✵✳ ❙❡❥❛ E/K ✉♠❛ ❝✉r✈❛ ❡❧í♣t✐❝❛✱ f ∈ K(E) ✉♠❛ ❢✉♥çã♦ ♣❛r ♥ã♦ ❝♦♥st❛♥t❡✳

  ✭❛✮ ❉❛❞♦ Q ∈ E(K)✳ ❊♥tã♦ h (P + Q) 6 2h (P ) + O(1),

  f f ♣❛r❛ t♦❞♦ P ∈ E(K).

  ✭❜✮ ❉❛❞♦ m ✐♥t❡✐r♦ ♣♦s✐t✐✈♦✱ t❡♠♦s

  2

  h f ([m]P ) = m h f (P ) + O(1), ♣❛r❛ t♦❞♦ P ∈ E(K),

  ♦♥❞❡ O(1) ❞❡♣❡♥❞❡ ❞❡ E✱ f ❡ m✳

  f (P

  ❉❡♠♦♥str❛çã♦✳ ✭❛✮ ❉❛❞♦ q✉❡ h − Q) > 0✱ ❡♥tã♦ ♣❡❧♦ t❡♦r❡♠❛ ❛♥t❡r✐♦r h (P + Q) 6 h (P + Q) + h (P (P ) + 2h (Q) + O(1).

  f f f f f

  − Q) = 2h ✭❜✮ Pr♦✈❡♠♦s ♣♦r ✐♥❞✉çã♦✳ P❛r❛ m = 1 ♦ r❡s✉❧t❛❞♦ é ❝❧❛r♦✱ ❞❛❞♦ q✉❡ ♥❡ss❡ ❝❛s♦ h ([1]P ) = h (P )

  f f

  ✳ ❙✉♣♦♥❤❛♠♦s ♣♦r ❤✐♣ót❡s❡ ❞❡ ✐♥❞✉çã♦✱ q✉❡ ♦ r❡s✉❧t❛❞♦ s❡❥❛ ✈á❧✐❞♦ ♣❛r❛ m − 1 ❡ m✱ ❝♦♠ m > 1✱ ❡ ✈❡❥❛♠♦s q✉❡ t❛♠❜é♠ ✈❛❧❡ ♣❛r❛ m + 1✳ ❚❡♠♦s

  ([m + 1]P ) + h ([m ([m]P ) + 2h (P ) + O(1),

  f f f f

  q✉❡ h − 1]P ) = 2h ❛ss✐♠ h ([m + 1]P ) = ([m ([m]P ) + 2h (P ) + O(1)

  f f f f

  −h − 1]P ) + 2h

  2

  2

  = ( + 2m + 2)h (P ) + O(1) (

  f

  −(m − 1) ♣♦r ❤✐♣ót❡s❡ ❞❡ ✐♥❞✉çã♦)

  2 = (m + 1) h (P ) + O(1). f

  ✼✶ ❖❜t❡♥❞♦ ❛ss✐♠ ♦ r❡s✉❧t❛❞♦ ❞❡s❡❥❛❞♦✳

  ❋✐♥❛❧♠❡♥t❡✱ ❡st❛♠♦s ❡♠ ❝♦♥❞✐çõ❡s ❞❡ ♣r♦✈❛r ♦ ❚❡♦r❡♠❛ ❞❡ ▼♦r❞❡❧❧✲❲❡✐❧✳ ❚❡♦r❡♠❛ ✸✳✸✶ ✭▼♦r❞❡❧❧✲❲❡✐❧✮✳ ❙❡❥❛ K ✉♠ ❝♦r♣♦ ❞❡ ♥ú♠❡r♦s ❡ E/K ✉♠❛ ❝✉r✈❛ ❡❧í♣t✐❝❛✳ ❊♥tã♦ ♦ ❣r✉♣♦ E(K) é ✜♥✐t❛♠❡♥t❡ ❣❡r❛❞♦✳ ❉❡♠♦♥str❛çã♦✳ ❊s❝♦❧❤❡♠♦s ✉♠❛ ❢✉♥çã♦ f ∈ K(E) ♣❛r ♥ã♦ ❝♦♥st❛♥t❡✱ ♣♦❞❡♠♦s ♣❡❣❛r ❛ ❢✉♥çã♦ ❝♦♦r❞❡♥❛❞❛ x ♥❛ ❡q✉❛çã♦ ❞❡ ❲❡✐❡rstr❛ss r❡❞✉③✐❞❛ q✉❡ ❞❡✜♥❡ ❛ ❝✉r✈❛ E✳ P❡❧❛ ✈❡rsã♦ ❢r❛❝❛ ❞♦ ❚❡♦r❡♠❛✱ ♣❛r❛ m > 2 ♦s ❣r✉♣♦s q✉♦❝✐❡♥t❡s E(K)/mE(K) sã♦ ✜♥✐t♦s✳ ❇❛st❛ t♦♠❛r m = 2✱ ❡ ✉s❛♥❞♦ ♦ ❈♦r♦❧ár✐♦ ❛♥t❡r✐♦r✱ ♣❛r❛ ❡ss❡ m ❡ ♣❛r❛ f✱ ❛ ❢✉♥çã♦ h : E(K)

  f

  → R s❛t✐s❢❛③ ✭✐✮ ❉❛❞♦ Q ∈ E(K)✳ ❊①✐st❡ ✉♠❛ ❝♦♥st❛♥t❡ c

  1 ✱ q✉❡ ❞❡♣❡♥❞❡ ❞❡ E, f ❡ Q✱ t❛❧ q✉❡

  h f (P + Q) 6 2h f (P ) + c

  1 , ∀P ∈ E(K).

  2

  ✭✐✐✮ ❊①✐st❡ ✉♠❛ ❝♦♥st❛♥t❡ c ✱ ❞❡♣❡♥❞❡♥❞♦ ❞❡ E ❡ f ♣❛r❛ ❛ q✉❛❧

  2

  h ([2]P ) > 2 h (P ) ,

  f f

  2 − c ∀P ∈ E(K).

  ✭✐✐✐✮ P❛r❛ q✉❛❧q✉❡r ❝♦♥st❛♥t❡ c

  3 ✱ ♦ ❝♦♥❥✉♥t♦ f (P ) 6 c

  3

  {P ∈ E(K) : h } é ✜♥✐t♦✳

  f

  P❡❧♦s r❡s✉❧t❛❞♦s ❛♥t❡r✐♦r❡s✱ ✈✐♠♦s q✉❡ ❡ss❛ ❢✉♥çã♦ h s❛t✐s❢❛③ t♦❞❛s ❡ss❛s ❝♦♥✲ ❞✐çõ❡s✱ ♠♦str❛♥❞♦ q✉❡ E(K) é ✜♥✐t❛♠❡♥t❡ ❣❡r❛❞♦ ♣❡❧♦ ❚❡♦r❡♠❛ ❞❛ ❉❡s❝✐❞❛✳ Pr♦✈❛♥❞♦ ❛ss✐♠ ♦ ❚❡♦r❡♠❛ ❞❡ ▼♦r❞❡❧❧✲❲❡✐❧✳

  P❡❧♦ q✉❡ ❛❝❛❜❛♠♦s ❞❡ ♣r♦✈❛r✱ ♦ ❣r✉♣♦ ❞♦s ♣♦♥t♦s K✲r❛❝✐♦♥❛✐s ❞❡ ✉♠❛ ❝✉r✈❛ ❡❧í♣✲ t✐❝❛ E/K é ✜♥✐t❛♠❡♥t❡ ❣❡r❛❞♦✱ ♦♥❞❡ K é ✉♠ ❝♦r♣♦ ❞❡ ♥ú♠❡r♦s✳ ❙❡♥❞♦ ❛ss✐♠✱ ♣♦❞❡♠♦s ❞❡❝♦♠♣♦r ❡ss❡ ❣r✉♣♦ ❡♠ ✉♠❛ s♦♠❛ ❞✐r❡t❛ ❡♥tr❡ ♦ s❡✉ s✉❜❣r✉♣♦ ❞❡ t♦rçã♦ ❡ ✉♠ ♥ú♠❡r♦ ✜♥✐t♦ ❞❡ ❝ó♣✐❛s ❞❡ Z✳ ❖ ♥ú♠❡r♦ ❞❡ ❝ó♣✐❛s ❞❡ Z q✉❡ ❛♣❛r❡❝❡♠ ♥❡ss❛ ❞❡❝♦♠♣♦s✐çã♦ é ✉♠ ✐♥✈❛r✐❛♥t❡ ❞❛ ❝✉r✈❛ ❡❧í♣t✐❝❛✱ ♦ q✉❛❧ é ✉♠ ✐♠♣♦rt❛♥t❡ ♣❡rs♦♥❛❣❡♠ ♥❛ ❈♦♥❥❡❝t✉r❛ ❞❡ ❇✐r❝❤ ❡ ❙✇✐♥♥❡rt♦♥✲❉②❡r✱ s♦❜r❡ ❛ q✉❛❧ ❢❛❧❛r❡♠♦s ♥♦ ♣ró①✐♠♦ ❝❛♣ít✉❧♦✳

  ✼✷ ●❡♦♠❡tr✐❝❛♠❡♥t❡✱ ♦ ❚❡♦r❡♠❛ ❞❡ ▼♦r❞❡❧❧✲❲❡✐❧ ❞✐③ q✉❡ ❡①✐st❡ ✉♠ ❝♦♥❥✉♥t♦ ✜♥✐t♦

  ❞❡ ♣♦♥t♦s K✲r❛❝✐♦♥❛✐s ❝♦♠ ♦s q✉❛✐s ♣♦❞❡♠♦s ♦❜t❡r t♦❞♦s ♦s ❞❡♠❛✐s ♣♦♥t♦s K✲r❛❝✐♦♥❛✐s ✉t✐❧✐③❛♥❞♦ ❞♦ ♣r♦❝❡ss♦ ❞❡ ❝♦r❞❛✲t❛♥❣❡♥t❡✱ q✉❡ é ♦ ♥♦♠❡ ❞❛❞♦ ❛♦ ♣r♦❝❡ss♦ ✉t✐❧✐③❛❞♦ ♥❛ ❞❡✜♥✐çã♦ ❞❛ ❧❡✐ ❞❡ ❣r✉♣♦ ❞❛ ❝✉r✈❛ ❡❧í♣t✐❝❛✳

  ❯♠ ♦✉tr♦ r❡s✉❧t❛❞♦ s♦❜r❡ ❝❧❛ss✐✜❝❛çã♦ ❞♦ ❝♦♥❥✉♥t♦ ❞♦s ♣♦♥t♦s K✲r❛❝✐♦♥❛✐s ❞❡ ❝✉r✈❛s ❛❧❣é❜r✐❝❛s ♣r♦❥❡t✐✈❛s ❡ s✉❛✈❡s✱ ❞❡✜♥✐❞❛s s♦❜r❡ ❝♦r♣♦s ❞❡ ♥ú♠❡r♦s K✱ é ♦ ❝❤❛♠❛❞♦ ❚❡♦r❡♠❛ ❞❡ ❋❛❧t✐♥❣s✳ ❊ss❡ t❡♦r❡♠❛ ❞✐③ q✉❡ s❡ ❛ ❝✉r✈❛ t✐✈❡r ❣ê♥❡r♦ ♠❛✐♦r ♦✉ ✐❣✉❛❧ ❛ ✷✱ ❡♥tã♦ ♦ ❝♦♥❥✉♥t♦ ❞❡ s❡✉s ♣♦♥t♦s K✲r❛❝✐♦♥❛✐s é ✜♥✐t♦✳ ❊ss❡ r❡s✉❧t❛❞♦ ❢♦✐ ❝♦♥❥❡❝t✉r❛❞♦ ♣♦r ▼♦r❞❡❧❧ ❡ ♣r♦✈❛❞♦ ♣♦r ❋❛❧t✐♥❣s ❡♠ ✶✾✽✸✱ ♦ q✉❡ ❝♦♥tr✐❜✉✐✉ ♣❛r❛ q✉❡ ❢♦ss❡ ❛❣r❛❝✐❛❞♦ ❝♦♠ ✉♠❛ ▼❡❞❛❧❤❛ ❋✐❡❧❞s ❡♠ ✶✾✽✻✳ ❊ss❡ t❡♦r❡♠❛✱ ❥✉♥t♦ ❛♦ ❚❡♦r❡♠❛ ❞❡ ▼♦r❞❡❧❧✲❲❡✐❧ ♥♦s ♣♦ss✐❜✐❧✐t❛ ❢❛③❡r ✉♠ q✉❛❞r♦ s♦❜r❡ ❛ ❡str✉t✉r❛ ❞♦ ❝♦♥❥✉♥t♦ ❞❡ss❡s ♣♦♥t♦s K✲r❛❝✐♦♥❛✐s✳

  ●ê♥❡r♦ ❞❡ C ❊str✉t✉r❛ ❞❡ C(K) g = 0 C(K) = ∅ ♦✉ C(K) é ✐♥✜♥✐t♦✳ g = 1 C(K)

  é ❛❜❡❧✐❛♥♦ ✜♥✐t❛♠❡♥t❡ ❣❡r❛♥❞♦✱ ♣♦❞❡♥❞♦ s❡r ✜♥✐t♦ ♦✉ ✐♥✜♥✐t♦ ✭▼♦r❞❡❧❧✲❲❡✐❧✮✳ g > 2 C(K)

  é ✜♥✐t♦ ✭❋❛❧t✐♥❣s✮✳

  ❈❛♣ít✉❧♦ ✹ ❆ ❈♦♥❥❡❝t✉r❛ ❞❡ ❇✐r❝❤ ❡ ❙✇✐♥♥❡rt♦♥✲❉②❡r ✭❇❙❉✮ ❡ ♦ Pr♦❜❧❡♠❛ ❞♦s ◆ú♠❡r♦s ❈♦♥❣r✉❡♥t❡s ✭P◆❈✮

  ❈♦♠♦ ✈✐♠♦s ♥♦ ❈❛♣ít✉❧♦ ✸✱ s❡ K é ✉♠ ❝♦r♣♦ ❞❡ ♥ú♠❡r♦s ❡ E/K é ✉♠❛ ❝✉r✈❛ ❡❧í♣t✐❝❛✱ ❡♥tã♦ ♦ ❝♦♥❥✉♥t♦ ❞❡ s❡✉s ♣♦♥t♦s K✲r❛❝✐♦♥❛✐s✱ ❝❤❛♠❛❞♦ ❣r✉♣♦ ❞❡ ▼♦r❞❡❧❧✲❲❡✐❧ ❞❡ E/K✱ é ✉♠ ❣r✉♣♦ ❛❜❡❧✐❛♥♦ ✜♥✐t❛♠❡♥t❡ ❣❡r❛❞♦✳

  ❙❡♥❞♦ ❛ss✐♠✱ ♣❡❧♦ t❡♦r❡♠❛ ❞❡ ❝❧❛ss✐✜❝❛çã♦ ❞♦s ❣r✉♣♦s ❛❜❡❧✐❛♥♦s ✜♥✐t❛♠❡♥t❡ ❣❡✲ r❛❞♦s✱ ♣♦❞❡♠♦s ❡s❝r❡✈❡r

  r

  E(K) ∼ = E(K) tor , ⊕ Z

  tor

  ❡♠ q✉❡ E(K) é ♦ s✉❜❣r✉♣♦ ❞❡ t♦rçã♦✱ q✉❡ é ♦ s✉❜❣r✉♣♦ ❞♦s ♣♦♥t♦s ❞❡ ♦r❞❡♠ ✜♥✐t❛ ❡ r é ✉♠ ✐♥t❡✐r♦ ♥ã♦ ♥❡❣❛t✐✈♦✱ q✉❡ é ✉♠ ✐♥✈❛r✐❛♥t❡ ❞❛ ❝✉r✈❛✳ ❆ ❡st❡ ✐♥✈❛r✐❛♥t❡ r ❝❤❛♠❛r❡♠♦s ♦ ♣♦st♦ ❛❧❣é❜r✐❝♦ ❞❛ ❝✉r✈❛ ❡❧í♣t✐❝❛✳

  ❏á s❛❜í❛♠♦s q✉❡ [

  E(K) = E[m](K),

  tor

m>1

  ❡ q✉❡ ❝❛❞❛ ✉♠ ❞♦s E[m] é ✜♥✐t♦✱ ♠❛s ❡ss❛ ✉♥✐ã♦ ❛ ♣r✐♥❝í♣✐♦ ♣♦❞❡r✐❛ s❡r ✐♥✜♥✐t❛✳ ▼❛s ❞❡♣♦✐s ❞❡ ♣r♦✈❛❞♦ ♦ ❚❡♦r❡♠❛ ❞❡ ▼♦r❞❡❧❧✲❲❡✐❧✱ t❡♠♦s q✉❡ ❡ss❛ ✉♥✐ã♦ t❛♠❜é♠ é ✜♥✐t❛✱ ✈✐st♦ q✉❡ ♦ s✉❜❣r✉♣♦ ❞❡ t♦rçã♦ ❞❡ ✉♠ ❣r✉♣♦ ❛❜❡❧✐❛♥♦ ✜♥✐t❛♠❡♥t❡ ❣❡r❛❞♦ é ✜♥✐t♦✳ ❙♦❜r❡ ♦ s✉❜❣r✉♣♦ ❞❡ t♦rçã♦ ❞❡ E(K)✱ ❇❛rr② ▼❛③✉r ♣r♦✈♦✉ q✉❡ s❡ E/Q é ✉♠❛ ❝✉r✈❛ ❡❧í♣t✐❝❛✱

  tor

  ❡♥tã♦ E(Q) é ✐s♦♠♦r❢♦ ❛ ✉♠ ❞♦s s❡❣✉✐♥t❡s ❣r✉♣♦s Z/nZ,

  ❝♦♠ 1 ≤ n ≤ 10 ♦✉ n = 12; Z/2Z ⊕ Z/2nZ, ❝♦♠ 1 ≤ n ≤ 4. ❆ss✐♠✱ t❡♠♦s ✉♠❛ ❝❧❛ss✐✜❝❛çã♦ ♣❛r❛ ♦s s✉❜❣r✉♣♦s ❞❡ t♦rçã♦ ❞❡ ❝✉r✈❛s ❡❧í♣t✐❝❛s r❛❝✐♦♥❛✐s✳

  ❉❛q✉✐ ❡♠ ❞✐❛♥t❡ ♥♦s ❝♦♥❝❡♥tr❛r❡♠♦s ❡♠ ❡st✉❞❛r ❛❧❣✉♠❛s q✉❡stõ❡s s♦❜r❡ ❝✉r✈❛s

  ✼✹ ❡❧í♣t✐❝❛s r❛❝✐♦♥❛✐s✳ ❈♦♠♦ ♦ ❚❡♦r❡♠❛ ❞❡ ▼❛③✉r ❝❧❛ss✐✜❝❛ ❛ ♣❛rt❡ ❞❡ t♦rçã♦ ❞♦ ❣r✉♣♦ ❞❡ ▼♦r❞❡❧❧✲❲❡✐❧✱ ♣♦❞❡♠♦s t❛♠❜é♠ ♥♦s ♣❡r❣✉♥t❛r s♦❜r❡ ❛s ♣♦ss✐❜✐❧✐❞❛❞❡s ♣❛r❛ ♦ ♣♦st♦ ❛❧❣é❜r✐❝♦ ❞❛ ❝✉r✈❛ ❡❧í♣t✐❝❛✳ ❈❛❧❝✉❧❛r ♦ ♣♦st♦ ❞❡ ✉♠❛ ❝✉r✈❛ ❡❧✐♣t✐❝❛ ❡♠ ❣❡r❛❧ é ✉♠ ♣r♦❜❧❡♠❛ ❞✐❢í❝✐❧✱ ❡ s♦❜r❡ ✐ss♦✱ ❡♠ ✶✾✻✺✱ ❇✳ ❏✳ ❇✐r❝❤ ❡ ❙✐r ❍✳ P✳ ❋✳ ❙✇✐♥♥❡rt♦♥✲❉②❡r ❝♦♥❥❡❝t✉r❛r❛♠ q✉❡ ♦ ♣♦st♦ ❛❧❣é❜r✐❝♦ ❞❡ ✉♠❛ ❝✉r✈❛ ❡❧í♣t✐❝❛ r❛❝✐♦♥❛❧ s❡r✐❛ ✐❣✉❛❧ ❛ ✉♠ ♦✉tr♦ ✐♥✈❛r✐❛♥t❡✱ ❛❣♦r❛ ❛♥❛❧ít✐❝♦✱ r❡❧❛❝✐♦♥❛❞♦ ❛ ▲✲sér✐❡ ❞❛ ❝✉r✈❛✱ q✉❡ ❞❡✜♥✐r❡♠♦s ♥❛ ♣ró①✐♠❛ s❡çã♦✱ ♦♥❞❡ ❢❛❧❛r❡♠♦s ❛ r❡s♣❡✐t♦ ❞❛ ❈♦♥❥❡❝t✉r❛ ❞❡ ❇✐r❝❤ ❡ ❙✇✐♥♥❡rt♦♥✲❉②❡r✳

  ❖ ❡st✉❞♦ ❞❛ ❛r✐t♠ét✐❝❛ ❞❛s ❝✉r✈❛s ❡❧í♣t✐❝❛s t❡♠ ♣♦ss✐❜✐❧✐t❛❞♦ ✉♠ ❣r❛♥❞❡ ❛✈❛♥ç♦ ❡♠ ✈ár✐❛s ár❡❛s ♥ã♦ ❛♣❡♥❛s ❞❛ ♠❛t❡♠át✐❝❛ ♣✉r❛ ❛ss✐♠ ❝♦♠♦ s✉❛s ❛♣❧✐❝❛çõ❡s✱ ❡♠ ❝r✐♣t♦✲ ❣r❛✜❛ ♣♦r ❡①❡♠♣❧♦✳ ➱ ❞❡ ✐♥t❡r❡ss❡ ❡♠ ●❡♦♠❡tr✐❛ ❆r✐t♠ét✐❝❛✱ ✉s❛r ❞❛s ❢❡rr❛♠❡♥t❛s ❞❛ ●❡♦♠❡tr✐❛ ❆❧❣é❜r✐❝❛ ♣❛r❛ ♦❜t❡r r❡s♣♦st❛s s♦❜r❡ q✉❡stõ❡s ❛r✐t♠ét✐❝❛s✳ ❯♠ ❜♦♠ ❡①❡♠♣❧♦ ❞♦ ✉s♦ ❞❡ss❛ t❡♦r✐❛ ♣❛r❛ ✜♠ ❞❡ r❡s♣♦♥❞❡r q✉❡stõ❡s ❛r✐t♠ét✐❝❛s é ♦ ❢❛♠♦s♦ Ú❧t✐♠♦ ❚❡✲ ♦r❡♠❛ ❞❡ ❋❡r♠❛t✱ q✉❡ ❢♦✐ ♣r♦✈❛❞♦ ❞❡♣♦✐s ❞❡ t♦❞❛ ✉♠❛ ❧✐♥❣✉❛❣❡♠ ❡ ❢❡rr❛♠❡♥t❛s t❡r❡♠ s✐❞♦ ❝r✐❛❞❛s ♣❛r❛ ♣r♦✈❛r ❛ ❈♦♥❥❡❝t✉r❛ ❞❡ ❚❛♥✐②❛♠❛✲❙❤✐♠✉r❛✱ ♣r♦✈❛❞❛ ♣♦r ❆♥❞r❡✇ ❲✐❧❡s ❡ ❘✐❝❤❛r❞ ❚❛②❧♦r ❡♠ ✶✾✾✺✳ ❆ ❈♦♥❥❡❝t✉r❛ ❞❡ ❋❡r♠❛t é ❝♦r♦❧ár✐♦ ❞♦ q✉❡ ❤♦❥❡ é ❝♦♥❤❡❝✐❞♦ ❝♦♠♦ ❚❡♦r❡♠❛ ❞❛ ▼♦❞✉❧❛r✐❞❛❞❡ ✭❱❡r ❬❲✐❧❡s✷❪✮✳

  ◆❡st❡ ❈❛♣ít✉❧♦✱ ❛❧é♠ ❞❡ ❢❛❧❛r ❞❛ ❈♦♥❥❡❝t✉r❛ ❞❡ ❇✐r❝❤ ❡ ❙✇✐♥♥❡rt♦♥✲❉②❡r ✭❇✳❙✳❉✮✱ ❡st✉❞❛r❡♠♦s t❛♠❜é♠ ♦ Pr♦❜❧❡♠❛ ❞♦s ◆ú♠❡r♦s ❈♦♥❣r✉❡♥t❡s✱ ❡ ❝♦♠♦ ❡ss❡ ♣r♦❜❧❡♠❛ ❡stá r❡❧❛❝✐♦♥❛❞♦ ❝♦♠ ❝✉r✈❛s ❡❧í♣t✐❝❛s ❡ ❝♦♠ ❛ ❝♦♥❥❡❝t✉r❛ ❇✳❙✳❉✳

  ✹✳✶ ❆ ❈♦♥❥❡❝t✉r❛ ❞❡ ❇✐r❝❤ ❡ ❙✇✐♥♥❡rt♦♥✲❉②❡r

  ❆ ✜♠ ❞❡ ❡♥✉♥❝✐❛r ❛ ❈♦♥❥❡❝t✉r❛ ❞❡ ❇✐r❝❤ ❡ ❙✇✐♥♥❡rt♦♥✲❉②❡r ♣r❡❝✐s❛♠♦s ❞❡ ❛❧❣✉✲ ♠❛s ❞❡✜♥✐çõ❡s ❡ ♥♦t❛çõ❡s✳ ❉❡✜♥✐çã♦ ✹✳✶ ✭▲✲sér✐❡✮✳ ❯♠❛ L✲sér✐❡ é ✉♠❛ sér✐❡ ❞❛ ❢♦r♠❛

  n

  X b

  L(s) = ,

  s

  n

  n>1 n

  ♦♥❞❡ b ∈ C✳ ❆❣♦r❛ ❝♦♥s✐❞❡r❡♠♦s ✉♠❛ ❝✉r✈❛ ❡❧í♣t✐❝❛ E/Q ❝♦♠ ❡q✉❛çã♦ ❞❡ ❲❡✐❡rtr❛ss r❡❞✉③✐❞❛

  2

  

3

E : y = x + ax + b,

  ❝♦♠ a, b ∈ Z ❡ ∆ s❡✉ ❞✐s❝r✐♠✐♥❛♥t❡✳ P❛r❛ ❝❛❞❛ p ♣r✐♠♦✱ ❝♦♥s✐❞❡r❡♠♦s ❛ r❡❞✉çã♦ ♠ó❞✉❧♦ p✱

  Z

  p

  → F z 7→ z,

  ✼✺ = Z/pZ

  p

  ♦♥❞❡ F é ♦ ❝♦r♣♦ ♣r✐♠♦ ❞❡ ❝❛r❛❝t❡ríst✐❝❛ p✳ P♦❞❡♠♦s ❝♦♥s✐❞❡r❛r ❛ r❡❞✉çã♦ ♠ó❞✉❧♦ p ❞❛ ❝✉r✈❛ E✱ ♦✉ s❡❥❛✱

  2

  

3

E : y = x + ax + b. p p

  ❚❡♠♦s q✉❡ ❛ ❝✉r✈❛ r❡❞✉③✐❞❛ E é s✉❛✈❡ s❡✱ ❡ s♦♠❡♥t❡ s❡✱ p ♥ã♦ ❞✐✈✐❞❡ ∆✳ P♦✐s ❛ss✐♠ t❡r❡♠♦s v (∆) = ord (∆) = 0.

  p p

  (F )

  p p p p p

  ❉❛❞♦ q✉❡ F é ✜♥✐t♦✱ ♦ ❝♦♥❥✉♥t♦ E ❞♦s ♣♦♥t♦s F ✲r❛❝✐♦♥❛✐s ❞❡ E é ✜♥✐t♦✳ ❈♦♠ ✐ss♦ ♣♦❞❡♠♦s ❞❡✜♥✐r a = p + 1 (F )

  

p p p

− |E |.

  ❆❣♦r❛ ❞❡✜♥✐♠♦s ❛ L✲sér✐❡ ❞❡ ✉♠❛ ❝✉r✈❛ E/Q ✈✐❛ ✉♠ ♣r♦❞✉t♦ ❞❡ ❊✉❧❡r✳ ❉❡✜♥✐çã♦ ✹✳✷ ✭▲✲sér✐❡ ❞❡ ✉♠❛ ❝✉r✈❛ ❡❧í♣t✐❝❛✮✳ ❙❡❥❛ E/Q ❝✉r✈❛ ❡❧í♣t✐❝❛✳ ❙✉❛ L✲sér✐❡ é ❞❡✜♥✐❞❛ ♣❡❧♦ ♣r♦❞✉t♦ ❞❡ ❊✉❧❡r

  Y Y

  1

  1 L(E, s) = ,

  −s −s 1−2s

  1 p 1 p + p

  p p

  − a − a

  p | ∆ p ∤ ∆

  ❝♦♠ s ∈ C✳ ❖❜s❡r✈❡♠♦s q✉❡ ❞❡✜♥✐♠♦s L(E, s) ❝♦♠♦ ✉♠ ♣r♦❞✉t♦ ✐♥✜♥✐t♦✳ P❛r❛ ♦❜t❡r♠♦s

  ✉♠❛ sér✐❡ ✐♥✜♥✐t❛ ❝♦♥s✐❞❡r❡ ❛ ❢✉♥çã♦ i(p) = 1, ❝❛s♦ p ∤ ∆ i(p) = 0, ❝❛s♦ p | ∆,

  ❡ ❛ s❡❣✉✐♥t❡ s❡q✉ê♥❝✐❛ ❞❡✜♥✐❞❛ r❡❝✉rs✐✈❛♠❡♥t❡✳ b

  1 = 1

  b p = a p , p ♣r✐♠♦

  m m− m−

  1

  2

  b p = b p , m > 2 · b p − i(p)p · b p b = b ,

  rs r s · b s❡ mdc(r, s) = 1.

  b P n

  , ❆ss✐♠✱ s❡ t♦♠❛r♠♦s L(s) = t❡♠♦s ♦ s❡❣✉✐♥t❡ r❡s✉❧t❛❞♦✳

  n>1

s

  n Pr♦♣♦s✐çã♦ ✹✳✸✳ L(E, s) = L(s)✳ ❉❡♠♦♥str❛çã♦✳ ❚❡♠♦s q✉❡

  −ms −ms −ms m m− m−

  1

  2

  b p = b p , m > 2 · p · b p · p − i(p)p · b p · p

  • i(p)p
  • i(p)p
  • b
  • i(p)b
  • b
  • i(p)b

  p

  m>0

  b

  p m

  · p

  −ms

  =

  1 (1

  − b

  p

  −s

  ❆ss✐♠✱

  1−2s

  ) . P♦rt❛♥t♦

  L(E, s) = Y

  p

  1 (1

  − b

  p

  p

  −s

  X

  1−2s ) = 1.

  ) (

  −s .

  · p

  1−2s

  = b

  1

  − b

  1

  )b

  p

  · p

  ❈♦♠♦ b

  

p p

−s

  1

  = 1 ✱ ❡♥tã♦

  X

  m>0

  b p

  m

  · p

  −ms

  (1 − b

  1−2s

  ❥á q✉❡ b p = a

  −s

  n

  n>1

  |b

  n

  n

  −s

  | =

  X

  n>1

  |b

  ||n

  ❡ s = α + iβ✱ t❡♠♦s

  −s

  | =

  X

  n>1

  |b

  n

  | · |e

  −α log n

  | · |e

  X

  −s

  p

  ! =

  ) =

  Y

  p

  X

  m>0

  b

  p m

  · p

  

−ms

  X

  n

  n>1

  b

  n

  n

  s = L(s).

  ❆❣♦r❛ s❡♥❞♦ L(E, s) = P

  n>1

  b

  n

  1

  · p

  −iβ log n |.

  − i(p)p

  p

  p

  −s

  X

  m>1

  b

  p m

  · p

  −ms

  1−2s

  −ms

  X

  m>0

  b

  p m

  · p

  −ms

  X

  m>2

  b

  = b

  · p

  · p

  m>2

  ✼✻

  X

  m>2

  b p

  m

  · p

  −ms

  =

  X

  b p · b p

  2

  m−

  1

  · p

  −ms

  − i(p)p

  X

  m>2

  b

  p m−

  p m

  −ms

  p

  · p

  − i(p)b

  1

  · p

  1−2s

  − i(p)b

  p

  · p

  1−3s

  p

  −s

  · p

  − b

  2 p · p −2s

  p

  · p

  1−3s

  1

  − b

  1

  · b

  −2s

  2 p

  (1 − b p p

  1−3s .

  −s

  1−2s

  ) = b

  2 p

  · p

  −2s

  − i(p)b

  1 · p 1−2s

  − i(p)b p · p

  ❊

  ) = b

  X

  m>0

  b p

  m

  · p

  −ms

  (1 − b

  p p −s

  1−2s

  • (1
  • i(p)p
  • i(p)p
  • i(p)p

  ✼✼ ❈♦♠♦ n é ✉♠ r❡❛❧ ♣♦s✐t✐✈♦✱ ❡♥tã♦

  −iβ log n

  |e | = 1, ❞❡ ♠♦❞♦ q✉❡ ❛ ❝♦♥✈❡r❣ê♥❝✐❛ ❞❛ L✲sér✐❡ só ❞❡♣❡♥❞❡ ❞❛ ♣❛rt❡ r❡❛❧ ❞❡ s✳ ❆❧é♠ ❞✐ss♦✱ s❡

  α = Re(s ) > Re(s ) = α ,

  1

  1

  2

  2

  ❡♥tã♦

  

−α log n −α log n

  1

  2 |e | < |e |.

  2

  1

  ❆ss✐♠✱ s❡ ❛ L✲sér✐❡ ❝♦♥✈❡r❣❡ ♣❛r❛ s = s ❡♥tã♦ ❝♦♥✈❡r❣❡ ♣❛r❛ s = s ✳

  3 ❆ sér✐❡ L(E, s) ❝♦♥✈❡r❣❡ ♥♦ s❡♠✐♣❧❛♥♦ ❝♦♠♣❧❡①♦ ❞❡t❡r♠✐♥❛❞♦ ♣♦r Re(s) > ✳ ◆❛

  2 ♣r♦✈❛ ❞❡ss❡ ❢❛t♦✱ é ✉s❛❞♦ ♦ ❚❡♦r❡♠❛ ❞❡ ❍❛ss❡ ♣❛r❛ ❞❛r ✉♠❛ ❡st✐♠❛t✐✈❛ ♣❛r❛ ♦s t❡r♠♦s ❞♦ ♣r♦❞✉t♦ q✉❡ ❞❡✜♥❡ L(E, s)✳ ▲♦❣♦ ❛❜❛✐①♦ ❡♥✉♥❝✐❛♠♦s ♦ ❚❡♦r❡♠❛ ❞❡ ❍❛ss❡ ❡ ♣r♦✈❛♠♦s ❛ ❝♦♥✈❡r❣ê♥❝✐❛ ❞❛ sér✐❡ ♥❛ r❡❣✐ã♦ ❡♠ q✉❡stã♦✳ ❚❡♦r❡♠❛ ✹✳✹ ✭❍❛ss❡✮✳ ❉❛❞♦ p ♣r✐♠♦ t❡♠♦s

  √

p p.

|a | 6 2

  ❉❡♠♦♥str❛çã♦✳ ❱❡r ✭❈❛♣ít✉❧♦ ✺✱ ❚❡♦r❡♠❛ ✶✳✶✱ ❬❙✐❧✈❡r♠❛♥❪✮✳

  3 Pr♦♣♦s✐çã♦ ✹✳✺✳ ❆ sér✐❡ L(E, s) ❝♦♥✈❡r❣❡ s❡ Re(s) > ✳

  2 ❉❡♠♦♥str❛çã♦✳ P❛r❛ ❡❢❡✐t♦ ❞❡ ♠♦str❛r ❛ ❝♦♥✈❡r❣ê♥❝✐❛✱ ♣♦❞❡♠♦s s✉♣♦r q✉❡ ♦s ❢❛t♦r❡s

  1 ❞❛❞♦s ♣♦r ✭q✉❡ ❛♣❛r❡❝❡♠ ❛♣❡♥❛s ✉♠ ♥ú♠❡r♦ ✜♥✐t♦ ❞❡ ✈❡③❡s✮ sã♦ ❞❛ ❢♦r♠❛

  −s

  1 p

  p

  − a

  1 ✱ ❞❡ ♠♦❞♦ q✉❡ ❛ ♣❛rt✐r ❞❛ r❡❝♦rrê♥❝✐❛

  −s 1−2s

  1 p p + p − a b = 1

  1

  b = a , p

  p p

  ♣r✐♠♦

  m m− 1 m−

  2

  b = b , m > 2

  p p p p

  · b − p · b b = b ,

  rs r s · b s❡ mdc(r, s) = 1.

  ♦❜t❡♠♦s ❛ L✲s❡r✐❡

  1 L(s) = = .

  X Y b n

  s −s 1−2s

  n 1 p + p

  p

  − a

  n>1 p

  3 ▼♦str❛r❡♠♦s q✉❡ L(s) ❝♦♥✈❡r❣❡ ♣❛r❛ Re(s) > ✳

  2

  , ∀ n > 1.

  n

  n+1 p − β n+1

p

  α

  p

  − β

  

p

  , ∀ n > 0. ❆❣♦r❛✱ ❝♦♠♦ c

  = α

  n

  n+1 p − β n+1 p

  α p − β

  p

  ✱ ❡♥tã♦ c

  n+1

  = α

  = α

  P♦rt❛♥t♦✱ c

  α p − β

  X

  n>0

  β

  n p

  z

  n

  =

  n>0

  n .

  α

  n+1 p − β n+1 p

  α

  p

  − β

  p

  · z

  n+2 p − β n+2 p

  p

  ·

  p

  p

  − α

  p β p

  · α

  n p − β n p

  α

  − β

  p

  p

  = b

  p

  · c

  n

  − p · c

  n−1

  − β

  α

  = (α

  β

  n+1 p − β n+1 p

  ) · (α

  p

  p

  ) − α

  n+1 p

  p

  n+1 p − β n+1 p

  p

  β

  n+1 p

  α

  p − β p

  = (α p + β p ) ·

  α

  X

  p

  ✼✽ ❙❡❥❛ p ♣r✐♠♦ ❡ ❝♦♥s✐❞❡r❡♠♦s

  n .

  =

  X

  n>0

  c

  n

  z

  ■r❡♠♦s ❡♥❝♦♥tr❛r ♦s ❝♦❡✜❝✐❡♥t❡s c

  p z + pz

  n

  ✳ ◆❡ss❡ ♠♦♠❡♥t♦ ♣r❡❝✐s❛r❡♠♦s ✉s❛r q✉❡ α

  p

  6= β

  p

  ✱ ♠❛s ♣r♦✈❛r❡♠♦s ✐ss♦ ♠❛✐s ❛❞✐❛♥t❡✳ ❚❡♠♦s

  1

  

2

  1 − b

  p z + pz

  z)(1 − β

  1 − b

  p

  z + pz

  2

  = (1 − α

  p

  p

  1

  z), ❝♦♠ α

  p , β p

  ∈ C✳ ❚❡♠♦s q✉❡ α

  p + β p = b p

  ❡ α

  p β p = p

  ✳ ❈♦♥s✐❞❡r❡♠♦s ❡♥tã♦✱ ❛ sér✐❡ ❞❡ ❚❛②❧♦r ❞❛❞❛ ♣♦r

  1 − b

  2

  α p − β

  ·

  1 − β

  p z

  = α

  p

  α p − β

  p

  X

  − β

  

n>0

  α

  n p

  z

  n

  − β

  p

  p

  p

  =

  p

  1 (1

  − α

  p z)(1

  − β

  p z)

  = α

  α

  α

  p

  − β

  p

  1 − α

  p z

  − β

  p

  • β
  • α

  ✼✾ ❉❛í✱ c = 1 c

  1 = α p + β p = b p

  2

  c

2 = b p p .

  · b − p · 1 = b p

  k

  ❉❡ ♠♦❞♦ q✉❡ ♣❡❧❛ r❡❝♦rrê♥❝✐❛ ❛❝✐♠❛ ❡ ♣❡❧❛ r❡❝♦rrê♥❝✐❛ q✉❡ ❞❡✜♥❡ ♦s b p ✱ ♦❜t❡♠♦s

  n

n+1 n+1

  α

  X

  p − β p i n−i n b p = c n = = α β . p p

  α

  

p p

  − β

  i=0

  √ p.

  p p

  ❆❣♦r❛ ✐r❡♠♦s ♠♦str❛r q✉❡ |α | = |β | = ❈♦♠ ❡❢❡✐t♦✱ ♣❡❧♦ ❚❡♦r❡♠❛ ❞❡ ❍❛ss❡✱ √ √

  2

  p p

  p p p p

  |b | = |a | 6 2 ✱ ❡♥tã♦ |b | < 2 ✭♣♦✐s b é ✐♥t❡✐r♦✮✳ P♦rt❛♥t♦✱ b p − 4p < 0✳

  2

  2

  z + pz = 0

  p

  ❈♦♠♦ ♦ ❞✐s❝r✐♠✐♥❛♥t❡ ❞❛ ❡q✉❛çã♦ 1 − b é b p − 4p < 0✱ ❡♥tã♦ ❛s

  1

  1 ❞✉❛s r❛í③❡s ❡ ❞❛ ❡q✉❛çã♦ ❢♦r♠❛♠ ✉♠ ♣❛r ❞❡ ❝♦♠♣❧❡①♦s ♥ã♦ r❡❛✐s ❡ ❝♦♥❥✉❣❛❞♦s✱ ❡♠

  α β

  p p

  1

  1 ♣❛rt✐❝✉❧❛r α p p ✳ P♦rt❛♥t♦✱ | p p

  6= β | = | |✳ ❆ss✐♠✱ |α | = |β |✳ α β

  p p

  √

  2

  = p = p

  p p p p p p p

  ❈♦♠♦ α · β ✱ ❡♥tã♦ |α | |α · β | = p ❡ |α | = |β | = ✳ ❉❡ ♠♦❞♦ q✉❡

  n n

  X X

  i n−1 n/2 n/2 n β p = (n + 1)p . p

  |b | 6 |α p p | =

  i=0 i=0 k 1 m k km

  = b k1

  n

  ❆❣♦r❛ s❡❥❛ n = p

  1 · · · p m ✳ ❚❡♠♦s q✉❡ b · · · b ✱ ❞❡ ♦♥❞❡ s❡❣✉❡ q✉❡ p m p

  1

km

k1 n

  |b | = |b | · · · |b |

  

p m

p

  1

k /2

  1 k m /2

  6 (k + 1)p + 1)p

  

1 m

1 · · · (k m m

  Y

  k i /2

  = (k + 1)p

  i

i

i=1 1/2

  = d(n) , · n

  ♦♥❞❡ d(n) é ♦ ♥ú♠❡r♦ ❞❡ ❞✐✈✐s♦r❡s ♣♦s✐t✐✈♦s ❞❡ n✳ P❛r❛ ✜♥❛❧✐③❛r ❛ ❞❡♠♦♥str❛çã♦✱ ✉s❛r❡♠♦s ♦ ❢❛t♦ ❞❡ q✉❡ ❛ ❢✉♥çã♦ ❩❡t❛ ❞❡ ❘✐❡♠❛♥♥✱

1 P

  ζ(s) = ✱ ❡stá ❞❡✜♥✐❞❛ ♣❛r❛ t♦❞♦ s ❝♦♠ Re(s) > 1✳

  k>1 s

  k

  ✽✵ ❙❡♥❞♦ α = Re(s)✱ t❡♠♦s

  X X

  X

  n −s −α |b |

  =

  n n

  |b · n | = |b | · n

  α

  n

  n>1 n>1 n>1

1/2

  X X

  X d(n) d(n) d(n) · n

  6 = = .

  α α−1/2 β

  n n n

  n>1 n>1 n>1

  ❆❣♦r❛✱ ❝♦♠♦  

  X X

  1 =  1 

  X d(n)

  β β

  n n

  n>1 n>1

k|n

  !

  X X

  1

  1 =

  ·

  β β

  k m

  n>1 km=n

  X X

  1

  1 =

  ·

  β β

  k m

  k>1 m>1

  ! !

  X X

  1

  1

  2

  = = ζ(β) , ·

  β β

  k m

  m>1 k>1

  d(n) b P P n q✉❡ ❡①✐st❡ ♣❛r❛ β > 1✳ ❊♥tã♦ n>1 ❝♦♥✈❡r❣❡ s❡ β > 1✳ P♦rt❛♥t♦✱ n>1 ❝♦♥✈❡r❣❡

  β s

  n n

  1

  1

  3 = α = β > 1 s❡ Re(s) − − ✱ ✐st♦ é✱ s❡ Re(s) > ✳

  2

  2

  2 ❆ ❝♦♥❥❡❝t✉r❛ ❞❡ ❇✐r❝❤ ❡ ❙✇✐♥♥❡rt♦♥✲❉②❡r é ✉♠ ♣r♦❜❧❡♠❛ q✉❡ ❞✐③ r❡s♣❡✐t♦ à ♦r❞❡♠

  ❞❛ L−sér✐❡ ❡♠ s = 1✳ ◆❛ é♣♦❝❛ ❡♠ q✉❡ ♦ ♣r♦❜❧❡♠❛ ❢♦✐ ❢♦r♠✉❧❛❞♦✱ ❡r❛ ❝♦♥❤❡❝✐❞♦ q✉❡ ❝❡rt❛s ❢❛♠í❧✐❛s ❞❡ ❝✉r✈❛s ❡❧í♣t✐❝❛s t✐♥❤❛♠ ❝♦♥t✐♥✉❛çã♦ ❛♥❛❧ít✐❝❛ ❛ t♦❞♦ ♣❧❛♥♦ ❝♦♠♣❧❡①♦✳ ❆t✉❛❧♠❡♥t❡✱ s❛❜❡✲s❡ q✉❡ ♣❛r❛ t♦❞❛s ❛s ❝✉r✈❛s r❛❝✐♦♥❛✐s✱ L(E, s) t❡♠ ❝♦♥t✐♥✉❛çã♦ ❛♥❛❧ít✐❝❛ ❛ t♦❞♦ ♣❧❛♥♦ ❝♦♠♣❧❡①♦✳ ❊ss❡ ❢❛t♦ é ❝♦♥s❡q✉ê♥❝✐❛ ❞♦ ❚❡♦r❡♠❛ ❞❛ ▼♦❞✉❧❛r✐❞❛❞❡✱ ❡ ♣❛r❛ ❞❡t❛❧❤❡s ❛ ❡ss❡ r❡s♣❡✐t♦ ❝✐t❛♠♦s ❛ r❡❢❡rê♥❝✐❛ ❬❲✐❧❡s✷❪✳

  ❆ ✈❡rsã♦ ❞❛ ❝♦♥❥❡❝t✉r❛ q✉❡ ❡♥✉♥❝✐❛r❡♠♦s ❛q✉✐ é ❝♦♥❤❡❝✐❞❛ ♣♦r ❛❧❣✉♥s ❝♦♠♦ ❛ ✈❡rsã♦ ❢r❛❝❛ ❞❛ ❈♦♥❥❡❝t✉r❛ ❞❡ ❇✐r❝❤ ❡ ❙✇✐♥♥❡rt♦✲❉②❡r✱ ✈✐st♦ q✉❡ ❡①✐st❡ ✉♠ ❝♦♠♣❧❡♠❡♥t♦ à ❡ss❛ ❝♦♥❥❡❝t✉r❛ q✉❡ ♣r❡❝✐s❛r✐❛ ❞❡ ♠✉✐t♦ ♠❛✐s ❞♦ q✉❡ ❢♦✐ ❡①♣♦st♦ ❛té ❛q✉✐ ♣❛r❛ q✉❡ ❢♦ss❡ ❡♥✉♥❝✐❛❞❛✳ ❈♦♥❥❡❝t✉r❛ ✹✳✻ ✭❇✳❙✳❉ ✲ ✈❡rsã♦ ❢r❛❝❛✮✳ ❙❡ r é ♦ ♣♦st♦ ❛❧❣é❜r✐❝♦ ❞❡ E/Q✱ ❡♥tã♦ L(E, s) t❡♠ ✉♠ ③❡r♦ ❞❡ ♦r❞❡♠ r ❡♠ s = 1✱ ♦✉ s❡❥❛✱ ❛ ❡①♣❛♥sã♦ ❞❡ ❚❛②❧♦r ❞❡ L(E, s) ❡♠ t♦r♥♦ ❞❡ s = 1 t❡♠ ❛ ❢♦r♠❛

  r r+1

  • L(E, s) = c (s + c (s

  r r+1

  − 1) − 1) t❡r♠♦s ❞❡ ❣r❛✉s ♠❛✐♦r❡s,

  ✽✶

  r

  ❡♠ q✉❡ c 6= 0✳ ❆ ♣❛rt✐r ❞❛ sér✐❡ ❞❡ ❚❛②❧♦r ❛❝✐♠❛✱ ❞❡✜♥✐♠♦s ♦ ♣♦st♦ ❛♥❛❧ít✐❝♦ ❞❡ E ❝♦♠♦ s❡♥❞♦

  ❛ ♦r❞❡♠ ❞❡ L(E, s) ❡♠ s = 1✳ P♦rt❛♥t♦✱ ❛ ❝♦♥❥❡❝t✉r❛ ❞✐③ q✉❡ ♦s ♣♦st♦s ❛❧❣é❜r✐❝♦ ❡ ❛♥❛❧í✲ t✐❝♦ ❞❡ E/Q ❝♦✐♥❝✐❞❡♠✳ ❉❡s❞❡ q✉❡ ❡ss❡ ♣r♦❜❧❡♠❛ ❢♦✐ ❢♦r♠✉❧❛❞♦✱ ❛❧❣✉♥s r❡s✉❧t❛❞♦s ♣❛r❝✐❛✐s ❢♦r❛♠ ❡♥❝♦♥✲ tr❛❞♦s✱ ♠❛s ❛té ❤♦❥❡ ♦ ♣r♦❜❧❡♠❛ ❡♠ s✐ ❝♦♥tí♥✉❛ ❡♠ ❛❜❡rt♦✱ s❡♥❞♦ q✉❡ ❡♠ ✷✵✵✵ ♦ ❈❧❛②

  ▼❛t❤❡♠❛t✐❝s ■♥st✐t✉t❡ ♦ ❧✐st♦✉ ❝♦♠♦ ✉♠ ❞♦s Pr♦❜❧❡♠❛s ❞♦ ▼✐❧ê♥✐♦✳ ❈♦♠ ♦ ❛❝ú♠✉❧♦ ❞❡ ❛❧❣✉♥s ❛✈❛♥ç♦s ♥❛ ❞✐r❡çã♦ ❞❛ ♣r♦✈❛ ❞❡st❛ ❝♦♥❥❡❝t✉r❛✱ ❝❤❡❣♦✉✲s❡

  ❛t✉❛❧♠❡♥t❡ ❛♦ s❡❣✉✐♥t❡ r❡s✉❧t❛❞♦✳ ❚❡♦r❡♠❛ ✹✳✼ ✭❑♦❧②✈❛❣✐♥✮✳ ❆ ✈❡rsã♦ ❢r❛❝❛ ❞❛ ❈♦♥❥❡❝t✉r❛ ❇✳❙✳❉ ✈❛❧❡ ♣❛r❛ ❝✉r✈❛s ❝♦♠ ♣♦st♦ ❛♥❛❧ít✐❝♦ ✵ ♦✉ ✶✳

  ◆❛ ♣ró①✐♠❛ s❡çã♦✱ ❡♥✉♥❝✐❛♠♦s ♦ Pr♦❜❧❡♠❛ ❞♦s ◆ú♠❡r♦s ❈♦♥❣r✉❡♥t❡s ❡ ♠♦str❛✲ ♠♦s q✉❛❧ ❛ s✉❛ r❡❧❛çã♦ ❝♦♠ ❛s ❝✉r✈❛s ❡❧í♣t✐❝❛s ❡ ❝♦♠ ❛ ❝♦♥❥❡❝t✉r❛ ❇✳❙✳❉✳

  ✹✳✷ ❖ Pr♦❜❧❡♠❛ ❞♦s ◆ú♠❡r♦s ❈♦♥❣r✉❡♥t❡s

  ◆❡st❛ s❡çã♦ ❢❛❧❛r❡♠♦s s♦❜r❡ ♦s ♥ú♠❡r♦s ❝♦♥❣r✉❡♥t❡s✱ s✉❛ r❡❧❛çã♦ ❝♦♠ ❛s ❝✉r✈❛s ❡❧í♣t✐❝❛s ❡ ❝♦♠ ❛ ❝♦♥❥❡❝t✉r❛ ❇✳❙✳❉✳ ▼❛s ❛✜♥❛❧ ♦ q✉❡ é ✉♠ ♥ú♠❡r♦ ❝♦♥❣r✉❡♥t❡❄

  ❯♠ ♥ú♠❡r♦ ❝♦♥❣r✉❡♥t❡ é ✉♠ ♥ú♠❡r♦ r❛❝✐♦♥❛❧ q✉❡ é ✐❣✉❛❧ à ♠❡❞✐❞❛ ❞❛ ár❡❛ ❞❡ ✉♠ tr✐â♥❣✉❧♦ r❡tâ♥❣✉❧♦ ❝♦♠ t♦❞♦s ♦s ❧❛❞♦s r❛❝✐♦♥❛✐s✱ ♦✉ s❡❥❛✱ n ∈ Q é ❝♦♥❣r✉❡♥t❡ s❡ ab

  2

  2

  2

  • b = c = n ❡①✐st❡♠ a, b, c ∈ Q ♣♦s✐t✐✈♦s✱ t❛✐s q✉❡ a ❡ ✳ ❖ ❡st✉❞♦ ❞♦s ♥ú♠❡r♦s

  2 ❝♦♥❣r✉❡♥t❡s r❡♠♦♥t❛ ❛♦s ❣r❡❣♦s ❡ q✉❡ ♠❛✐s ❛ ❢r❡♥t❡ ❢♦✐ s✐st❡♠❛t✐③❛❞❛ ♣❡❧♦s ár❛❜❡s✳

  2

  2

  2

  2

  2

  20

  3

  41

  24

  35

  2

  2

  2

  • 3
  • = + 4 = 5 = ❊①❡♠♣❧♦ ✹✳✽✳ ❈♦♠♦ ✱ 3 ❡

  2

  6

  5

  12

  2

  337 ✱ ❡ 5, 6 ❡ 7 sã♦ ❛s r❡s♣❡❝t✐✈❛s ♠❡❞✐❞❛s ❞❛s ár❡❛s ❞♦s tr✐â♥❣✉❧♦s r❡tâ♥❣✉❧♦s ❛ss♦❝✐✲

  60 ❛❞♦s à ❡ss❛s tr✐♣❧❛s ♣✐t❛❣ór✐❝❛s r❛❝✐♦♥❛✐s✱ t❡♠♦s q✉❡ 5, 6 ❡ 7 sã♦ ♥ú♠❡r♦s ❝♦♥❣r✉❡♥t❡s✳

  ❖ ❢❛t♦ ❞❡ q✉❡ 7 é ❝♦♥❣r✉❡♥t❡ ❢♦✐ ♣r♦✈❛❞♦ ♣♦r ❊✉❧❡r✳ ❋❡r♠❛t ♣r♦✈♦✉ ❡♠ ✶✻✹✵✱ q✉❡ ♦ ♥ú♠❡r♦ ✶ ♥ã♦ é ❝♦♥❣r✉❡♥t❡✳ Pr♦♣♦s✐çã♦ ✹✳✾✳ ✶ ♥ã♦ é ❝♦♥❣r✉❡♥t❡✳ ❉❡♠♦♥str❛çã♦✳ ❙✉♣♦♥❤❛♠♦s ♣♦r ❛❜s✉r❞♦ q✉❡ ✶ s❡❥❛ ❝♦♥❣r✉❡♥t❡✱ ♦✉ s❡❥❛✱ q✉❡ ❡①✐st❛♠ r❛✲

  2

  2

  2

  2

  • b = c ❝✐♦♥❛✐s a/d, b/d, c/d t❛✐s q✉❡ a ❡ ab = 2d ✱ ❝♦♠ t♦❞♦s ♦s ✐♥t❡✐r♦s a, b, c ❡ d ♣♦s✐t✐✈♦s✳ ▼♦str❛r❡♠♦s q✉❡ ❡ss❡ s✐st❡♠❛ ❞❡ ❡q✉❛çõ❡s ♥ã♦ ♣♦ss✉✐ s♦❧✉çã♦ ✐♥t❡✐r❛✳

  ✽✷

  2

  2

  2

  2

  ❈♦♥s✐❞❡r❡ e = mdc(a, b)✳ ❈♦♠♦ e | a, e | b ❡♥tã♦ e | c ❡ e | 2d ✱ ♣♦rt❛♥t♦ e | c ❡ e | d✳ P❛r❛ ✈❡r ✐ss♦ ❜❛st❛ ♦❧❤❛r ❛ ❢❛t♦r❛çã♦ ❞❡ e, c ❡ d ❡♠ ❢❛t♦r❡s ♣r✐♠♦s✳ ❙❡

  m n l m 1 k n 1 k l 1 k

  e = p , c = p ❡ d = p ❡♥tã♦

  1 · · · p k 1 · · · p k 1 · · · p k

  6 2m i 2n i , i = 1, . . . , k,

  i i i

  ❡ ❛ss✐♠ m ✱ ♠♦str❛♥❞♦ q✉❡ e | c✳ ❆❣♦r❛ s❡ t♦❞♦s ♦s p ❢♦r❡♠ í♠♣❛r❡s✱ ✉s❛♥❞♦ ♦

  6 n

  6 = 2 1 + 2l

  1

  1

  1

  ♠❡s♠♦ ❛r❣✉♠❡♥t♦ t❡r❡♠♦s q✉❡ e | d✱ ♠❛s s❡ ♣♦r ❡①❡♠♣❧♦ p ✱ t❡♠♦s 2m ✱

  1

  1

  ❞❛í m ❥á q✉❡ ❛♠❜♦s sã♦ ✐♥t❡✐r♦s✱ ❡ ♣♦rt❛♥t♦ e | d✳ P♦rt❛♥t♦✱ ♣♦❞❡♠♦s s✉♣♦r ❛ ❡①✐s✲ tê♥❝✐❛ ❞❛ s♦❧✉çã♦ ✐♥t❡✐r❛✱ ❝♦♠ mdc(a, b) = 1✱ ❜❛st❛♥❞♦ ♣❛r❛ ✐ss♦ ❞✐✈✐❞✐r t♦❞♦s ♦s t❡r♠♦s ❞❛ s♦❧✉çã♦ ♦r✐❣✐♥❛❧ ♣♦r e✳

  6 l

  2

  ❉❛❞♦ q✉❡ ab = 2d ❡ a ❡ b sã♦ ♣r✐♠♦s ❡♥tr❡ s✐✱ ❡♥tã♦ a ♦✉ b é ♣❛r✱ ♠❛s ♥ã♦ ❛♦

  2

  2

  2

  • b = c ♠❡s♠♦ t❡♠♣♦✳ P♦rt❛♥t♦✱ a é í♠♣❛r✱ ✐♠♣❧✐❝❛♥❞♦ q✉❡ c é í♠♣❛r✳ ❈♦♠♦ ab é ♦ ❞♦❜r♦ ❞❡ ✉♠ q✉❛❞r❛❞♦ ❡ ❛♠❜♦s ♥ã♦ ♣♦ss✉❡♠ ❞✐✈✐s♦r❡s ♣r✐♠♦s ❡♠ ❝♦♠✉♠✱ ❡♥tã♦ ✉♠ ❞❡❧❡s é ✉♠ q✉❛❞r❛❞♦ ❡ ♦ ♦✉tr♦ é ♦ ❞♦❜r♦ ❞❡ ✉♠ q✉❛❞r❛❞♦✳ P♦❞❡♠♦s ❛ss✉♠✐r s❡♠ ♣❡r❞❛ ❞❡ ❣❡♥❡r❛❧✐❞❛❞❡ q✉❡

  2

  2

  a = 2k , ❡ b = l

  ♦♥❞❡ k ❡ l sã♦ ✐♥t❡✐r♦s✱ s❡♥❞♦ l í♠♣❛r✳ ❙✉❜st✐t✉✐♥❞♦ ♦ ✈❛❧♦r ❞❡ a ♥❛ ♣r✐♠❡✐r❛ ❡q✉❛çã♦ ❡♥❝♦♥tr❛♠♦s

  2

  

2

  2

  4k + b = c , ❞❡ ♦♥❞❡ c + b c

  − b

  4 = k .

  ·

  2

  2

  c+b c−b

  ❈♦♠♦ b ❡ c sã♦ ❛♠❜♦s í♠♣❛r❡s ❡ r❡❧❛t✐✈❛♠❡♥t❡ ♣r✐♠♦s✱ ❡♥tã♦ ❡ sã♦ r❡❧❛t✐✈❛♠❡♥t❡

  2

  2

  ♣r✐♠♦s✳ P♦rt❛♥t♦✱ c + b c

  4 − b

  4

  = r = s , ❡

  2

  2 ♣❛r❛ r ❡ s ✐♥t❡✐r♦s ♣♦s✐t✐✈♦s r❡❧❛t✐✈❛♠❡♥t❡ ♣r✐♠♦s✳ ❉❡ss❛s ❞✉❛s ú❧t✐♠❛s ❡q✉❛çõ❡s t✐r❛♠♦s

  4

  4

  4

  4

  b = r + s , − s ❡ c = r

  ❞❡ ♦♥❞❡

  2

  2

  2

  2

  2 l = b = (r + s )(r ).

  − s

  2

  2

  2

  2

  − s

  • s ❆❣♦r❛✱ ❝♦♠♦ l é í♠♣❛r✱ q✉❛❧q✉❡r ❢❛t♦r ❡♠ ❝♦♠✉♠ ❡♥tr❡ r ❡ r t❡r✐❛

  

2

  2

  q✉❡ s❡r ♣r✐♠♦ ❡ t❡r✐❛ q✉❡ ❞✐✈✐❞✐r ❛ s♦♠❛ 2r ❡ ❛ ❞✐❢❡r❡♥ç❛ 2s ✱ s❡♥❞♦ ♣♦rt❛♥t♦ ✉♠ ❢❛t♦r

  2

  2

  2

  2

  2

  

2

  , s ) = 1 + s ❞❡ mdc(r ✳ ❆ss✐♠✱ r ❡ r − s sã♦ r❡❧❛t✐✈❛♠❡♥t❡ ♣r✐♠♦s✳ P♦rt❛♥t♦✱ ❝♦♠♦

  2

  2

  2

  2

  • s ♦ ♣r♦❞✉t♦ ❡♥tr❡ r ❡ r − s é ✉♠ q✉❛❞r❛❞♦ í♠♣❛r✱ ❝❛❞❛ ✉♠ ❞❡❧❡s t❛♠❜é♠ é ✉♠

  ✽✸ q✉❛❞r❛❞♦ í♠♣❛r✱ ✐st♦ é✱

  2

  2

  2

  2

  2

  2

  r + s = t = u , ❡ r − s

  2

  2

  2

  = u ❡♠ q✉❡ t ❡ u sã♦ í♠♣❛r❡s r❡❧❛t✐✈❛♠❡♥t❡ ♣r✐♠♦s✳ P♦rt❛♥t♦ r − s ≡ 1(mod 4)✱ ♦ q✉❡ ❢♦rç❛ r s❡r í♠♣❛r ❡ s s❡r ♣❛r✱ ♣♦✐s ❝❛s♦ ❝♦♥trár✐♦ t❡rí❛♠♦s ❛ ❝♦♥❣r✉ê♥❝✐❛ −1 ≡ 1(mod 4)

  2

  2

  2

  2

  t + u t + u

  2 t − u

  r = + = ,

  2

  2

  2 ♦♥❞❡ t + u, t − u sã♦ ✐♥t❡✐r♦s✱ ✈✐st♦ q✉❡ t ❡ u sã♦ í♠♣❛r❡s✳

  ❋✐♥❛❧♠❡♥t❡✱ t♦♠❛♥❞♦ t + u t

  ′ ′ − u ′

  a = , b = = r, ❡ c

  2

  2 t❡♠♦s

  

′2 ′2 ′2

a + b = c .

  

′ ′

  , b ) = 1 ❉♦ ❢❛t♦ ❞❡ q✉❡ mdc(t, u) = 1✱ t❡♠♦s mdc(a ✳ ▼❛✐s ❞♦ q✉❡ ✐ss♦✱

  2

  2

  2

  2

  t 2s s

  ′ ′ − u

  a b = = = 2 ,

  4

  4

  2

  ′ ′ ′ ′

  , b , c , d ) ❝♦♠ s/2 ✐♥t❡✐r♦✳ ▼♦str❛♥❞♦ q✉❡ (a t❛♠❜é♠ é s♦❧✉çã♦ ♣❛r❛ ❛ ♥♦ss❛ ♣r✐♠✐r❛

  ′

  4

  4

  

4

  = r 6 r < r +s = c ❡q✉❛çã♦✳ ▼❛s ❝♦♠♦ 0 < c ✱ ♣♦❞❡♠♦s ❛♣❧✐❝❛r ♦ ♠❡s♠♦ ♣r♦❝❡❞✐♠❡♥t♦

  , b , c , d ) <

  n n n n n+1

  ❡ ❡♥❝♦♥tr❛r ✉♠❛ s❡q✉ê♥❝✐❛ ❞❡ s♦❧✉çõ❡s ✐♥t❡✐r❛s (a t❛✐s q✉❡ 0 < · · · < c c <

  n

  · · · < c✱ ♦ q✉❡ é ❝❧❛r❛♠❡♥t❡ ❝♦♥tr❛❞✐çã♦✳ P♦rt❛♥t♦✱ ♥ã♦ ❡①✐st❡ s♦❧✉çã♦ ✐♥t❡✐r❛ ♣❛r❛ ❛ ♣r✐♠❡✐r❛ ❡q✉❛çã♦ ❝♦♥s✐❞❡r❛❞❛✱ ♠♦str❛♥❞♦ q✉❡ ✶ ♥ã♦ é ❝♦♥❣r✉❡♥t❡✳ xy

  2

  2

  2

  • y = z ❆❣♦r❛ ♦❜s❡r✈❡♠ q✉❡ s❡ q é ❝♦♥❣r✉❡♥t❡✱ ❝♦♠ q = ❡ x ✱ ❡♥tã♦

  2 ax · ay

  2

  2

  q = t♦♠❛♥❞♦ ax, ay, az t❡♠♦s q✉❡ a ✳ P♦rt❛♥t♦✱ s❡♥❞♦ Q ♦ ❝♦♥❥✉♥t♦ ❞♦s r❛❝✐♦♥❛✐s

  2

  2

  2

  = : q q✉❛❞r❛❞♦s ♥ã♦ ♥✉❧♦s✱ ✐st♦ é✱ Q {q ∈ Q − {0}}✱ ♣♦❞❡♠♦s ❞❡✜♥✐r ❛ r❡❧❛çã♦

  2 x , x = qy.

  ∼ y ⇔ ∃ q ∈ Q ❊ss❛ é ✉♠❛ r❡❧❛çã♦ ❞❡ ❡q✉✐✈❛❧ê♥❝✐❛✱ ❡♠ q✉❡ t♦❞❛ ❝❧❛ss❡ ❞❡ ❡q✉✐✈❛❧ê♥❝✐❛ t❡♠ ✉♠ r❡♣r❡s❡♥✲ t❛♥t❡ ✐♥t❡✐r♦ ❡ ❧✐✈r❡ ❞❡ q✉❛❞r❛❞♦s✱ ❞❡ ♠♦❞♦ q✉❡ s❡ ✉♠ ❡❧❡♠❡♥t♦ ❞❛ ❝❧❛ss❡ é ❝♦♥❣r✉❡♥t❡✱ ❡♥tã♦ t♦❞♦s ♦ sã♦✳ ❉❡ss❡ ♠♦❞♦✱ ♣♦❞❡♠♦s ♥♦s ♣r❡♦❝✉♣❛r ❛♣❡♥❛s ❝♦♠ ♦s ✐♥t❡✐r♦s ❧✐✈r❡s ❞❡ q✉❛❞r❛❞♦s✳ P♦rt❛♥t♦ ♣♦❞❡♠♦s ♥♦s ♣❡r❣✉♥t❛r q✉❛♥❞♦ ✉♠ ✐♥t❡✐r♦ é ♦✉ ♥ã♦ ✉♠ ♥ú♠❡r♦ ❝♦♥❣r✉❡♥t❡✱ ❡ ❡ss❡ é ❝♦♥❤❡❝✐❞♦ ❝♦♠♦ ♦ Pr♦❜❧❡♠❛ ❞♦s ◆ú♠❡r♦s ❈♦♥❣r✉❡♥t❡s✿ ❊♥❝♦♥tr❛r ✉♠ ❛❧❣♦r✐t♠♦ ♣❛r❛ ❞❡t❡r♠✐♥❛r q✉❛♥❞♦ ✉♠ ✐♥t❡✐r♦ ♣♦s✐t✐✈♦ n é ♦✉ ♥ã♦ ❝♦♥❣r✉❡♥t❡✳

  ✽✹ ❖ ♥♦ss♦ ♣r✐♥❝✐♣❛❧ ♦❜❥❡t✐✈♦ é ❝❤❡❣❛r ❛♦ s❡❣✉✐♥t❡ r❡s✉❧t❛❞♦✱ q✉❡ r❡❧❛❝✐♦♥❛ ♦ ♣r♦✲

  ❜❧❡♠❛ ❞❡ ❞❡t❡r♠✐♥❛r s❡ ✉♠ ✐♥t❡✐r♦ n é ❝♦♥❣r✉❡♥t❡ ❝♦♠ ✉♠❛ ❝❡rt❛ ❢❛♠í❧✐❛ ❞❡ ❝✉r✈❛s ❡❧í♣t✐❝❛s✳ ❚❡♦r❡♠❛ ✹✳✶✵✳ ❙❡ n é ✉♠ ✐♥t❡✐r♦ ♣♦s✐t✐✈♦✱ ❡♥tã♦ n é ❝♦♥❣r✉❡♥t❡ s❡✱ ❡ s♦♠❡♥t❡ s❡✱ ❛ ❝✉r✈❛ ❡❧í♣t✐❝❛

  2

  3

  2 E n : y = x x,

  − n t❡♠ ♣♦st♦ ❛❧❣é❜r✐❝♦ ♣♦s✐t✐✈♦✳ ❉❛q✉✐ ❡♠ ❞✐❛♥t❡✱ ✐r❡♠♦s s✉♣♦r a < b < c q✉❛♥❞♦ ♥♦s r❡❢❡r✐r♠♦s à tr✐♣❧❛ (a, b, c)✳

  Pr♦♣♦s✐çã♦ ✹✳✶✶✳ ❙❡❥❛ n > 1 ✉♠ ✐♥t❡✐r♦ ❧✐✈r❡ ❞❡ q✉❛❞r❛❞♦s✳ ❙❡❥❛♠ 0 < a < b < c r❛❝✐♦♥❛✐s✳ ❊♥tã♦ ❡①✐st❡ ✉♠❛ ❜✐❥❡çã♦ ❡♥tr❡ ♦s ❝♦♥❥✉♥t♦s

  2

  2

  2

  2

  • b = c , 2n = ab {(a, b, c) : a } ❡ {d ∈ Q : d, d + n, d − n ∈ Q },

  ❞❛❞❛ ♣♦r

  2

  (a, b, c) , 7→ d = (c/2)

  ❝✉❥❛ ✐♥✈❡rs❛ é √ √ √ √ √ d d + n d d + n + d d). 7→ ( − − n, − n, 2

  2

  2

  2

  • b = c ❉❡♠♦♥str❛çã♦✳ ❙❡ a ❡ 2n = ab✱ ❡♥tã♦

  2

  2

  2

  (a = c ± b) ± 2ab = c ± 4n. ▲♦❣♦✱

  2

  a ± b

  2

  = (c/2) ± n.

  2

  2

  ❚♦♠❛♥❞♦ d = (c/2) ✱ ♦❜t❡♠♦s

  2 d, d + n, d .

  − n ∈ Q √ √

  2

  d + n d ❘❡❝✐♣r♦❝❛♠❡♥t❡✱ s❡ t✐✈❡r♠♦s d, d + n, d − n ∈ Q ❡♥tã♦ a = − − n, b = √ √ √ d + n + d d

  − n ❡ c = 2 s❛t✐s❢❛③❡♠

  2

  2

  2 a < b < c + b = c .

  ❡ a ▼❛✐s ❞♦ q✉❡ ✐ss♦✱ ❞❡ ❢❛t♦ ✉♠❛ ❛♣❧✐❝❛çã♦ é ✐♥✈❡rs❛ ❞❛ ♦✉tr❛

  √ √ √ √ √ d d + n d d + n + d d) 7→ ( − − n, − n, 2 7→ d.

  ✽✺ ❈♦♠♦ ❝♦♥s❡q✉ê♥❝✐❛ ❞❡ss❛ ♣r♦♣♦s✐çã♦✱ t❡♠♦s q✉❡ n é ❝♦♥❣r✉❡♥t❡ s❡✱ ❡ s♦♠❡♥t❡ s❡✱

  2

  ❡①✐st❡ ✉♠ r❛❝✐♦♥❛❧ d t❛❧ q✉❡ d, d + n, d − n ∈ Q ✳

  2

  3

  2

  = x x Pr♦♣♦s✐çã♦ ✹✳✶✷✳ ❙❡❥❛ (a, b) ∈ Q × Q ✉♠❛ s♦❧✉çã♦ ❞❛ ❡q✉❛çã♦ y t❛❧ q✉❡

  − n

  2

  a t❡♥❤❛ ❞❡♥♦♠✐♥❛❞♦r ♣❛r✳ ❊♥tã♦✱ ❡①✐st❡ ✉♠ tr✐â♥❣✉❧♦ r❡tâ♥❣✉❧♦ ❞❡ ár❡❛ n ❡ ❧❛❞♦s ∈ Q

  √ √ √ √ √ a + n a a + n + a a. − − n, − n, 2 b

  √ a ❉❡♠♦♥str❛çã♦✳ ❙❡❥❛ u = ∈ Q, u > 0 ❡ v = ✳ ❊♥tã♦ u

  2

  

2

  b b

  2

  2

  2

  v = = = a , − n

  2

  u a

  2

  3

  2 = x x.

  ❥á q✉❡ (a, b) é s♦❧✉çã♦ ❞❛ ❡q✉❛çã♦ y − n ❆❣♦r❛✱ s❡❥❛ t ♦ ❞❡♥♦♠✐♥❛❞♦r ❞❡ u ♥❛ s✉❛ ❢♦r♠❛ r❡❞✉③✐❞❛✳ ❆ss✐♠✱

  2

  2

  2

  2

  

4

  2

  4

  2

  2

  2

  (t v) + (t n) = t v + t u = (t

  a) ,

  2

  2

  2

  2

  2

  2

  2

  v, t n, t

  a) n v, t n, t

  a) = 1 ❡ ♣♦rt❛♥t♦ (t é ✉♠❛ tr✐♣❧❛ ♣✐t❛❣ór✐❝❛ ❝♦♠ t ♣❛r ❡ mdc(t ✳

  2

  2

  2

  n v a ❈♦♠♦ t é ♣❛r✱ ❡♥tã♦ t ❡ t sã♦ í♠♣❛r❡s✳ ❙❡❥❛♠ A, B, C > 0 ✐♥t❡✐r♦s t❛✐s q✉❡

  2

  2

  2

  2

  2 t n = 2C, t a + t v = 2A, t a v = 2B.

  − t

  2

  ❖❜t❡♠♦s✱ mdc(A, B) = 1 ❡ AB = C ✳ P♦rt❛♥t♦✱ ❡①✐st❡♠ ✐♥t❡✐r♦s α, β > 0 t❛✐s q✉❡

  2

  2 A = α , B = β .

  ❉❡ ♦♥❞❡ t❡♠♦s

  2

  2

  2

  t a = A + B = α + β

  2

  2

  2

  t v = A − B = α − β

  2

  2

  2

  2

  2

  2

  2

  (t n) = (t v) a) = (2αβ) .

  − (t

  2 n = 2αβ.

  P♦rt❛♥t♦✱ t ❉❡ ♦♥❞❡ t✐r❛♠♦s q✉❡ ♦ tr✐â♥❣✉❧♦ r❡tâ♥❣✉❧♦ ❞❡ ❧❛❞♦s 2α 2β

  , , 2u t t 2αβ

  2

  = n t❡♠ ár❡❛ ✳

  t

  ✽✻ P❡❧❛ ♣r♦♣♦s✐çã♦ ❛♥t❡r✐♦r✱ ❡ss❛ tr✐♣❧❛ ❡stá ❛ss♦❝✐❛❞❛ ❛

  

2

  2u

  2 d = = u = a.

  2 ▼♦str❛♥❞♦ q✉❡ ❡①✐st❡ ✉♠ tr✐â♥❣✉❧♦ r❡tâ♥❣✉❧♦ ❞❡ ❧❛❞♦s

  √ √ √ √ √ a + n a a + n + a a − − n, − n, 2

  ❡ ár❡❛ n✳ ❱✐st♦ q✉❡ 2α

  √ √ a + n a , − − n = t

  √ √ 2β a + n + a , − n = ❡ t

  √ 2 a = 2u.

  ❆❣♦r❛ ♠❛✐s ❛❧❣✉♥s r❡s✉❧t❛❞♦s ♣❛r❛ ❡♥tã♦ ♣r♦✈❛r♠♦s ♦ ❚❡♦r❡♠❛ ♦❜❥❡t✐✈♦ ❞❡st❛ s❡çã♦✳ Pr♦♣♦s✐çã♦ ✹✳✶✸✳ ❈♦♥s✐❞❡r❡ ❛ ❛♣❧✐❝❛çã♦ ❞❡ r❡❞✉çã♦

  2

  2 P (Q) (F ) p

  → P P = [a, b, c]

  7→ P = [a, b, c],

  1 = P

  2

  ♦♥❞❡ a, b, c ∈ Z ❡ mdc(a, b, c) = 1✳ ❊♥tã♦ P s❡✱ ❡ s♦♠❡♥t❡ s❡✱ p ❞✐✈✐❞❡ s✐♠✉❧t❛♥❡❛✲

  1 c

  2 2 c 1 ) 1 b

  2 2 b 1 ) 1 c

  2 2 c 1 )

  ♠❡♥t❡ (b − b ✱ (a − a ❡ (a − a ✳

  1 = P

  2 1 , b 1 , c 1 ) 2 , b 2 , c 2 )

  ❉❡♠♦♥str❛çã♦✳ ❱❡♠♦s q✉❡ P s❡✱ ❡ s♦♠❡♥t❡ s❡✱ ♦s ✈❡t♦r❡s (a ❡ (a

  p

  sã♦ F ✲❧✐♥❡❛r♠❡♥t❡ ❞❡♣❡♥❞❡♥t❡s✱ q✉❡ é ❡q✉✐✈❛❧❡♥t❡ ❛ p ❞✐✈✐❞✐r s✐♠✉❧t❛♥❡❛♠❡♥t❡ ♦s três ♥ú♠❡r♦s ♥❛ ♣r♦♣♦s✐çã♦✳

  (F )

  n p

  Pr♦♣♦s✐çã♦ ✹✳✶✹✳ ❙❡ p ≡ 3(mod 4) ❡ p ♥ã♦ ❞✐✈✐❞❡ n✱ ❡♥tã♦ |E | = p + 1✳ ❉❡♠♦♥str❛çã♦✳ Pr✐♠❡✐r❛♠❡♥t❡ ♦❜s❡r✈❛♠♦s q✉❡ ♦s ♣♦♥t♦s [0, 0, 1], [n, 0, 1], [−n, 0, 1] ❡ O

  (F ) (F )

  n p n p

  sã♦ ♣♦♥t♦s ❞✐st✐♥t♦s ❞❡ E ✳ ❆ss✐♠✱ ❢❛❧t❛ ❝♦♥t❛r ❛ q✉❛♥t✐❞❛❞❡ ❞❡ ♣♦♥t♦s (x, y) ∈ E

  p

  t❛✐s q✉❡ x 6= 0, ±n✳ ❆♦ t♦❞♦ t❡♠♦s p − 3 ❡❧❡♠❡♥t♦s ❡♠ F ❞✐st✐♥t♦s ❞❡ 0, ±n✳ P❡❧♦ ❢❛t♦

  3

  

2

  x ❞❡ q✉❡ p ≡ 3(mod 4) ❡ q✉❡ f(x) = x − n é ❢✉♥çã♦ í♠♣❛r✱ ❡♥tã♦ ✉♠ ❡ s♦♠❡♥t❡ ✉♠ ❡♥tr❡ f(x) ❡ f(−x) = −f(x) é q✉❛❞r❛❞♦ ♠ó❞✉❧♦ p✳ ❙❡❥❛ q✉❛❧ ❢♦r ♦ ❝❛s♦✱ t❡r❡♠♦s q✉❡ ✉♠

  1/2 1/2

  ) ) (F ) ❡ s♦♠❡♥t❡ ✉♠ ❡♥tr❡ (x, ±f(x) ❡ (−x, ±f(−x) s❡rã♦ ♣❛r❡s ❞❡ ♣♦♥t♦s ❡♠ E n p ✳

  n (F p ) P♦rt❛♥t♦✱ |E | = p − 3 + 4 = p + 1. n

  ❚❡♠♦s ♠❛✐s ♦ s❡❣✉✐♥t❡ r❡s✉❧t❛❞♦ s♦❜r❡ ❛s ❝✉r✈❛s E ✳

  ✽✼ (Q)

  n tor Pr♦♣♦s✐çã♦ ✹✳✶✺✳ |E | = 4.

  (Q)

  n tor

  ❉❡♠♦♥str❛çã♦✳ ❚❡♠♦s q✉❡ ♦s s❡❣✉✐♥t❡s ♣♦♥t♦s ❡stã♦ ❡♠ E ✿ O✱ [0, 0, 1]✱ [n, 0, 1] ❡ [

  −n, 0, 1]✳ ❙❡♥❞♦ q✉❡ ❡ss❡s três ú❧t✐♠♦s t❡♠ ♦r❞❡♠ ✷✳ (Q)

  ❙❡♥❞♦ ❛ss✐♠✱ t❡♠♦s q✉❡ |E n tor | > 4✳ ❙✉♣♦♥❤❛♠♦s q✉❡ ♥ã♦ ✈❛❧❡ ❛ ✐❣✉❛❧❞❛❞❡✱

  (Q) ✐st♦ é✱ |E n tor

  | > 4✳ P♦rt❛♥t♦✱ ❡①✐st❡ ✉♠ ❡❧❡♠❡♥t♦ Q ❞❡ ♦r❞❡♠ N > 2✳ P♦rt❛♥t♦✱ N é í♠♣❛r ♦✉ ❡①✐st❡ ✉♠ ♣♦♥t♦ P ❞❡ ♦r❞❡♠ ❡①❛t❛♠❡♥t❡ ✹✳

  ◆♦ ♣r✐♠❡✐r♦ ❝❛s♦✱ ❝♦♥s✐❞❡r❡ S ♦ s✉❜❣r✉♣♦ ❣❡r❛❞♦ ♣♦r Q✳ ❏á ♥♦ s❡❣✉♥❞♦ ❝❛s♦✱ ♣❡❧♦ ♠❡♥♦s ✉♠ ❞♦s ❡❧❡♠❡♥t♦s ❞❡♥tr❡ ♦s três ❞❡ ♦r❞❡♠ ✷ q✉❡ ❝✐t❛♠♦s ❛❝✐♠❛✱ ♥ã♦ ❡stã♦ ♥♦ s✉❜❣r✉♣♦ ❣❡r❛❞♦ ♣♦r P ✳ ❙✉♣♦♥❤❛ q✉❡ ❡ss❡ ❡❧❡♠❡♥t♦ ❞❡ ♦r❞❡♠ ✷ s❡❥❛ R✳ ❆ss✐♠✱ ♣❛r❛ ❡ss❡

  = Z/2Z ❝❛s♦ ❝♦♥s✐❞❡r❛♠♦s S ♦ ♣r♦❞✉t♦ ❞♦s s✉❜❣r✉♣♦s ❣❡r❛❞♦s ♣♦r P ❡ R✳ ▲♦❣♦ S ∼ ×Z/4Z. , . . . , P

  1 m

  P❛r❛ ❛♠❜♦s ♦s ❝❛s♦s✱ s❡❥❛ S = {P } ❝♦♠ m = N ♦✉ m = 8✳ = [x , y , z ] , y , z , y , z ) =

  i i i i i i i i i i

  P❛r❛ ❝❛❞❛ i, j ∈ {1, . . . , m}✱ s❡❥❛ P ❝♦♠ x ∈ Z ❡ mdc(x

  1 ❡ ❞❡✜♥❛

  3 P = (y z z , x z z , x y y ) . i j i j j i j i i j i j j i

  × P − y − x − x ∈ R

  i j i j ij

  ❈❛s♦ P 6= P ✱ ❡♥tã♦ P × P 6= 0✳ P♦rt❛♥t♦ ❝♦♥s✐❞❡r❛♠♦s m ♦ ♠❞❝ ❞❛s ❝♦♦r❞❡✲ = P

  ♥❛❞❛s ❞❡ P i j ✳ P❡❧❛ Pr♦♣♦s✐çã♦ ✹✳✶✸✱ ♣❛r❛ p ♣r✐♠♦✱ P i j s❡✱ ❡ s♦♠❡♥t❡ s❡✱ p|m ij ✳ × P

  ❆❣♦r❛ s❡❥❛ p > 2 ♣r✐♠♦ q✉❡ ♥ã♦ ❞✐✈✐❞❡ m ❡ t❛❧ q✉❡ p > m ij ✳ P♦rt❛♥t♦✱ P i 6= P j ✳

  n (F p )

  ❊♠ ♣❛rt✐❝✉❧❛r✱ S é ✐s♦♠♦r❢♦ à ✉♠ s✉❜❣r✉♣♦ ❞❡ E ✳ ❉❡ ♠♦❞♦ q✉❡ ♣❛r❛ q✉❛s❡ t♦❞♦s

  n (F p )

  ♦s ♣r✐♠♦s p✱ m ❞✐✈✐❞❡ |E |✱ ♦✉ s❡❥❛✱ ♣❛r❛ t♦❞♦s ♣r✐♠♦s ♠❛✐♦r❡s q✉❡ ✉♠ ❝❡rt♦ ♥ú♠❡r♦

  n (F p )

  t❡♠✲s❡ q✉❡ m ❞✐✈✐❞❡ |E |✳ (F )

  n p

  ❆❣♦r❛✱ ♣❡❧❛ ♣r♦♣♦s✐çã♦ ❛♥t❡r✐♦r✱ s❡ p ≡ 3 (mod 4) ❡♥tã♦ |E | = p + 1✱ ❞❡ ♠♦❞♦ q✉❡ p ≡ −1(mod m) ♣❛r❛ q✉❛s❡ t♦❞♦ p q✉❡ é ❝ô♥❣r✉♦ ❛ ✸ ♠ó❞✉❧♦ ✹✳

  ▼❛s ♣❡❧♦ ❚❡♦r❡♠❛ ❞❛s ♣r♦❣r❡ssõ❡s ❛r✐t♠ét✐❝❛s ❞❡ ❉✐r✐❝❤❧❡t✱ ❞❛❞♦s r, s 6= 0 t❛✐s q✉❡ mdc(r, s) = 1✱ ❡①✐t❡♠ ✐♥✜♥✐t♦s ♣r✐♠♦s ❞❛ ❢♦r♠❛ rd + s ❝♦♠ d > 1✳ ❆ss✐♠✱ ♣❛r❛ ♦ ❝❛s♦ ❡♠ q✉❡ m = 8 ❜❛st❛ t♦♠❛r♠♦s r = 8 ❡ s = 3 q✉❡ t❡r❡♠♦s ❛ ❡①✐stê♥❝✐❛ ❞❡ ✐♥✜♥✐t♦s ♣r✐♠♦s p

  ❞❛ ❢♦r♠❛ 8d + 3✱ ♦✉ s❡❥❛✱ ✐♥✜♥✐t♦s ♣r✐♠♦s t❛✐s q✉❡ p = 8d + 3 6≡ −1 (mod 8). ❏á ♣❛r❛ ♦ ❝❛s♦ ❡♠ q✉❡ m é í♠♣❛r ❡ ✸ ♥ã♦ ❞✐✈✐❞❡ m✱ ❜❛st❛ t♦♠❛r♠♦s r = 4m ❡ s = 3

  ♦♥❞❡ t❡r❡♠♦s ❛ ❡①✐stê♥❝✐❛ ❞❡ ✐♥✜♥✐t♦s ♣r✐♠♦s p ❛ ❢♦r♠❛ 4md + 3✱ ♦✉ s❡❥❛✱ ✐♥✜♥✐t♦s ♣r✐♠♦s t❛✐s q✉❡ p = 4md + 3

  ✽✽ P♦r ú❧t✐♠♦✱ s❡ m é í♠♣❛r ❡ ✸ ❞✐✈✐❞❡ m✱ t♦♠❛♠♦s r = 12 ❡ s = 7✱ ❞❡ ♠♦❞♦ q✉❡ ❡①✐st❡♠ ✐♥✜♥✐t♦s ♣r✐♠♦s ❞❛ ❢♦r♠❛ 12d + 7✱ q✉❡ ♥❡ss❡ ❝❛s♦ s❛t✐s❢❛③❡♠ p = 12d + 7

  6≡ −1 (mod m). ❖❜s❡r✈❡♠ q✉❡ ❡♠ t♦❞♦s ❡ss❡s ❝❛s♦s✱ ♦s ♣r✐♠♦s sã♦ t❛✐s q✉❡ p ≡ 3 (mod 4)✳ P♦rt❛♥t♦ ❛ ❝♦♥❝❧✉sã♦ ❞❡ q✉❡ p ≡ −1(mod m) ♣❛r❛ q✉❛s❡ t♦❞♦ p ♣r✐♠♦ ❝♦♠ p ❝ô♥❣r✉♦ ❛ 3 ♠ó❞✉❧♦ 4 ❡♥tr❛ ❡♠ ❝♦♥tr❛❞✐çã♦ ❝♦♠ ♦ ❚❡♦r❡♠❛ ❞❛s ♣r♦❣r❡ssõ❡s ❛r✐t♠ét✐❝❛s ❞❡ ❉✐r✐❝❤❧❡t✳ ❉❡ ♠♦❞♦

  (Q)

  n tor

  q✉❡ t❛✐s ♣♦♥t♦s Q ♦✉ P ♥ã♦ ♣♦❞❡♠ ❡①✐st✐r✱ ♠♦str❛♥❞♦ q✉❡ |E | = 4✳ ❋✐♥❛❧♠❡♥t❡ ✐r❡♠♦s ♣r♦✈❛r ♦ s❡❣✉✐♥t❡ r❡s✉❧t❛❞♦✳

  ❚❡♦r❡♠❛ ✹✳✶✻✳ ❯♠ ♥ú♠❡r♦ ✐♥t❡✐r♦ n > 0 é ❝♦♥❣r✉❡♥t❡✱ s❡ ❡ s♦♠❡♥t❡ s❡✱ ♦ ♣♦st♦ ❞❡ E (Q)

  n é ♣♦s✐t✐✈♦✳

  ❉❡♠♦♥str❛çã♦✳ ❙✉♣♦♥❤❛♠♦s q✉❡ n s❡❥❛ ❝♦♥❣r✉❡♥t❡ ❡ ❧✐✈r❡ ❞❡ q✉❛❞r❛❞♦s✱ ❡ s❡❥❛ (a, b) ∈ E (Q)

  n

  ❞❛ ❢♦r♠❛

  2

  2

  2

  z z (x ) − y a = ,

  ❡ b = ·

  2

  2

  4

  2

  2

  2

  • y = z t❛✐s q✉❡ 0 < x < y < z✱ x ❡ 2n = xy✳

  2

  ◆❡st❡ ❝❛s♦✱ t❡♠♦s q✉❡ a ∈ Q ❝♦♠ ❞❡♥♦♠✐♥❛❞♦r ♣❛r✳ ❙❡ (a, b) t✐✈❡ss❡ ♦r❞❡♠ ✜♥✐t❛✱ ❡♥tã♦ ♣❡❧♦ t❡♦r❡♠❛ ❛♥t❡r✐♦r✱ (a, b) ♣r❡❝✐s❛r✐❛ s❡r ✉♠ ❞♦s ❡❧❡♠❡♥t♦s ❞❡ ♦r❞❡♠ 2✳ ❙❡♥❞♦ ❛ss✐♠✱ s✉❛ ♣r✐♠❡✐r❛ ❝♦♦r❞❡♥❛❞❛ ❞❡✈❡r✐❛ s❡r 0, −n, ♦✉ n✳ ❈♦♠♦ n é ❧✐✈r❡ ❞❡ q✉❛❞r❛❞♦s✱

  2

  ❡♥tã♦ 0, −n, n 6∈ Q ✳ ❉❡ ♠♦❞♦ q✉❡ (a, b) ♣r❡❝✐s❛ s❡r ✉♠ ❡❧❡♠❡♥t♦ ❞❡ ♦r❞❡♠ ✐♥✜♥✐t❛✱ ❡ ♣❡❧♦ (Q)

  n

  ❚❡♦r❡♠❛ ❞❡ ▼♦r❞❡❧❧✲❲❡✐❧ ✐ss♦ s✐❣♥✐✜❝❛ q✉❡ ♦ ♣♦st♦ ❛❧❣é❜r✐❝♦ ❞❡ E é ♣❡❧♦ ♠❡♥♦s ✶✳ (Q)

  n

  ❘❡❝✐♣r♦❝❛♠❡♥t❡✱ ❞❛❞♦ P ∈ E ❞❡ ♦r❞❡♠ ✐♥✜♥✐t❛ ❡♥tã♦ ♣❡❧❛ ❢ór♠✉❧❛ ❞❡ ❞✉♣❧✐✲ ❝❛çã♦

  4

  2

  2

  x + n − 2nx a = x([2]P ) = .

  2

  (2y) ❊ ❝♦♠♦ ♦ ♥ú♠❡r♦ a s❛t✐s❢❛③ ❛s ❤✐♣ót❡s❡s ❞❛ Pr♦♣♦s✐çã♦ ✹✳✶✷✱ n é ❝♦♥❣r✉❡♥t❡✳

  ❊ss❡ r❡s✉❧t❛❞♦ ♥♦s ❞✐③ q✉❡ ♣❛r❛ ❞❡❝✐❞✐r s❡ ✉♠ ❞❛❞♦ ✐♥t❡✐r♦ ♣♦s✐t✐✈♦ n é ❝♦♥❣r✉❡♥t❡ (Q)

  n

  é s✉✜❝✐❡♥t❡ s❛❜❡r s❡ ♦ ♣♦st♦ ❞❛ ❝✉r✈❛ E é ♣♦s✐t✐✈♦✱ q✉❡ ❡q✉✐✈❛❧❡ ❛ s❛❜❡r s❡ ❡st❡ ❣r✉♣♦ é ✐♥✜♥✐t♦✳

  ❊♥✉♥❝✐❛r❡♠♦s ❛❣♦r❛ ✉♠ t❡♦r❡♠❛ q✉❡ ❞á ❝♦♥❞✐çõ❡s ♥❡❝❡ssár✐❛s ♣❛r❛ ❞❡t❡r♠✐♥❛r q✉❛♥❞♦ ✉♠ ♥ú♠❡r♦ n ❧✐✈r❡ ❞❡ q✉❛❞r❛❞♦s é ❝♦♥❣r✉❡♥t❡✱ ❡ q✉❡ s❡ ❈♦♥❥❡❝t✉r❛ ❞❡ ❇✐r❝❤ ❡

  n

  ❙✇✐♥♥❡rt♦♥✲❉②❡r ❢♦r ✈❡r✐✜❝❛❞❛ ❛♦ ♠❡♥♦s ♣❛r❛ ❛s ❝✉r✈❛s E ✱ ❡♥tã♦ ❛ ❝♦♥❞✐çã♦ ❞♦ t❡♦r❡♠❛ é s✉✜❝✐❡♥t❡✳ ❙❡❣✉❡ ♦ ❚❡♦r❡♠❛✳ ❚❡♦r❡♠❛ ✹✳✶✼ ✭❚❡♦r❡♠❛ ❞❡ ❚✉♥♥❡❧❧✮✳ ❙❡❥❛ n ✉♠ ✐♥t❡✐r♦ ♣♦s✐t✐✈♦ ❧✐✈r❡ ❞❡ q✉❛❞r❛❞♦s✳ ❉❡✜♥❛

  

3

  2

  2

  2 A = # : n = 2x + y + 8z n

  ✽✾

  3

  2

  2

  2 B = # : n = 2x + y + 32z n

  {(x, y, z) ∈ Z },

  3

  2

  2

  2 C = # : n = 8x + 2y + 16z n

  {(x, y, z) ∈ Z },

  3

  2

  2

  2 D = # : n = 8x + 2y + 64z n

  {(x, y, z) ∈ Z }.

  = 2B

  n n

  ❙❡ n é í♠♣❛r ❡ ✉♠ ♥ú♠❡r♦ ❝♦♥❣r✉❡♥t❡✱ ❡♥tã♦ A ✳ ❙❡ n é ♣❛r ❡ ❝♦♥❣r✉❡♥t❡✱ ❡♥tã♦ C = 2D

  

n n ✳ P♦r ♦✉tr♦ ❧❛❞♦✱ s❡ ❛ ✈❡rsã♦ ❢r❛❝❛ ❞❡ ❇❙❉ ✈❛❧❡ ♣❛r❛ E n ✱ ❡♥tã♦ s❡ n é í♠♣❛r ❡

  A = 2B = 2D

  n n ♦✉ s❡ n é ♣❛r ❡ C n n ✱ ❡♥tã♦ n é ❝♦♥❣r✉❡♥t❡✳

  ❆ss✐♠✱ s❡ ❛ ✈❡rsã♦ ❢r❛❝❛ ❞❛ ❈♦♥❥❡❝t✉r❛ ❇❙❉ ❢♦r ✈❡r✐✜❝❛❞❛✱ ❡♥tã♦ ♣❛r❛ ❞❡❝✐❞✐r s❡ ✉♠ ✐♥t❡✐r♦ ♣♦s✐t✐✈♦ ❧✐✈r❡ ❞❡ q✉❛❞r❛❞♦s é ❝♦♥❣r✉❡♥t❡✱ ❜❛st❛ ✈❡r✐✜❝❛r ❛s ✐❣✉❛❧❞❛❞❡s ❡♥tr❡ ❛s ❝❛r❞✐♥❛❧✐❞❛❞❡s ❞♦s ❝♦♥❥✉♥t♦s ❞❡✜♥✐❞♦s ♥♦ t❡♦r❡♠❛✱ ❝❛❞❛ ✉♠ ❞♦s q✉❛✐s é ❝❧❛r❛♠❡♥t❡ ✜♥✐t♦✳

  ◆❛ ♣r♦✈❛ ❞♦ ❚❡♦r❡♠❛ ❞❡ ❚✉♥♥❡❧❧✱ sã♦ ✉s❛❞♦s ♠✉✐t♦s r❡s✉❧t❛❞♦s s♦❜r❡ ❢♦r♠❛s ♠♦❞✉❧❛r❡s q✉❡ ♥ã♦ s❡rã♦ ✈✐st♦s ❛q✉✐✱ ♠❛s q✉❡ ♣♦❞❡♠ s❡r ❡♥❝♦♥tr❛❞♦s ❡♠ ❬❑♦❜❧✐t③❪✳ ◆♦ q✉❡ s❡❣✉❡✱ ❢❛❧❛r❡♠♦s ❞❡ ❢♦r♠❛ r❡s✉♠✐❞❛ s♦❜r❡ ❛ ❝♦♥❡①ã♦ ❡①✐st❡♥t❡ ❡♥tr❡ ❛ ❈♦♥❥❡❝t✉r❛ ❇❙❉

  , B , C

  n n n n

  ❡ ♦s ❝♦❡✜❝✐❡♥t❡s A ❡ D ✱ ❞❡✜♥✐❞♦s ♥♦ ❚❡♦r❡♠❛ ❞❡ ❚✉♥♥❡❧❧✳ ❈♦♠❡ç❛♠♦s ❢❛❧❛♥❞♦ ❛ r❡s♣❡✐t♦ ❞❡ ✉♠ r❡s✉❧t❛❞♦ ❞❡ ✶✾✼✼ ❞❡✈✐❞♦ ❛♦ ❆✳ ❲✐❧❡s ❡ ❏✳ ❈♦❛t❡s✱ q✉❡ ❞✐③ q✉❡ ♣❛r❛ ✉♠❛ ❝❡rt❛ ❢❛♠í❧✐❛ ❞❡ ❝✉r✈❛s ❡❧í♣t✐❝❛s r❛❝✐♦♥❛✐s✱ t❡♠✲s❡ ✉♠ r❡s✉❧t❛❞♦ ♣❛r❝✐❛❧ ♣❛r❛ ❛ ❈♦♥❥❡❝t✉r❛ ❇❙❉✳ ❖ r❡s✉❧t❛❞♦ ❣❛r❛♥t❡ q✉❡ ♣❛r❛ ❝✉r✈❛s ❡❧í♣t✐❝❛s r❛❝✐♦♥❛✐s E/Q ❝♦♠ ♠✉❧t✐♣❧✐❝❛çã♦ ❝♦♠♣❧❡①❛✱ s❡ E t❡♠ ✐♥✜♥✐t♦s ♣♦♥t♦s r❛❝✐♦♥❛✐s ❡♥tã♦ s✉❛ L✲sér✐❡ ❛✈❛❧✐❛❞❛ ❡♠ ✶ é ✐❣✉❛❧ ❛ ✵✱ ♦✉ s❡❥❛✱ s❡ r > 0 ❡♥tã♦ L(E, 1) = 0✳

  ❆❣♦r❛✱ s❡♥❞♦ n ✉♠ ✐♥t❡✐r♦ ♣♦s✐t✐✈♦ ❧✐✈r❡ ❞❡ q✉❛❞r❛❞♦s✱ ❚✉♥♥❡❧❧ ♣r♦✈♦✉ q✉❡ s❡ n ❢♦r , 1) = 0 = 0

  í♠♣❛r ✭r❡s♣❡❝t✐✈❛♠❡♥t❡ ♣❛r✮ ❡♥tã♦ L(E n s❡✱ ❡ s♦♠❡♥t❡ s❡✱ α n ✭r❡s♣❡❝t✐✈❛♠❡♥t❡

  ′

  2

  3

  2 ′

  α = 0 n : y = x x k ✮✱ ♦♥❞❡ E − n ❡ q✉❛♥❞♦ k é í♠♣❛r✱ α ✭r❡s♣❡❝t✐✈❛♠❡♥t❡ α ✮ é ♦

  n/2 k

  k ✲és✐♠♦ ❝♦❡✜❝✐❡♥t❡ ❞❛ sér✐❡

  X

  2

  2

  2

  2

  2

  1

  2x +y +32z 2x +y +8z

  2 X

  q q ( − r❡s♣❡❝t✐✈❛♠❡♥t❡

  2

  x,y,z∈Z x,y,z∈Z

  X

  2

  2

  2

  2

  2

  1

  2 X

  4x +y +32z 4x +y +8z q q ).

  −

  2

  x,y,z∈Z x,y,z∈Z

  ❯♠❛ ♦✉tr❛ ❝♦✐s❛ q✉❡ s❡ s❛❜❡ s♦❜r❡ ❛s ❝✉r✈❛s ❡❧í♣t✐❝❛s E n é q✉❡ ❡st❛s t❡♠ ♠✉❧t✐✲

  n n (Q)

  ♣❧✐❝❛çã♦ ❝♦♠♣❧❡①❛✱ ❡ ❛ss✐♠ s❡♥❞♦✱ ❝❤❛♠❛♥❞♦ ❞❡ r ♦ ♣♦st♦ ❛❧❣é❜r✐❝♦ ❞❡ E t❡♠✲s❡ ♦ s❡❣✉✐♥t❡ n > 0 = , 1) = 0.

  n n

  é ❝♦♥❣r✉❡♥t❡ ⇐⇒ r ⇒ L(E , 1) = 0

  n

  ❆❣♦r❛✱ s❡ n ❢♦r í♠♣❛r ❡ ❝♦♥❣r✉❡♥t❡✱ t❡♠♦s L(E ❡ ♣♦rt❛♥t♦✱

  ✾✵ 0 = α

  n

  1

  3

  2

  2

  2

  3

  2

  2

  2

  = # : n = 2x + y + 32z : n = 2x + y + 8z {(x, y, z) ∈ Z } − · #{(x, y, z) ∈ Z }

  2 A

  n = B = A = 2B . n n n

  − ⇒

  2 ❆❣♦r❛✱ s❡ n ❢♦r ♣❛r ❡ ❝♦♥❣r✉❡♥t❡✱ t❡♠♦s q✉❡ n = 2k ♦♥❞❡ k é í♠♣❛r ✭❥á q✉❡ n é

  ′ ′

  , 1) = 0 = α = 0

  n

  ❧✐✈r❡ ❞❡ q✉❛❞r❛❞♦s✮✱ ♣♦rt❛♥t♦ L(E ❡ α k ✱ ❞❛í

  n/2 ′

  0 = α

  k

  1

  3

  2

  2

  2

  3

  2

  2

  2

  = # : k = 4x + y + 32z : k = 4x + y + 8z {(x, y, z) ∈ Z } − · #{(x, y, z) ∈ Z }

  2

  1

  3

  2

  2

  2

  3

  2

  2

  2

  = # : n = 8x + 2y + 64z : n = 8x + 2y + 16z {(x, y, z) ∈ Z } − · #{(x, y, z) ∈ Z }

  2 C

  n = D = C = 2D . n n n

  − ⇒

  2 ❊ ❛❣♦r❛✱ ✈❡❥❛♠♦s ❛ r❡❝í♣r♦❝❛ s✉♣♦♥❞♦ ❛ ✈❛❧✐❞❛❞❡ ❞❛ ❈♦♥❥❡❝t✉r❛ ❇❙❉✳

  n

  ❙✉♣♦♥❞♦ q✉❡ ❇❙❉ ✈❛❧❡ ♣❛r❛ ❛s ❝✉r✈❛s E ✱ t❡♠♦s q✉❡ r > 0 , 1) = 0.

  n n

  ⇔ L(E

  n = 2B n n = 0 n , 1) = 0

  ❙❡ n ❢♦r í♠♣❛r ❡ A ✱ ❡♥tã♦ α ❡ ❝♦♥s❡q✉❡♥t❡♠❡♥t❡ L(E ✱ ❞❡ ♦♥❞❡ ❝♦♥❝❧✉✐♠♦s q✉❡ n é ❝♦♥❣r✉❡♥t❡✳

  ′

  = 2D = 0

  n n

  ❆❣♦r❛✱ s❡ n ❢♦r ♣❛r ❡ C ✱ ❡♥tã♦ α ✭n/2 é í♠♣❛r ♥❡ss❡ ❝❛s♦✮ ❡

  n/2

  , 1) = 0

  n

  ❝♦♥s❡q✉❡♥t❡♠❡♥t❡ L(E ❡ ♣♦rt❛♥t♦ n é ❝♦♥❣r✉❡♥t❡✳ ❆ss✐♠✱ ❛❝❛❜❛♠♦s ❞❡ ♠♦str❛r , B , C

  n n n n

  ❝♦♠♦ ♦s ❝♦❡✜❝✐❡♥t❡s A ❡ D ❛♣❛r❡❝❡♠ ♥♦ ❚❡♦r❡♠❛ ❞❡ ❚✉♥♥❡❧❧✱ ❡ ❝♦♠♦ ❡stã♦ r❡❧❛❝✐♦♥❛❞♦s ❝♦♠ ❛ ❈♦♥❥❡❝t✉r❛ ❇❙❉✳ ❯s❛♥❞♦ ♦ ❚❡♦r❡♠❛ ❞❡ ❚✉♥♥❡❧❧ ❡ s✉♣♦♥❞♦ ✈❡r❞❛❞❡✐r❛ ❛ ❈♦♥❥❡❝t✉r❛ ❇❙❉ t❡♠✲s❡ ♦ s❡❣✉✐♥t❡ r❡s✉❧t❛❞♦✳

  n

  Pr♦♣♦s✐çã♦ ✹✳✶✽✳ ❙❡ n ≡ 5, 6 ♦✉ 7 (mod 8) ❡ ❛ ❈♦♥❥❡❝t✉r❛ ❇❙❉ ✈❛❧❡ ♣❛r❛ E ✱ ❡♥tã♦ n é ❝♦♥❣r✉❡♥t❡✳ ❉❡♠♦♥str❛çã♦✳ ❙❡ n ≡ 5 ♦✉ 7 (mod 8)✱ ❡♥tã♦ n é í♠♣❛r✳ ❉❛❞♦ ✉♠ ✐♥t❡✐r♦ k✱ t❡♠♦s q✉❡

  2

  k ≡ 0, 1, ♦✉ 4 (mod 8)✱ ❞❡ ♠♦❞♦ q✉❡

  2

  2

  2

  2

  2

  2

  2

  2

  2x + y + 8z + y + 32z + y ≡ 2x ≡ 2x ≡ 0, 1, 2, 3, 4 ♦✉ 6 (mod 8).

  2

  2

  2

  2

  2

  2

  • y + 8z = n + y + 32z = n P♦rt❛♥t♦✱ ❛s ❡q✉❛çõ❡s 2x ❡ 2x ♥ã♦ ♣♦ss✉❡♠ s♦❧✉çã♦

  = 0 = 2B

  n n

  ✐♥t❡✐r❛✱ ❞❡ ♠♦❞♦ q✉❡ A ✳

  ✾✶ ❆❣♦r❛✱ s❡ n ≡ 6 (mod 8)✱ ❡♥tã♦ n é ♣❛r ❡

  2

  2

  2

  2

  2

  2

  2

  8x + 2y + 16z + 2y + 64z ≡ 8x ≡ 2y ≡ 0 ♦✉ 2 (mod 8),

  2

  2

  

2

  2

  2

  2

  • 2y + 16z = n + 2y + 64z = n ♠♦str❛♥❞♦ q✉❡ ❛s ❡q✉❛çõ❡s 8x ❡ 8x ♥ã♦ ♣♦ss✉❡♠

  = 0 = 2D

  n n

  s♦❧✉çã♦ ✐♥t❡✐r❛✱ ♣♦rt❛♥t♦✱ C ✳

  n

  ❙✉♣♦♥❞♦ ❛ ✈❛❧✐❞❛❞❡ ❞❛ ❈♦♥❥❡❝t✉r❛ ❇❙❉ ♣❛r❛ E ❝♦♠ n ≡ 5, 6 ♦✉ 7 (mod 8)✱ ♣❡❧♦ ❚❡♦r❡♠❛ ❞❡ ❚✉♥♥❡❧❧✱ n é ❝♦♥❣r✉❡♥t❡✳

  ❈❛❜❡ ♦❜s❡r✈❛r q✉❡✱ ♣❡❧♦ ❚❡♦r❡♠❛ ❞❡ ❚✉♥♥❡❧❧✱ ♥ã♦ ♣r❡❝✐s❛♠♦s s✉♣♦r ❛ ✈❛❧✐❞❛❞❡ ❞❛ ❈♦♥❥❡❝t✉r❛ ❇❙❉ ♣❛r❛ ❞❡t❡r♠✐♥❛r q✉❛♥❞♦ ✉♠ ✐♥t❡✐r♦ ♣♦s✐t✐✈♦ n ❧✐✈r❡ ❞❡ q✉❛❞r❛❞♦s ◆➹❖ é ❝♦♥❣r✉❡♥t❡✱ ❡ ❝♦♠ ✐ss♦✱ ✜♥❛❧✐③❛r❡♠♦s ✉s❛♥❞♦ ♦ ❝r✐tér✐♦ ❞❡ ❚✉♥♥❡❧❧ ♣❛r❛ ✈❡r✐✜❝❛r q✉❡ ♦s ♥ú♠❡r♦s ✶✱ ✷✱ ✸ ❡ ✶✵ ♥ã♦ sã♦ ❝♦♥❣r✉❡♥t❡s✳

  2

  3

  : y = x

  1 1 1 ✳ ❈♦♠

  • n = 1✱ E − x✳ ❈♦♠♦ ✶ é í♠♣❛r✱ ❜❛st❛ ✈❡r✐✜❝❛r♠♦s q✉❡ A 6= 2B

  3

  2

  2

  2

  = # : 2x + y + 8z = 1 = # ❡❢❡✐t♦✱ s❡♥❞♦ A

  1 {(x, y, z) ∈ Z } ❡ B 1 {(x, y, z) ∈

  3

  2

  2

2 Z : 2x + y + 32z = 1

  }✱ ❡ ❝♦♠♦ ❛s ú♥✐❝❛s s♦❧✉çõ❡s ✐♥t❡✐r❛s ♣❛r❛ ❛s ❡q✉❛çõ❡s

  2

  2

  2

  2

  2

  2

  2x + y + 8z = 1 + y + 32z = 1 ❡ 2x sã♦ (0, ±1, 0) ❡ (0, ±1, 0) r❡s♣❡❝t✐✈❛♠❡♥t❡✱

  ❡♥tã♦ A = 2 .

  1

  1

  6= 4 = 2B ▼♦str❛♥❞♦ ♥♦✈❛♠❡♥t❡ q✉❡ ✶ ♥ã♦ é ❝♦♥❣r✉❡♥t❡✳

  2

  3

  : y = x

  2

  • n = 2✱ E − 4x✳ ❈♦♠♦ ❛s ú♥✐❝❛s s♦❧✉çõ❡s ✐♥t❡✐r❛s ♣❛r❛ ❛s ❡q✉❛çõ❡s

  2

  2

  2

  2

  2

  2

  8x +2y +16z = 2 +2y +64z = 2 ❡ 8x sã♦ (0, ±1, 0) ❡ (0, ±1, 0) r❡s♣❡❝t✐✈❛♠❡♥t❡✱

  ❡♥tã♦ C = 2 ,

  2

  2

  6= 4 = 2D ❞❡ ♦♥❞❡ s❡❣✉❡ q✉❡ ✷ ♥ã♦ é ❝♦♥❣r✉❡♥t❡✳

  2

  3

  2

  2

  2

  : y = x + y + 8z = 3

  3

  • n = 3✱ E − 9x✳ P❛r❛ ❛ ❡q✉❛çã♦ 2x ✱ ❛s ú♥✐❝❛s s♦❧✉çõ❡s

  2

  2

  2

  = 4 +y +32z = ✐♥t❡✐r❛s sã♦ (−1, ±1, 0) ❡ (1, ±1, 0)✱ ❛ss✐♠✱ A

  3 ✳ ❊ ♣❛r❛ ❛ ❡q✉❛çã♦ 2x

  3 = 4 . t❡♠♦s ❛s ♠❡s♠❛s s♦❧✉çõ❡s ✐♥t❡✐r❛s✱ ❞❡ ♦♥❞❡ s❡❣✉❡ q✉❡ B

  3 ❡ ♣♦rt❛♥t♦✱ A

  3

  3

  6= 2B

  2

  3

  2

  2

  2 10 : y = x

  • 2y + 16z = 10
    • n = 10✱ E − 100x✳ ❆s s♦❧✉çõ❡s ♣❛r❛ ❛s ❡q✉❛çõ❡s 8x

  2

  2

  2

  • 2y + 64z = 10 ❡ 8x ❝♦✐♥❝✐❞❡♠ ❡ sã♦ (−1, ±1, 0) ❡ (1, ±1, 0)✱ ❞❡ ♦♥❞❡ s❡❣✉❡ q✉❡ C

  10 = 4

  10

  6= 8 = 2D ✳ ❈♦♠ ✐ss♦✱ ❞♦s ✐♥t❡✐r♦s ❞❡ ✶ ❛ ✶✵✱ t❡♠♦s ♦ s❡❣✉✐♥t❡ q✉❛❞r♦✿ ✶✱ ✷✱ ✸✱ ✹✱ ✽✱ ✾ ❡ ✶✵ ♥ã♦ sã♦ ❝♦♥❣r✉❡♥t❡s✱ ❥á ✺✱ ✻ ❡ ✼ sã♦ ❝♦♥❣r✉❡♥t❡s ✭♦ ✹ ❡ ♦ ✾ sã♦ q✉❛❞r❛❞♦s r❛❝✐♦♥❛✐s✱ ❞❡ ♦♥❞❡ s❡❣✉❡ q✉❡ ❡stã♦ ♥❛ ♠❡s♠❛ ❝❧❛ss❡ ❞♦ ✶ ♥❛ r❡❧❛çã♦ q✉❡ ❛ss♦❝✐❛ ❞♦✐s r❛❝✐♦♥❛✐s ♥ã♦ ♥✉❧♦s a ❡ b s❡ ♦ q✉♦❝✐❡♥t❡ a/b é ✉♠ q✉❛❞r❛❞♦ r❛❝✐♦♥❛❧✱ s❡♥❞♦ q✉❡ ❡st❛r ♥❛ ♠❡s♠❛ ❝❧❛ss❡ ❢❛③ ❝♦♠

  ✾✷ q✉❡ ❛♠❜♦s ♦✉ s❡❥❛♠ ❝♦♥❣r✉❡♥t❡s ♦✉ ♥ã♦ s❡❥❛♠ ❝♦♥❣r✉❡♥t❡s✱ ❥á ♦ ✽ ❡stá ♥❛ ♠❡s♠❛ ❝❧❛ss❡ q✉❡ ♦ ✷✱ ♠♦str❛♥❞♦ q✉❡ ✽ ♥ã♦ é ❝♦♥❣r✉❡♥t❡✮✳

  ❘❡❢❡rê♥❝✐❛s

  ❬❆t✐②❛❤❪ ❆❚■❨❆❍✱ ▼✳ ❋✳✱ ▼❆❈❉❖◆❆▲❉✱ ■✳ ●✳✱ ■♥tr♦❞✉❝t✐♦♥ t♦ ❈♦♠♠✉t❛t✐✈❡ ❆❧❣❡❜r❛✱ ❆❞❞✐s♦♥✲❲❡s❧❡② P✉❜❧✐s❤✐♥❣ ❈♦♠♣❛♥②✱ ✶✾✻✾✳

  ❬❇✉tt❪ ❇❯❚❚✱ ❑✳✱ ❊❧❧✐♣t✐❝ ❈✉r✈❡s ❛♥❞ t❤❡ ▼♦r❞❡❧❧✲❲❡✐❧ ❚❤❡♦r❡♠✱ ✷✵✶✻✳ ❬❈♦♥❝❡✐çã♦❪ ❈❖◆❈❊■➬➹❖✱ ❘✳ P✳✱ ❆r✐t♠ét✐❝❛ ❞❛s ❈✉r✈❛s ❞❡ ●ê♥❡r♦ ✵ ❡ ✶ s♦❜r❡ ♦s ❝♦r♣♦s

  F ❡ Q✱ ❯♥✐✈❡rs✐❞❛❞❡ ❋❡❞❡r❛❧ ❞❡ P❡r♥❛♠❜✉❝♦✱ ❉✐ss❡rt❛çã♦✱ ✷✵✵✸✳

  ❬❈♦♥r❛❞❪ ❈❖◆❘❆❉✱ ❑✳✱ ❚❤❡ ❈♦♥❣r✉❡♥t ◆✉♠❜❡r Pr♦❜❧❡♠✱ ❍❛r✈❛r❞ ❈♦❧❧❡❣❡ ▼❛t❤❡♠❛t✐❝❛❧ ❘❡✈✐❡✇✱ ✷ ✭✷✮✱ ♣♣✳ ✺✽✲✼✸✱ ✷✵✵✽✳

  ❬❋✉❧t♦♥❪ ❋❯▲❚❖◆✱ ❲✳✱ ❆❧❣❡❜r❛✐❝ ❈✉r✈❡s✿ ❆♥ ■♥tr♦❞✉❝t✐♦♥ t♦ ❆❧❣❡❜r❛✐❝ ●❡♦♠❡tr②✱ ✷✵✵✽✳ ❬❍❛rts❤♦r♥❡❪ ❍❆❘❚❙❍❖❘◆❊✱ ❘✳✱ ❆❧❣❡❜r❛✐❝ ●❡♦♠❡tr②✱ ●❚▼ ✺✷✱ ❙♣r✐♥❣❡r✱ ✶✾✼✼✳ ❬❑♦❜❧✐t③❪ ❑❖❇▲■❚❩✱ ◆✳✱ ■♥tr♦❞✉❝t✐♦♥ t♦ ❊❧❧✐♣t✐❝ ❈✉r✈❡s ❛♥❞ ▼♦❞✉❧❛r ❋♦r♠s✱ ●❚▼ ✾✼✱

  ❙♣r✐♥❣❡r✱ ✶✾✾✸✳ ❬▲❛♥❣❪ ▲❆◆●✱ ❙✳✱ ❆❧❣❡❜r❛✐❝ ◆✉♠❜❡r ❚❤❡♦r②✱ ❙❡❝♦♥❞ ❊❞✐t✐♦♥✱ ●❚▼ ✶✶✵✱ ❙♣r✐♥❣❡r✱ ✶✾✾✹✳ ❬▼♦r❛♥❞✐❪ ▼❖❘❆◆❉■✱ P✳✱ ❋✐❡❧❞ ❛♥❞ ●❛❧♦✐s ❚❤❡♦r②✱ ●❚▼ ✶✻✼✱ ❙♣r✐♥❣❡r✱ ✶✾✾✻✳ ❬◆❡✉❦✐r❝❤❪ ◆❊❯❑■❘❈❍✱ ❏✳✱ ❆❧❣❡❜r❛✐❝ ◆✉♠❜❡r ❚❤❡♦r②✱ ●❚▼ ✸✷✷✱ ❙♣r✐♥❣❡r✱ ✶✾✾✾✳ ❬P❛❝❤❡❝♦❪ P❆❈❍❊❈❖✱ ❆✳✱ ◆ú♠❡r♦s ❝♦♥❣r✉❡♥t❡s ❡ ❝✉r✈❛s ❡❧ít✐❝❛s✳ ▼❛t❡♠át✐❝❛ ❯♥✐✈❡rs✐✲ tár✐❛✱ ♥✳ ✷✷✴✷✸✱ ♣♣✳ ✶✽✲✷✾✱ ✶✾✾✼✳ ❬Pér❡③❪ P➱❘❊❩✱ ❆✳ ❩✳✱ ❚❡♦r❡♠❛ ❞❡ ▼♦r❞❡❧❧✲❲❡✐❧✱ ❯♥✐✈❡rs✐❞❛❞ ❆✉tó♥♦♠❛ ❞❡ ▼❛❞r✐❞✱

  ❚r❛❜❛❥♦ ❞❡ ✜♥ ❞❡ ♠ást❡r✱ ✷✵✶✵✳ ❬❙✐❧✈❡r♠❛♥❪ ❙■▲❱❊❘▼❆◆✱ ❏✳ ❍✳✱ ❚❤❡ ❆r✐t❤♠❡t✐❝ ♦❢ ❊❧❧✐♣t✐❝ ❈✉r✈❡s✱ ✷♥❞ ❊❞✐t✐♦♥✱ ●❚▼

  ✶✵✻✱ ❙♣r✐♥❣❡r✱ ✷✵✵✾✳ ❬❚✉♥♥❡❧❧❪ ❚❯◆◆❊▲▲✱ ❏✳ ❇✳✱ ❆ ❝❧❛ss✐❝❛❧ ❉✐♦♣❤❛♥t✐♥❡ ♣r♦❜❧❡♠ ❛♥❞ ♠♦❞✉❧❛r ❢♦r♠s ♦❢ ✇❡✐❣❤t

  ✸✴✷✳ ■♥✈❡♥t✐♦♥❡s ▼❛t❤❡♠❛t✐❝❛❡✳ ✼✷ ✭✷✮✱ ♣♣✳ ✸✷✸✲✸✸✹✱ ✶✾✽✸✳ ❬❲✐❧❡s✶❪ ❲■▲❊❙✱ ❆✳✱ ❚❤❡ ❇✐r❝❤ ❛♥❞ ❙✇✐♥♥❡rt♦♥✲❉②❡r ❈♦♥❥❡❝t✉r❡✳ ❚❤❡ ▼✐❧❧❡♥♥✐✉♠ ♣r✐③❡

  ✾✹ ❬❲✐❧❡s✷❪ ❲■▲❊❙✱ ❆✳✱ ▼♦❞✉❧❛r ❡❧❧✐♣t✐❝ ❝✉r✈❡s ❛♥❞ ❋❡r♠❛t✬s ❧❛st t❤❡♦r❡♠✱ ❆♥♥✳ ▼❛t❤✳ ✶✹✷✱

  ♣♣✳ ✹✹✸✲✺✺✶✱ ✶✾✾✺✳

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