✣❸■ ❍➴❈ ◗❯➮❈ ●■❆ ❍⑨ ◆❐■ ❚❘×❮◆● ✣❸■ ❍➴❈ ❑❍❖❆ ❍➴❈ ❚Ü ◆❍■➊◆ ✖✖✖✖✖✖✖✖✕ ❱➹ ❚Ò◆● ▲■◆❍ ❱➋ ❉❸◆● ❈❍❯❽◆ ❊❉❲❆❘❉❙ ❱⑨ ▼❐❚ ❱⑨■ Ù◆● ❉Ö◆● ▲❯❾◆ ❱❿◆ ❚❍❸❈ ❙➒ ❚❖⑩◆ ❍➴❈ ❍⑨ ◆❐■ ✲ ✷✵✶✹ ✣❸■ ❍➴❈ ◗❯➮❈ ●■❆ ❍⑨ ◆❐■ ❚❘×❮◆● ✣❸■ ❍➴❈ ❑❍❖❆ ❍➴❈ ❚Ü ◆❍■➊◆ ✖✖✖✖✖✖✖✖✕ ❱➹ ❚Ò◆● ▲■◆❍ ❱➋ ❉❸◆● ❈❍❯❽◆ ❊❉❲❆❘❉❙ ❱⑨ ▼❐❚ ❱⑨■ Ù◆● ❉Ư◆● ❈❤✉②➯♥ ♥❣➔♥❤✿ ✣❸■ ❙➮ ❱⑨ ▲Þ ❚❍❯❨➌❚ ❙➮ ▼➣ sè✿ ✻✵✹✻✵✶✵✹ ▲❯❾◆ ❱❿◆ ❚❍❸❈ ❙➒ ❚❖⑩◆ ❍➴❈ ◆❣÷í✐ ữợ Põ ự ✲ ✷✵✶✹ ▼ư❝ ❧ư❝ ▲í✐ ❝↔♠ ì♥ ▲í✐ ♠ð ✤➛✉ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✷ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✸ ✶ ❑✐➳♥ t❤ù❝ ❝❤✉➞♥ ❜à ỵ tt ữớ t ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶✳✷ ❉↕♥❣ ▼♦♥t❣♦♠❡r② ❝õ❛ ✤÷í♥❣ ❝♦♥❣ ❡❧❧✐♣t✐❝ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✷ ❉↕♥❣ ❝❤✉➞♥ ❊❞✇❛r❞s ❝❤♦ ✤÷í♥❣ ❝♦♥❣ ❡❧❧✐♣t✐❝ ✷✳✶ ✷✳✷ ✻ ✶✷ ✶✺ ❉↕♥❣ ❝❤✉➞♥ ❊❞✇❛r❞s ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶✺ ✷✳✶✳✶ ❉↕♥❣ ❝❤✉➞♥ ❊❞✇❛r❞s ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶✺ ✷✳✶✳✷ ❍❛✐ ❝ỉ♥❣ t❤ù❝ ❝ë♥❣ ✤✐➸♠ tr➯♥ ✤÷í♥❣ ❝♦♥❣ ❊❞✇❛r❞s ✳ ✳ ✳ ✷✵ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✷✼ ◆❤â♠ ❝→❝ ✤✐➸♠ tr➯♥ ✤÷í♥❣ ❝♦♥❣ ❊❞✇❛r❞s ❝✉ë♥ ✸ ▼ët sè ù♥❣ ❞ư♥❣ ❝õ❛ ✤÷í♥❣ ❝♦♥❣ ❞↕♥❣ ❝❤✉➞♥ ❊❞✇❛r❞s ✸✳✶ ❈→❝ ✤✐➸♠ ❝â ❝➜♣ ♥❤ä tr➯♥ ✤÷í♥❣ ❝♦♥❣ ❊❞✇❛r❞s ❝✉ë♥ ✸✳✷ ◆❤â♠ ①♦➢♥ ❝õ❛ ✤÷í♥❣ ❝♦♥❣ ❊❞✇❛r❞s tr➯♥ ✸✳✸ ✹✶ ✳ ✳ ✳ ✳ ✳ ✳ ✹✶ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✹✻ Ù♥❣ ❞ư♥❣ ❝õ❛ ✤÷í♥❣ ❝♦♥❣ ❊❞✇❛r❞s tr♦♥❣ ♠➟t ♠➣ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✺✽ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✻✶ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✻✷ ❑➳t ❧✉➟♥ ❚➔✐ ❧✐➺✉ t❤❛♠ ❦❤↔♦ ✶ Q ▲í✐ ỡ ữủ t ữợ sỹ ữợ t t ❣✐→♦✱ ❚✐➳♥ s➽ P❤â ✣ù❝ ❚➔✐✱ ●✐↔♥❣ ✈✐➯♥ ❑❤♦❛ ❚♦→♥✲❈ì✲❚✐♥ ❤å❝✱ ❚r÷í♥❣ ✣↕✐ ❤å❝ ❑❤♦❛ ❤å❝ ❚ü ♥❤✐➯♥✱ ✣↕✐ ❤å❝ ◗✉è❝ ❣✐❛ ❍➔ ♥ë✐✳ ❚❤➛② ✤➣ ❣✐➔♥❤ ♥❤✐➲✉ t❤í✐ ❣✐❛♥ ữợ tr ỳ t ❝õ❛ tæ✐ tr♦♥❣ s✉èt q✉→ tr➻♥❤ ❧➔♠ ❧✉➟♥ ✈➠♥✳ ◗✉❛ ❧✉➟♥ ✈➠♥ ♥➔②✱ tỉ✐ ♠✉è♥ ❜➔② tä ❧á♥❣ ❜✐➳t ì♥ s➙✉ s➢❝ ✤➳♥ ❚❤➛② ❣✐→♦ ❝õ❛ ♠➻♥❤✳ ❚æ✐ ①✐♥ ❣û✐ ❧í✐ ❝↔♠ ì♥ s➙✉ s➢❝ ✤➳♥ ❝→❝ ▲➣♥❤ ✤↕♦ ❱✐➺♥ ❑❤♦❛ ❤å❝ ✲ ❈ỉ♥❣ ♥❣❤➺ ▼➟t ♠➣✱ ❇❛♥ ❈ì ❨➳✉ ❈❤➼♥❤ P❤õ✱ ▲➣♥❤ ✤↕♦ P❤➙♥ ✈✐➺♥ ◆❣❤✐➯♥ ❝ù✉ ❑❤♦❛ ❤å❝ ▼➟t ♠➣ ✈➔ t➜t ❝↔ ❝→❝ ❈ỉ✱ ❈❤ó ✈➔ ❆♥❤✱ ỗ tr ỡ t tố ụ ữ õ õ ỳ ỵ qỵ ú tổ t ❚ỉ✐ ❝ơ♥❣ ①✐♥ ❣û✐ ❧í✐ ❝↔♠ ì♥ s➙✉ s➢❝ tỵ✐ P●❙✳❚❙✳ ▲➯ ▼✐♥❤ ❍➔ ✈➔ ❝→❝ ❚❤➛②✱ ❈æ tr♦♥❣ ❑❤♦❛ ❚♦→♥✲❈ì✲❚✐♥ ❤å❝✱ ❚r÷í♥❣ ✣↕✐ ❤å❝ ❑❤♦❛ ❤å❝ ❚ü ◆❤✐➯♥✱ ✣↕✐ ❤å❝ ◗✉è❝ ●✐❛ ❍➔ ♥ë✐✱ ❝ơ♥❣ ♥❤÷ t➜t ❝↔ ♥❤ú♥❣ ❚❤➛②✱ ❈æ ✤➣ t❤❛♠ ❣✐❛ ❣✐↔♥❣ ❞↕② ❦❤â❛ ❈❛♦ ❤å❝ ổ õ ỳ ữợ ✈➔ ❝æ♥❣ ❧❛♦ ❞↕② ❞é ❝õ❛ ❝→❝ ❚❤➛②✱ ❈æ t❤➻ tỉ✐ ❝ơ♥❣ ❦❤ỉ♥❣ ❤♦➔♥ t❤➔♥❤ ✤÷đ❝ ❧✉➟♥ ✈➠♥ ♥➔②✳ ▲í✐ ❝✉è✐ ❝ị♥❣✱ tỉ✐ ♠✉è♥ ❣û✐ ❧í✐ ❝↔♠ ì♥ s➙✉ s➢❝ ✤➳♥ ❇è✱ ▼➭ ✈➔ ❣✐❛ ✤➻♥❤ tỉ✐✱ ♥❤ú♥❣ ♥❣÷í✐ ✤➣ t✐♥ t÷ð♥❣ s➙✉ s➢❝✱ ✤➣ ❧✉ỉ♥ ❝ê ✈ơ ✤ë♥❣ ✈✐➯♥ ✈➔ ❝❤✐❛ s➫ ♠å✐ ❦❤â ❦❤➠♥ ❣✐ó♣ tỉ✐ ❤♦➔♥ t❤➔♥❤ ❧✉➟♥ ✈➠♥ ♥➔②✳ ❚ỉ✐ ❝ơ♥❣ ①✐♥ ❝↔♠ ì♥ t➜t ❝↔ ♥❤ú♥❣ ❛♥❤ ❡♠ ❜↕♥ ❜➧ ❧✉æ♥ ❜➯♥ ❝↕♥❤ tæ✐ tr♦♥❣ tr♦♥❣ s✉èt ❦❤â❛ ❤å❝ ♥➔②✳ ❚æ✐ ①✐♥ ❝❤➙♥ t❤➔♥❤ ❝↔♠ ì♥ t➜t ❝↔✦ ❍➔ ◆ë✐✱ t❤→♥❣ ✶✷ ♥➠♠ ✷✵✶✹ ❍å❝ ✈✐➯♥ ❱ã ❚ị♥❣ ▲✐♥❤ ✷ ▲í✐ ♠ð ✤➛✉ ❚r♦♥❣ ♥❤ú♥❣ t trữợ tr ▼✐❧❧❡r ✤➣ ✤ë❝ ❧➟♣ ✤➲ ①✉➜t ✈✐➺❝ sû ❞ư♥❣ ✤÷í♥❣ ❝♦♥❣ ❡❧❧✐♣t✐❝ ❝❤♦ ❝→❝ ❤➺ ♠➟t ♠➣ ❦❤â❛ ❝æ♥❣ ❦❤❛✐✳ ❚ø ✤â ✤➳♥ ♥❛② ❤➺ ♠➟t ✤÷í♥❣ ❝♦♥❣ ❡❧❧✐♣t✐❝ ✤➣ ✤÷đ❝ ♥❣❤✐➯♥ ❝ù✉ s➙✉ rë♥❣ ✈➔ trð ♥➯♥ ♣❤ê ❜✐➳♥ ❝ị♥❣ ✈ỵ✐ ❝→❝ ❤➺ ♠➟t ♠➣ ❦❤â❛ ❝ỉ♥❣ ❦❤❛✐ ❦❤→❝✱ ❝❤➥♥❣ ❤↕♥ ♥❤÷ ❘❙❆✱ ❉✐❢❢✐❡ ✕ ❍❡❧❧♠❛♥ ✈➔ ❊❧●❛♠❛❧✳ ❉♦ ÷✉ t❤➳ ❧➔ ❝â ❝ï ❝õ❛ ❝→❝ t❤❛♠ ❜✐➳♥ ♥❤ä ❤ì♥ s♦ ✈ỵ✐ ❝→❝ ❤➺ ♠➟t ♠➣ ❦❤â❛ ❝ỉ♥❣ ❦❤❛✐ ❦❤→❝ ❦❤✐ ①➨t ð ❝ò♥❣ ♠ët ♠ù❝ ❛♥ t♦➔♥ ♥➯♥ ❤➺ ♠➟t ✤÷í♥❣ ❝♦♥❣ ❡❧❧✐♣t✐❝ ❧➔ r➜t ❤➜♣ ❞➝♥ ✤è✐ ✈ỵ✐ ❝→❝ ù♥❣ ❞ư♥❣ ♠➔ ❝â t➔✐ ♥❣✉②➯♥ ❤↕♥ ❝❤➳✳ ❱➔♦ ♥➠♠ ✷✵✵✼✱ ❍❛r♦❧❞ ❊❞✇❛r❞s tr♦♥❣ ❬✼❪ ✤➣ ✤➲ ①✉➜t ởt t ợ ữớ t tờ qt õ ởt t ỗ tø ❊✉❧❡r ✈➔ ●❛✉ss✱ ❊❞✇❛r❞s ✤➣ ❣✐ỵ✐ t❤✐➺✉ ♠ët ♣❤➨♣ ❝ë♥❣ ✤✐➸♠ tr➯♥ ✤÷í♥❣ ❝♦♥❣ x2 + y = c2 (1 + x2 y ) tr➯♥ ♠ët tr÷í♥❣ k ❝â ✤➦❝ sè ❦❤→❝ ✷✳ ▼➦❝ ❞ò ❜➔✐ ❜→♦ ❝õ❛ ❍✳ ❊❞✇❛r❞s ❦❤ỉ♥❣ t➟♣ tr✉♥❣ ✈➔♦ ✈✐➺❝ →♣ ❞ư♥❣ ❞↕♥❣ ✤÷í♥❣ ❝♦♥❣ ♥➔② tr♦♥❣ ♠➟t ♠➣✱ ♥❤÷♥❣ ❞➛♥ ❞➛♥✱ ✈ỵ✐ ♥❤ú♥❣ ♥❣❤✐➯♥ ❝ù✉ s❛✉ ✤â✱ ❞↕♥❣ ❝❤✉➞♥ t➢❝ ♥➔② ✤➣ t❤➸ ❤✐➺♥ ❝→❝ t➼♥❤ ❝❤➜t ♠➟t ♠➣ ✤→♥❣ ♠♦♥❣ ♠✉è♥ ✈➔ ❤ú✉ ➼❝❤ tr♦♥❣ ♥é ❧ü❝ tr→♥❤ ✤➸ ❧ë t❤æ♥❣ t✐♥✳ ❚✐➳♣ s❛✉ ❊❞✇❛r❞s✱ ❇❡r♥st❡✐♥✱ ▲❛♥❣✱ ❇✐r❦❡r ✈➔ ❝→❝ ❝ë♥❣ sü tr♦♥❣ ❬✶✱ ✷✱ ✹✱ ✺❪ ✤➣ tê♥❣ q✉→t õ ự rs ởt ợ ữớ rë♥❣ ❤ì♥ ax2 + y = + dx2 y ✈ỵ✐ a = d, a, d ∈ k \ {0, 1}✳ ◆❤ú♥❣ t→❝ ❣✐↔ ♥➔② ✤➣ ❦➳t ❤ñ♣ þ t÷ð♥❣ ①➙② ❞ü♥❣ ♣❤➨♣ ❝ë♥❣ ✤✐➸♠ ❝õ❛ ❊❞✇❛r❞s ✈➔ ♣❤➨♣ ❝ë♥❣ ✤✐➸♠ ✤è✐ ♥❣➝✉ ❞♦ ❍✐s✐❧✱ ❲♦♥❣✱ ❈❛rt❡r ✈➔ ❉❛✇s♦♥ ✤➲ ①✉➜t tr♦♥❣ ❬✾❪ ✤➸ ✤÷❛ r❛ ♠ët ❝ỉ♥❣ t❤ù❝ ❞✉② ♥❤➜t ❝❤♦ ❝↔ ✈✐➺❝ ❝ë♥❣ ✤✐➸♠ ❧➝♥ ♥❤➙♥ ✤æ✐ ✤✐➸♠✳ ✣➙② ❧➔ ♠ët ♣❤→t tr✐➸♥ q✉❛♥ trå♥❣ ❜ð✐ ❦❤ỉ♥❣ ❝❤➾ ♠❛♥❣ ❧↕✐ ❝❤♦ ♥❤â♠ ✤✐➸♠ tr➯♥ ❝→❝ ✤÷í♥❣ ❝♦♥❣ ❊❞✇❛r❞s ❝✉ë♥ ♥â✐ ❝❤✉♥❣ ✈➔ ❝→❝ ✤÷í♥❣ ❝♦♥❣ ❊❞✇❛r❞s ♥â✐ r✐➯♥❣ ♠ët ✸ ▲í✐ ♠ð ✤➛✉ ✹ ❝➜✉ tró❝ ♥❤â♠✱ ♠➔ ❝æ♥❣ t❤ù❝ ❝ë♥❣ ✤✐➸♠ ❞✉② ♥❤➜t ♥➔② ❧➔ ❝ì sð ♥➲♥ t↔♥❣ ✈ú♥❣ ❝❤➢❝ ❝❤♦ ✈✐➺❝ sû ❞ư♥❣ ❞↕♥❣ ❝❤✉➞♥ ❊❞✇❛r❞s tr♦♥❣ ♠➟t ♠➣ ♥❤➡♠ ❝❤è♥❣ ❧↕✐ ❝→❝ t➜♥ ❝ỉ♥❣ ❦➯♥❤ ❦➲✳ ❍ì♥ ♥ú❛✱ tr♦♥❣ ♥❤✐➲✉ tr÷í♥❣ ❤đ♣✱ ♣❤➨♣ ❝ë♥❣ ✤✐➸♠ ❞♦ ❝→❝ t→❝ ❣✐↔ tr➯♥ ✤÷❛ r❛ ❝â sè ❧÷đ♥❣ ♥❤ú♥❣ t➼♥❤ t♦→♥ ❝ì ❜↔♥ ✭♣❤➨♣ ♥❤➙♥ ✈➔ ♣❤➨♣ ❝ë♥❣ tr♦♥❣ tr÷í♥❣ ❝ì sð✮ ➼t ❤ì♥✱ ❞➝♥ ✤➳♥ ✈✐➺❝ t➼♥❤ t♦→♥ tr♦♥❣ t❤ü❝ t➳ s➩ ♥❤❛♥❤ ❤ì♥ s ợ rstrss ỗ tớ t ụ ỹ tữớ ợ ữớ rs õ ợ ữớ t rstrss tr trữớ Q ợ õ trữợ r ❧✉➟♥ ✈➠♥ ♥➔②✱ ❝❤ó♥❣ tỉ✐ tr➻♥❤ ❜➔② ❧↕✐ ✤à♥❤ ♥❣❤➽❛ ✤÷í♥❣ ❝♦♥❣ ❊❞✇❛r❞s ✈➔ ✤÷í♥❣ ❝♦♥❣ ❊❞✇❛r❞s ❝✉ë♥ t❤❡♦ ♥❣❤✐➯♥ ❝ù✉ ❝õ❛ ❇❡rst❡✐♥ ✈➔ ❝→❝ ❝ë♥❣ sü✳ ❈❤ó♥❣ tỉ✐ ❝ơ♥❣ ✤✐ ✈➔♦ ❝❤✐ t✐➳t ✈✐➺❝ ①➙② ❞ü♥❣ ♣❤➨♣ ❝ë♥❣ ✤✐➸♠ tr➯♥ ❝→❝ ❞↕♥❣ ✤÷í♥❣ ❝♦♥❣ ♥➔②✱ ✈➔ tø ✤➜② ✤✐ t➼♥❤ ❝→❝ ♥❤â♠ ①♦➢♥ ❝â t❤➸ ❝â ❝õ❛ ❝❤ó♥❣ tr➯♥ trữớ Q ố ỗ õ ❝❤÷ì♥❣✿ ❈❤÷ì♥❣ ✶✿ ❑✐➳♥ t❤ù❝ ❝❤✉➞♥ ❜à✳ ❚r♦♥❣ ❝❤÷ì♥❣ ♥➔②✱ ❝❤ó♥❣ tỉ✐ tr➻♥❤ ❜➔② ♠ët sè ❦✐➳♥ t❤ù❝ ❝❤✉➞♥ ❜à ỵ tt ữớ t tờ qt ỗ ❝→❝ ✤à♥❤ ♥❣❤➽❛✱ ❦➳t q✉↔ ❝ì ❜↔♥✱ ✈✐➺❝ ①➙② ❞ü♥❣ tr ữớ t ỗ tớ ú tỉ✐ ❝ơ♥❣ tr➻♥❤ ❜➔② ✈➲ ❞↕♥❣ ▼♦♥t❣♦♠❡r② ❝õ❛ ✤÷í♥❣ ❝♦♥❣ ❡❧❧✐♣t✐❝ ✈➔ ✈✐➺❝ ❜✐➳♥ ✤ê✐ q✉❛ ❧↕✐ ❣✐ú❛ ❞↕♥❣ ▼♦♥t❣♦♠❡r② ✈➔ ❞↕♥❣ ❲❡✐❡rstr❛ss✳ ❈❤÷ì♥❣ ✷✿ ❉↕♥❣ ❝❤✉➞♥ ❊❞✇❛r❞s ❝❤♦ ✤÷í♥❣ t ữỡ ỗ P ởt tr ❜➔② ✈➲ ❞↕♥❣ ❝❤✉➞♥ ❊❞✇❛r❞s ✈➔ ❞↕♥❣ tê♥❣ q✉→t ❤ì♥ ❧➔ ❝→❝ ✤÷í♥❣ ❝♦♥❣ ❊❞✇❛r❞s ❝✉ë♥✳ ❈❤ó♥❣ tỉ✐ ❝ơ♥❣ tr➻♥❤ ❜➔② ♠è✐ q✉❛♥ ❤➺ t÷ì♥❣ ✤÷ì♥❣ s♦♥❣ ❤ú✉ t➾ ❣✐ú❛ ♠ët ✤÷í♥❣ ❝♦♥❣ ❊❞✇❛r❞s ❝✉ë♥ ✭tr÷í♥❣ ❤đ♣ r✐➯♥❣ ❧➔ ✤÷í♥❣ rs ợ ữớ rstrss õ ✤÷í♥❣ ❝♦♥❣ ❞↕♥❣ ▼♦♥t❣♦♠❡r② ♥â✐ r✐➯♥❣✳ ❚r♦♥❣ ♣❤➛♥ ♥➔② ❝❤ó♥❣ tỉ✐ ❝ơ♥❣ tr➻♥❤ ❜➔② ❝❤✐ t✐➳t ❤❛✐ ❝ỉ♥❣ t❤ù❝ ❝ë♥❣ ✤✐➸♠ tr➯♥ ✤÷í♥❣ ❝♦♥❣ ❊❞✇❛r❞s ✈➔ ❝❤➾ r❛ ♥❤÷đ❝ ✤✐➸♠ ❝õ❛ ❤❛✐ ❝æ♥❣ t❤ù❝ ♥➔②✳ P❤➛♥ ❤❛✐ tr➻♥❤ ❜➔② ✈➲ ❝æ♥❣ t❤ù❝ ❝ë♥❣ ✤✐➸♠ ✤➛② ✤õ ✈➔ ❞✉② ♥❤➜t tr➯♥ ữớ rs ợ ữủ ð ❞↕♥❣ ①↕ ↔♥❤ tr♦♥❣ P1 × P1 ✳ ❚➼♥❤ ✤ó♥❣ ✤➢♥ ❝õ❛ ♣❤➨♣ ❝ë♥❣ ✤✐➸♠ ♥➔② ✤÷đ❝ ❝❤ù♥❣ ♠✐♥❤ q ỵ ứ õ rót r❛ ▲í✐ ♠ð ✤➛✉ ✺ ❤➺ q✉↔ q✉❛♥ trå♥❣ ❧➔ t➟♣ ❝→❝ ✤✐➸♠ tr➯♥ ✤÷í♥❣ ❝♦♥❣ ❊❞✇❛r❞s ❝✉ë♥ ✭✤÷í♥❣ ❝♦♥❣ ❊❞✇❛r❞s✮ ❧➔ ♠ët ♥❤â♠ ❛❜❡♥✱ ❤ì♥ ♥ú❛ ♥❤â♠ ♥➔② ợ õ tr ữớ t ▼♦♥t❣♦♠❡r② t÷ì♥❣ ù♥❣✳ ❈❤÷ì♥❣ ✸✿ ▼ët sè ù♥❣ ❞ư♥❣ ❝õ❛ ữớ rs ữỡ ỗ P❤➛♥ ♠ët ❝❤ó♥❣ tỉ✐ t➼♥❤ ❝→❝ ✤✐➸♠ ❝â ❝➜♣ ♥❤ä✱ ❝ö t❤➸ ❧➔ ❝→❝ ✤✐➸♠ ❝➜♣ ✷✱ ✸✱ ✹✱ ✽ tr➯♥ ✤÷í♥❣ ❝♦♥❣ ❊❞✇❛r❞s ❝✉ë♥✳ P❤➛♥ ❤❛✐ ❝❤ó♥❣ tỉ✐ tr➻♥❤ ❜➔② ✤✐➲✉ ❦✐➺♥ ❝õ❛ t❤❛♠ sè d ✤➸ ✤÷í♥❣ ❝♦♥❣ rs tr Q õ õ trữợ ứ ✤â✱ ♥❤÷ ♠ët ❤➺ q✉↔✱ ❝❤ó♥❣ tỉ✐ ①➙② ❞ü♥❣ ♠ët ợ ữớ t rstrss ợ õ ✤➣ ❝❤♦ t❤➸ ❤✐➺♥ q✉❛ ❍➺ q✉↔ ✸✳✶✷✳ ❈✉è✐ ❝ò♥❣✱ tr♦♥❣ ♣❤➛♥ ❜❛ ❝❤ó♥❣ tỉ✐ ✤÷❛ r❛ ♠ët ✈➔✐ ♥❤➟♥ ①➨t ✈➲ ❦❤↔ ♥➠♥❣ ù♥❣ ❞ư♥❣ ✤÷í♥❣ ❝♦♥❣ ❊❞✇❛r❞s tr♦♥❣ ♠➟t ♠➣✳ ❚➜t ❝↔ t➼♥❤ t♦→♥ tr♦♥❣ ❧✉➟♥ ✈➠♥ ❝❤ó♥❣ tổ ữủ tỹ ợ ◆ë✐✱ t❤→♥❣ ✶✷ ♥➠♠ ✷✵✶✹ ❍å❝ ✈✐➯♥ ❱ã ❚ò♥❣ ▲✐♥❤ ❈❤÷ì♥❣ ✶ ❑✐➳♥ t❤ù❝ ❝❤✉➞♥ ❜à ❚r♦♥❣ ❝❤÷ì♥❣ ♥➔②✱ ❝❤ó♥❣ tổ tr ởt số tự ỵ tt ✤÷í♥❣ ❝♦♥❣ ❡❧❧✐♣t✐❝ tê♥❣ q✉→t✳ ◆❣♦➔✐ r❛✱ ❝❤ó♥❣ tỉ✐ ❝ơ♥❣ tr➻♥❤ ❜➔② ✈➲ ❞↕♥❣ ▼♦♥t❣♦♠❡r② ❝õ❛ ✤÷í♥❣ ❝♦♥❣ ❡❧❧✐♣t✐❝✳ ◆❤ú♥❣ ❦➳t q✉↔ ❝❤➼♥❤ ✤÷đ❝ ❧➜② tø ❝→❝ t➔✐ ❧✐➺✉ ❬✽✱ ỵ tt ữớ ❡❧❧✐♣t✐❝ ❈❤♦ K ❧➔ ♠ët tr÷í♥❣ ❝â ✤➦❝ sè tị② þ✳ ✣à♥❤ ♥❣❤➽❛ ✶✳✶✳ ▼ët ❬✽✱ ✣à♥❤ ♥❣❤➽❛ ✸✳✶❪ ✤÷í♥❣ ❝♦♥❣ ❡❧❧✐♣t✐❝ E tr➯♥ tr÷í♥❣ K ✤÷đ❝ ✤à♥❤ ♥❣❤➽❛ ❜ð✐ ♣❤÷ì♥❣ tr➻♥❤ E : y + a1 xy + a3 y = x3 + a2 x2 + a4 x + a6 , ✈ỵ✐ a1 , a2 , a3 , a4 , a6 ∈ K ✈➔ ∆ = 0✱ tr♦♥❣ ✤â ∆ ❧➔ ❜✐➺t t❤ù❝ ✭✶✳✶✮ ❝õ❛ E ✤÷đ❝ ✤à♥❤ ♥❣❤➽❛ ♥❤÷ s❛✉✿ ∆ = −d22 d8 − 8d34 − 27d26 + 9d2 d4 d6 d2 = a1 + 4a2 d4 = 2a4 + a1 a3 d6 = a23 + 4a6 d8 = a2 a6 + 4a2 a6 − a1 a3 a4 + a2 a2 − a2 ◆➳✉ L ❧➔ ♠ët tr÷í♥❣ ♠ð rë♥❣ ❝õ❛ K t❤➻ t➟♣ ❝→❝ ✤✐➸♠ L − ❤ú✉ t➾ tr➯♥ E ❧➔ E(L) = {(x, y) ∈ L × L : y + a1 xy + a3 y − x3 − a2 x2 − a4 x − a6 = 0} ∪ {∞} ✻ ❈❤÷ì♥❣ ✶✳ ❑✐➳♥ t❤ù❝ ❝❤✉➞♥ ❜à ∞ tr♦♥❣ ✤â ❧➔ ✼ ✤✐➸♠ t↕✐ ✈ỉ ❤↕♥✳ ❱➼ ❞ư ✶✳✷✳ ❍➻♥❤ ✶✳✶✿ y = x3 − x ❈❤♦ E ❍➻♥❤ ✶✳✷✿ y = x3 + x ❧➔ ♠ët ✤÷í♥❣ ❝♦♥❣ ❡❧❧✐♣t✐❝ tr➯♥ tr÷í♥❣ K ❝â ♣❤÷ì♥❣ tr➻♥❤ t ữợ E : y + a1 xy + a3 y = x3 + a2 x2 + a4 x + a6 ❑❤✐ ✤â ♣❤÷ì♥❣ tr➻♥❤ ①↕ ↔♥❤ ❝õ❛ E s➩ ❧➔ E¯ : y z + a1 xyz + a3 yz = x3 + a2 x2 z + a4 xz + a6 z , ✈➔ ✤✐➸♠ P ♥➳✉ ✤✐➸♠ tr➯♥ P z=0 z=0 s õ tồ t ữợ õ tồ t ữợ ❧➔ ❝õ❛ ♥â s➩ ❧➔ ✈ỵ✐ E (x : y : 1)✳ P ◆❣÷đ❝ ❧↕✐✱ ♥➳✉ ✤✐➸♠ ❝❤➼♥❤ ❧➔ ✤✐➸♠ ∞, ❉➵ t❤➜②✱ (x, y) t❤➻ ❞↕♥❣ ①↕ ↔♥❤ t÷ì♥❣ ù♥❣ P t❤➻ ❞↕♥❣ ❛❢❢✐♥❡ t÷ì♥❣ ù♥❣ ❝õ❛ ♥â s➩ ❧➔ t❤➻ ✤✐➸♠ (x : y : z) ❝â tå❛ ✤ë ①↕ ↔♥❤ (x/z, y/z)✳ (x : y : z) ❚r♦♥❣ tr÷í♥❣ ❤đ♣ ✈➔ t❛ ❝â ❞↕♥❣ ①↕ ↔♥❤ ❝õ❛ ✤✐➸♠ ✈ỉ ❝ị♥❣ ❧➔ P = (0 : y : 0) = (0 : : 0)✳ ❚❛ ❝â ♠ët số ú ỵ ú ỵ P❤÷ì♥❣ tr➻♥❤ ❲❡✐❡rstr❛ss tê♥❣ q✉→t✱ ❤❛② ✤➸ ✤ì♥ ❣✐↔♥✱ t❛ ❣å✐ ❧➔ P❤÷ì♥❣ tr➻♥❤ ❲❡✐❡rstr❛ss✳ ✶✳ P❤÷ì♥❣ tr➻♥❤ ✭✶✳✶✮ ✤÷đ❝ ❣å✐ ❧➔ ❈❤÷ì♥❣ ✶✳ ❑✐➳♥ t❤ù❝ ❝❤✉➞♥ ❜à ✷✳ ❚❛ ♥â✐ E ✤÷đ❝ ✽ ✤à♥❤ ♥❣❤➽❛ tr➯♥ ♣❤÷ì♥❣ tr➻♥❤ ✤à♥❤ ♥❣❤➽❛ ❝õ❛ E K ✤à♥❤ ♥❣❤➽❛ tr➯♥ t❤➻ E K E ❜ð✐ ✈➻ ❝→❝ ❤➺ sè a1 , a2 , a3 , a4 , a6 ❧➔ ❝→❝ ♣❤➛♥ tû t❤✉ë❝ K✳ tr♦♥❣ ❘ã r➔♥❣ ❧➔ ♥➳✉ ❝ô♥❣ ✤à♥❤ ♥❣❤➽❛ tr➯♥ ởt trữớ rở tũ K ỵ ❦✐➺♥ ∆ = ✤↔♠ ❜↔♦ ✤÷í♥❣ ❝♦♥❣ ❡❧❧✐♣t✐❝ ♥❣❤➽❛ ổ tỗ t tr E E ✏trì♥✑✱ ✤✐➲✉ ♥➔② ❝â ♠➔ t↕✐ ✤â ✤÷í♥❣ ❝♦♥❣ ❝â ♥❤✐➲✉ ❤ì♥ ♠ët ✤÷í♥❣ t❤➥♥❣ t✐➳♣ t✉②➳♥✳ ✹✳ ✣✐➸♠ ∞ ❧➔ ✤✐➸♠ ❞✉② ♥❤➜t tr➯♥ ✤÷í♥❣ t❤➥♥❣ t↕✐ ✈ỉ ❤↕♥ ♠➔ t❤ä❛ ♠➣♥ ❞↕♥❣ ①↕ ↔♥❤ ❝õ❛ ♣❤÷ì♥❣ tr➻♥❤ ❲❡✐❡rstr❛ss✳ ✺✳ ❈→❝ ✤✐➸♠ L − ❤ú✉ t➾ tr➯♥ E ❧➔ ❝→❝ ✤✐➸♠ ✤÷í♥❣ ❝♦♥❣ ✈➔ ❝â ❝→❝ tå❛ ✤ë ♠ët ✤✐➸♠ L − ❤ú✉ ✣à♥❤ ♥❣❤➽❛ ✶✳✹✳ x, y t❤✉ë❝ (x, y) t❤ä❛ ♠➣♥ ♣❤÷ì♥❣ tr➻♥❤ ❝õ❛ L✳ ✣✐➸♠ t↕✐ ổ ữủ t ố ợ trữớ ♠ð rë♥❣ ❍❛✐ ✤÷í♥❣ ❝♦♥❣ ❡❧❧✐♣t✐❝ E1 ✈➔ E2 L ❝õ❛ K✳ ✤à♥❤ ♥❣❤➽❛ tr➯♥ K ✈➔ ✤÷đ❝ ❝❤♦ ❜ð✐ ❝→❝ ♣❤÷ì♥❣ tr➻♥❤ ❲❡✐❡rstr❛ss E1 : y + a1 xy + a3 y = x3 + a2 x2 + a4 x + a6 E2 : y + a ¯1 xy + a ¯3 y = x3 + a ¯2 x2 + a ¯4 x + a ¯6 ✤÷đ❝ õ tr K tỗ t u, r, s, t ∈ K, u = s❛♦ ❝❤♦ ♣❤➨♣ ✤ê✐ ❜✐➳♥ (x, y) → (u2 x + r, u3 y + u2 sx + t) ❜✐➳♥ ✤ê✐ ♣❤÷ì♥❣ tr➻♥❤ E1 t❤➔♥❤ ♣❤÷ì♥❣ tr➻♥❤ ✭✶✳✷✮ E2 ✳ ❇➙② ❣✐í✱ ❣✐↔ sû t❛ ❝â ♣❤÷ì♥❣ tr➻♥❤ ❲❡✐❡rstr❛ss E : y + a1 xy + a3 y = x3 + a2 x2 + a4 x + a6 ①→❝ ✤à♥❤ tr➯♥ K ✈ỵ✐ ❝❤❛r(K) = 2, 3✳ ❑❤✐ ✤➜② t❛ ❝â t❤➸ t❤ü❝ ❤✐➺♥ ♣❤➨♣ ✤ê✐ ❜✐➳♥ ♥❤÷ s❛✉✿ ❚❛ ✈✐➳t ♣❤÷ì♥❣ tr➻♥❤ ✭✶✳✶✮ t❤➔♥❤ a1 x a3 + y+ 2 = x3 + a2 + a21 a1 a3 x + a4 + x+ a23 + a6 ❈❤÷ì♥❣ ✸✳ ▼ët sè ù♥❣ ❞ư♥❣ ❝õ❛ ✤÷í♥❣ ❝♦♥❣ ❞↕♥❣ ❝❤✉➞♥ ❊❞✇❛r❞s ①→❝ ✤à♥❤ ✤✐➲✉ ❦✐➺♥ ❝❤♦ t❤❛♠ sè ❤â❛ ❝õ❛ d t s❛✉✮✳ ❚ø ❤➺ t❤ù❝ t t❤❡♦ (x3 , y3 ) ❧➔ ✤✐➸♠ ❝➜♣ ❍ì♥ ♥ú❛✱ ♥➳✉ ✈ỵ✐ ❣✐↔ t❤✐➳t x3 = ±1 d = ✣✐➲✉ ♥➔② ❞➝♥ ✤➳♥ d = −(2y3 + 1)/x23 y32 t❛ ♥❤➟♥ ✤÷đ❝ ❧➔ d= ❱➻ ✹✾ (1 + t2 )3 (1 − 4t + t2 ) (1 − t)6 (1 + t)2 ♥➯♥ t❤❡♦ ▼➺♥❤ ✤➲ ✸✳✶ t❛ ❝â y3 + = 0✱ t❤➻ s✉② r❛ x3 = 0✳ ❙✉② r❛ t = ±1✳ d = −(2y3 + 1)/(x23 y32 ) = 1✱ tr♦♥❣ ✤à♥❤ ♥❣❤➽❛ ✤÷í♥❣ ❝♦♥❣ ❊❞✇❛r❞s✳ ❉♦ ✤â tr→✐ x3 = ±1✳ t = 0✳ ◆❣÷đ❝ ❧↕✐✱ ❣✐↔ sû t❛ ❝â d= ✈ỵ✐ t ∈ Q \ {0, ±1}✳ (t2 + 1)3 (t2 − 4t + 1) , (t − 1)6 (t + 1)2 ✣➦t t2 − x3 = , t +1 (t − 1)2 y3 = − t +1 ❑❤✐ ✤â (t − 1)4 (t − 1)2 ((t + 1)2 + (t − 1)2 ) (t2 − 1)2 + = (t2 + 1)2 (t2 + 1)2 (t2 + 1)2 2(t − 1)2 (t2 + 1) 2(t − 1)2 = = = −2y3 (t2 + 1) t +1 x23 + y32 = ❇➯♥ ❝↕♥❤ ✤â t❛ t❤➜② d= 2(t − 1)2 − (t2 + 1) (t2 + 1)2 (t2 + 1)2 −2y3 − · · = t2 + (t2 − 1)2 (t − 1)4 x23 y32 ▼➦t ❦❤→❝✱ ❞♦ d=0 ♥ú❛✱ ❣✐↔ t❤✐➳t t = 0, ±1 ◆❤÷ ✈➟② t❛ t❤➜② tr♦♥❣ Q ♥➯♥ tø ♣❤÷ì♥❣ tr➻♥❤ tr➯♥ s✉② r❛ ❞➝♥ ✤➳♥ x3 , y3 ✈➔ d ❦➳t ❧✉➟♥ ✤÷í♥❣ ❝♦♥❣ ❊❞✇❛r❞s y3 = −1/2✳ ❍ì♥ x3 = 0, y3 = −2, −1, tọ tt ỵ ✤â t❛ ❝â E ✤à♥❤ ♥❣❤➽❛ ❜ð✐ ♣❤÷ì♥❣ tr➻♥❤ E : x2 + y = + dx2 y ❝â ♠ët ✤✐➸♠ ❝➜♣ ❧➔ ✤✐➸♠ P = (x3 , y3 ) ✈➔ ♥❤â♠ ①♦➢♥ Et♦r (Q) ∼ = Z/12Z✳ ❇➙② ❣✐í✱ ❝❤ó♥❣ tỉ✐ ♠✉è♥ ①➙② ❞ü♥❣ ♠ët ữớ rs + dx2 y ợ d = 0, tr➯♥ Q ❝â ♥❤â♠ ①♦➢♥ ✤➥♥❣ ❝➜✉ ✈ỵ✐ Z/8Z✳ E : x2 +y = ●✐↔ sû t❛ ❝â ❈❤÷ì♥❣ ✸✳ ▼ët sè ù♥❣ ❞ư♥❣ ❝õ❛ ✤÷í♥❣ ❝♦♥❣ ❞↕♥❣ ❝❤✉➞♥ ❊❞✇❛r❞s ✤÷í♥❣ ❝♦♥❣ E ♥❤÷ ♠♦♥❣ ♠✉è♥✳ ❱➻ ❝❤➾ ❝â ❞✉② ♥❤➜t ✶ ✤✐➸♠ ❝➜♣ ✷ ♥➯♥ Z/8Z E ❝ô♥❣ ♣❤↔✐ ❝â t➼♥❤ ❝❤➜t ♥➔②✳ ❚❤❡♦ ▼➺♥❤ ✤➲ ✸✳✶✱ t❛ t❤➜② ❧➔ (0, −1)✱ (1, 0) ✈➔ d ✈➔ ✤✐➸♠ ♥➔② ❧➔ ❞✉② ♥❤➜t ❦❤✐ ✈➔ ữỡ ỡ ỳ ợ (−1, 0)✳ ❇➙② ❣✐í ❣✐↔ sû P d ✺✵ t❤➻ E E ❝â ✶ ✤✐➸♠ ❝➜♣ ✷ ❦❤æ♥❣ ♣❤↔✐ ❧➔ sè ❝❤➼♥❤ ❝❤➾ ❝â ✤ó♥❣ ❤❛✐ ✤✐➸♠ ❝➜♣ ✹ ❧➔ ✤✐➸♠ ❝➜♣ ✽ ❧➔ ❝õ❛ E✱ tø ▼➺♥❤ ✤➲ ✸✳✶✱ t❛ ❝â P = (α, ±α) ✈ỵ✐ α ∈ Q \ {0} t❤ä❛ ♠➣♥ ♣❤÷ì♥❣ tr➻♥❤ dx4 − 2x2 + = 0✳ ❱➻ P ✤✐➸♠ tr➯♥ ✤÷í♥❣ ❝♦♥❣ E tr➻♥❤ ✤÷í♥❣ ❝♦♥❣ s✉② r❛ ♥➯♥ t❛ ♣❤↔✐ ❝â d = 1✱ ❧➔ α = ±1✱ ✈➻ ♥➳✉ ♥❣÷đ❝ ❧↕✐ t tứ ữỡ tr ợ tt d = ◆❣♦➔✐ r❛✱ t❛ ❝ô♥❣ ❝â d = (2α2 −1)/α4 ✈➔ dα2 + α12 = 2✳ ❚❛ ✈✐➳t dx4 −2x2 +1 = dx4 −(dα2 + α12 )x2 +1 = (x−α)(x+α) = (x − α)(x + α)(dx2 − α12 )✳ ❉♦ d ❦❤ỉ♥❣ ♣❤↔✐ α2 ❧➔ sè ❝❤➼♥❤ ♣❤÷ì♥❣ ♥➯♥ dx − ❧➔ ❜➜t ❦❤↔ q✉② tr♦♥❣ Q✱ ❞♦ ✤â ♣❤÷ì♥❣ tr➻♥❤ α dx4 − 2x2 + = ❝❤➾ ❝â ❤❛✐ ♥❣❤✐➺♠ ❧➔ α ✈➔ −α✳ ❚❤❡♦ ▼➺♥❤ ✤➲ ✸✳✶✱ t❛ t❤➜② E dx2 (x − α)(x + α) − (α, α), (α, −α), (−α, −α) ✈➔ (−α, α)✳ Et♦r (Q) ∼ = Z/8Z t❤➻ d ♣❤↔✐ t❤ä❛ ♠➣♥ ✤✐➲✉ ❝❤➾ ❝â ✹ ✤✐➸♠ ❝➜♣ ✽ ❧➔ ◆❤÷ ✈➟②✱ ♥➳✉ ♣❤÷ì♥❣ tr♦♥❣ Q ✈➔ d = (2α2 − 1)/α4 ✈ỵ✐ ❦✐➺♥ ❦❤ỉ♥❣ ❝❤➼♥❤ α ∈ Q \ {0, ±1}✳ ◆❤ú♥❣ ❧➟♣ ❧✉➟♥ ð tr➯♥ ❧➔ ♣❤➛♥ ❝❤ù♥❣ ỵ s ỵ ✸✳✼✳ ✣÷í♥❣ ❝♦♥❣ ❊❞✇❛r❞s E : x ❝â ♥❤â♠ ①♦➢♥ Et♦r(Q) ✤➥♥❣ ❝➜✉ ✈ỵ✐ Z/8Z ♥➳✉ ✈➔ ❝❤➾ ♥➳✉ d ❦❤ỉ♥❣ ♣❤↔✐ ❧➔ sè ❝❤➼♥❤ ♣❤÷ì♥❣ tr♦♥❣ Q ✈➔ d = (2α2 − 1)/α4 ✈ỵ✐ α ∈ Q \ {0, ±1}✳ ❈❤ù♥❣ ♠✐♥❤✳ ❈❤ó♥❣ t❛ ❝❤➾ ❝➛♥ ❝❤ù♥❣ ♠✐♥❤ ✤✐➲✉ ❦✐➺♥ ✤õ✳ ●✐↔ sû t❛ ❝â d = (2α2 − 1)/α4 d = 0, 1✳ ✈ỵ✐ α ∈ Q \ {0, ±1} ✈➔ d + y = + dx2 y , d ∈ Q \ {0, 1} ❦❤ỉ♥❣ ♣❤↔✐ ❧➔ sè ❝❤➼♥❤ ♣❤÷ì♥❣✳ ❙✉② r❛ ❑❤✐ ✤â ✤÷í♥❣ ❝♦♥❣ ❊❞✇❛r❞s ①→❝ ✤à♥❤ tr➯♥ Q E : x2 + y = + dx2 y ❝â ❞✉② ♥❤➜t ♠ët ✤✐➸♠ ❝➜♣ ✷✱ ❤❛✐ ✤✐➸♠ ❝➜♣ ✹ t❤❡♦ ▼➺♥❤ ✤➲ ✸✳✶✳ ❍ì♥ ♥ú❛✱ t❤ä❛ ♠➣♥ ♣❤÷ì♥❣ tr➻♥❤ (α, ±α) dx4 − 2x2 + = ❧➔ ❝→❝ ✤✐➸♠ ❝â ❝➜♣ ✽ tr➯♥ ♣❤÷ì♥❣ tr➻♥❤ dx4 − 2x2 + = ✤â t➜t ❝↔ ♥❤ú♥❣ ✤✐➸♠ ❝➜♣ ✽ tr➯♥ E✳ ❱➻ d ♥➯♥✱ ❝ô♥❣ tø ▼➺♥❤ ✤➲ ✸✳✶✱ s✉② r❛ ❦❤æ♥❣ ♣❤↔✐ ❧➔ sè ❝❤➼♥❤ ♣❤÷ì♥❣ ♥➯♥ ❝❤➾ ❝â ❤❛✐ ♥❣❤✐➺♠ tr♦♥❣ E ❧➔ α Q (, ), (, ), (, ) ỵ ▼❛③✉r t❛ ♥❤➟♥ ✤÷đ❝ Et♦r (Q) ∼ = Z/8Z α ✈➔ −α✳ ❉♦ (−α, α)✳ ❚ø ✈➔ ❈❤÷ì♥❣ ✸✳ ▼ët sè ù♥❣ ❞ư♥❣ ❝õ❛ ✤÷í♥❣ ❝♦♥❣ ❞↕♥❣ ❝❤✉➞♥ ❊❞✇❛r❞s ✺✶ ỵ ữủ ự ỷ ữỡ ữ tr➯♥✱ t❛ ✤✐ ①➙② ❞ü♥❣ ❝→❝ ✤÷í♥❣ ❝♦♥❣ ❊❞✇❛r❞s tr➯♥ Q õ õ ợ Z/2ZìZ/8Z sỷ E : x2 +y = 1+dx2 y , d = 0, 1✱ ❧➔ ✤÷í♥❣ ❝♦♥❣ t❤ä❛ ♠➣♥ ②➯✉ ❝➛✉✳ ❉♦ tr♦♥❣ Z/2Z × Z/8Z ❝â t➜t ❝↔ ❜è♥ ♣❤➛♥ tû ❝➜♣ ✹✱ ✈➔ ♥❤➙♥ ✤æ✐ ❝õ❛ ❝→❝ ♣❤➛♥ tû ♥➔② ❧➔ ♥❤÷ ♥❤❛✉✱ ✈➟② ♥➯♥ ✤÷í♥❣ ❝♦♥❣ E ❝ô♥❣ ♣❤↔✐ ❝â ❜è♥ ✤✐➸♠ ❝â ❝➜♣ ✹ ♠➔ ✈✐➺❝ ♥❤➙♥ ✤ỉ✐ ♠é✐ ✤✐➸♠ ❝❤♦ t❛ ❝ị♥❣ ♠ët ❦➳t q✉↔✳ ❚ø ▼➺♥❤ ✤➲ ✸✳✶ t❛ t❤➜② ✤÷í♥❣ ❝♦♥❣ E ❧✉ỉ♥ ❝â ❤❛✐ ✤✐➸♠ ❝➜♣ ✹ ❧➔ (1, 0), (−1, 0) ✈➔ ♥❤➙♥ ✤ỉ✐ ❝õ❛ ❝❤ó♥❣ ❜➡♥❣ (0, −1)✱ ❞♦ ✤â ❤❛✐ ✤✐➸♠ ❝➜♣ ✹ ❝á♥ ❧↕✐ (0, −1)✳ ❱✐➺❝ t➼♥❤ t♦→♥ ❝→❝ ✤✐➸♠ ❝➜♣ ✹ tr♦♥❣ ▼➺♥❤ √ √ d), (1 : 0)) ✈➔ ((1 : − d), (1 : 0))✳ ♥➔② ♣❤↔✐ ❧➔ ((1 : ❝ơ♥❣ ♣❤↔✐ ♥❤➙♥ ✤ỉ✐ ❧➯♥ ❜➡♥❣ ✤➲ ✸✳✶ ✤➣ ❝❤➾ r❛ ❤❛✐ ✤✐➸♠ ✣✐➲✉ tữỡ ữỡ ợ sỷ t õ d ♣❤↔✐ ❧➔ sè ❝❤➼♥❤ ♣❤÷ì♥❣ tr♦♥❣ P = ((x8 : 1), (y8 : 1)) x8 , y8 = ✈➔ 2P ❧➔ ♠ët ✤✐➸♠ ❝➜♣ ✽ tr➯♥ ♣❤↔✐ ❜➡♥❣ E✳ Q✳ ❑❤✐ ✤â tø ▼➺♥❤ ✤➲ √ ((±1 : 1), (0 : 1)) ❤♦➦❝ ((1 : ± d), (1 : 0)) t tứ trữớ ủ t ữợ ✤➙②✳ ❚r÷í♥❣ ❤đ♣ P = ((x : 1), (y8 : 1)) ♥❤➙♥ ✤æ✐ t❤➔♥❤ ((±1 : 1), (0 : 1))✿ ✤â✱ ❝ô♥❣ tø ✈✐➺❝ t➼♥❤ t♦→♥ ✤✐➸♠ ❝➜♣ ✽ tr♦♥❣ ▼➺♥❤ ✤➲ ✸✳✶✱ t❛ ❝â ♣❤÷ì♥❣ tr➻♥❤ dx4 − 2x2 + = ❧↕✐ t❤➻ ♣❤÷ì♥❣ tr➻♥❤ d=1 ✈➔ y8 = ±x8 ✳ dx48 − 2x28 + = ❚❛ s✉② r❛ ❦➨♦ t❤❡♦ d = 1✱ x8 = ±1 d = (2x28 − 1)/x48 ✱ = (dx48 + 1)/x28 = dx28 + 1/x28 t❤ä❛ ♠➣♥ ữủ t ợ tt tr ♥❣❤➽❛ ✤÷í♥❣ ❝♦♥❣ ❊❞✇❛r❞s✳ P❤÷ì♥❣ tr➻♥❤ ❝❤♦ t❛ x8 ❑❤✐ dx48 − 2x28 + = ✈➔ ❤ì♥ ♥ú❛ t❛ ✈✐➳t ✤÷đ❝ )x + x28 = (dx4 − dx28 x2 ) − ( x2 − 1) x8 = (x2 − x28 )(dx2 − ) x8 √ √ = (x − x8 )(x + x8 )( dx − )( dx + ), x8 x8 dx4 − 2x2 + = dx4 − (dx28 + ❞♦ d ❜➡♥❣ ❧➔ sè ❝❤➼♥❤ ♣❤÷ì♥❣✳ ❚ø ✤➙② t❛ ♥❤➟♥ ✤÷đ❝ ❝→❝ ✤✐➸♠ ❝➜♣ ✽ ♠➔ ♥❤➙♥ ✤æ✐ ((±1 : 1), (0 : 1)) ❧➔ ((±x8 : 1), (±x8 : 1)) ❈→❝ ✤✐➸♠ ♥➔② ❧➔ ❦❤→❝ ♥❤❛✉ tø♥❣ ✤æ✐ ♠ët ✈➻ s✉② r❛ dx48 = ❦➨♦ t❤❡♦ 2x28 = 2✱ √ √ ((1 : ±x8 d), (1 : ±x8 d))✳ √ ♥➳✉ ♥❣÷đ❝ ❧↕✐✱ t❛ ❝â ±x8 =1✱ ✈➔ ✈➔ tø ♣❤÷ì♥❣ tr➻♥❤ ❝õ❛ ✤÷í♥❣ ❝♦♥❣ ❞➝♥ ✤➳♥ ❈❤÷ì♥❣ ✸✳ ▼ët sè ù♥❣ ❞ư♥❣ ❝õ❛ ✤÷í♥❣ ❝♦♥❣ ❞↕♥❣ ❝❤✉➞♥ ❊❞✇❛r❞s y8 = 0✱ ♠➙✉ t❤✉➝♥ ợ tt rữớ ủ 0)) y8 = P = ((x8 : 1), (y8 : 1)) ♥❤➙♥ ✤æ✐ t❤➔♥❤ ✺✷ √ ((1 : ± d), (1 : ❑❤✐ ✤â✱ ❝ô♥❣ tø ✈✐➺❝ t➼♥❤ t♦→♥ ✤✐➸♠ ❝➜♣ ✽ tr♦♥❣ ▼➺♥❤ ✤➲ ✸✳✶✱ t❛ ❝â √ y8 = ±1/( dx8 )✱ tù❝ ❧➔ √ √ P = ((x8 : 1), (±1/( dx8 ) : 1)) = ((x8 : 1), (1 : ± dx8 ))✳ ❚÷ì♥❣ tü ♥❤÷ x8 t❤ä❛ ♠➣♥ ♣❤÷ì♥❣ tr➻♥❤ tr➯♥ t❛ ❝ơ♥❣ ❝❤➾ r❛ ✤÷đ❝ dx4 − 2dx2 + = x8 = 0, ±1 2d = (dx48 + 1)/x28 = dx28 + 1/x28 ✈➔ ✈➔ d = 1/(2x28 − 1x48 ) = 1/(x28 (2 − x28 ))✱ ✈➔ ❤ì♥ ♥ú❛ t❛ ✈✐➳t ✤÷đ❝ )x + x28 = (dx4 − dx28 x2 ) − ( x2 − 1) x8 = (x2 − x28 )(dx2 − ) x8 √ √ = (x − x8 )(x + x8 )( dx − )( dx + ), x8 x8 dx4 − 2dx2 + = dx4 − (dx28 + d ❧➔ sè ❝❤➼♥❤ ♣❤÷ì♥❣✳ ❚ø ✤➙② t❛ ♥❤➟♥ ✤÷đ❝ ❝→❝ ✤✐➸♠ ❝➜♣ ✽ ♠➔ ♥❤➙♥ ✤æ✐ ❜➡♥❣ √ √ √ ((1 : ± d), (1 : 0)) ❧➔ ((±x8 : 1), (1 : ± dx8 )) ✈➔ ((1 : ± dx8 ), (±x8 : 1))✳ ❈→❝ √ ✤✐➸♠ ♥➔② ❧➔ ❦❤→❝ ♥❤❛✉ tø♥❣ ✤ỉ✐ ♠ët ✈➻ ♥➳✉ ♥❣÷đ❝ ❧↕✐✱ t❛ ❝â ± dx8 = 1✱ s✉② ❞♦ r❛ dx48 = ❦➨♦ t❤❡♦ ❝♦♥❣ ❞➝♥ ✤➳♥ x8 = 0✱ 2dx28 = 2✱ tù❝ ❧➔ dx28 = ♠➙✉ t❤✉➝♥ ✈ỵ✐ ❣✐↔ t❤✐➳t ✈➔ tø ♣❤÷ì♥❣ tr➻♥❤ ❝õ❛ ✤÷í♥❣ x8 = 0✳ õ ỵ s ỵ ✤÷í♥❣ ❝♦♥❣ ❊❞✇❛r❞s E : x2 + y2 = + dx2 y , d = 0, ✤à♥❤ ♥❣❤➽❛ tr➯♥ Q ♠➔ ❝â ♥❤â♠ ①♦➢♥ Et♦r (Q) ✤➥♥❣ ợ Z/2Z ì Z/8Z s tở ởt tr trữớ ủ s ỵ ữớ ❝♦♥❣ E ❝â ♥❤â♠ ①♦➢♥ Et♦r(Q) ✤➥♥❣ ❝➜✉ ✈ỵ✐ Z/2Z × Z/8Z ✈➔ ❝â ❝→❝ ✤✐➸♠ ❝➜♣ ❧➔ (x8, ±x8) ♥❤➙♥ ✤æ✐ t❤➔♥❤ ((±1 : 1), (0 : 1)) ❦❤✐ ✈➔ ❝❤➾ ❦❤✐ d = (2x28 − 1)/x48 ❧➔ ởt số ữỡ tr Q ợ x8 Q \ {0, ±1}✳ ✷✳ ✣÷í♥❣ ❝♦♥❣ E ❝â ♥❤â♠ ①♦➢♥ Etr(Q) ợ Z/2Z ì Z/8Z õ √ ❝→❝ ✤✐➸♠ ❝➜♣ ❧➔ (x8, ±1/(x8 d)) ♥❤➙♥ ✤æ✐ t❤➔♥❤ ((1 : ± d), (1 : 0)) ❦❤✐ ✈➔ ❝❤➾ ❦❤✐ d = 1/(x28(2 − x28)) ❧➔ ♠ët số ữỡ tr Q ợ x8 Q \ {0, ±1}✳ ❈❤÷ì♥❣ ✸✳ ▼ët sè ù♥❣ ❞ư♥❣ ❝õ❛ ✤÷í♥❣ ❝♦♥❣ ❞↕♥❣ ❝❤✉➞♥ ❊❞✇❛r❞s ✺✸ ❈❤ù♥❣ ♠✐♥❤✳ ✶✳ ✣✐➲✉ ❦✐➺♥ sỷ E õ Etr(Q) ợ Z/2ZìZ/8Z ✈➔ P = ((x8 : 1), (y8 : 1)) ❧➔ ♠ët ✤✐➸♠ ❝➜♣ ✽ t❤ä❛ ♠➣♥ 2P = ((±1 : 1), (0 : 1))✳ ▲➟♣ ❧✉➟♥ ❤♦➔♥ t♦➔♥ t÷ì♥❣ tü ♥❤÷ ✤➣ tr➻♥❤ ❜➔② ð tr➯♥✱ t❛ t❤✉ ✤÷đ❝ ✤✐➲✉ ỵ ữỡ õ ●✐↔ sû x8 ∈ Q \ {0, ±1} d = 0, ✈➔ ✈➔ d = (2x28 − 1)/x48 E : x2 + y = + dx2 y ❧➔ ♠ët sè ❝❤➼♥❤ ❧➔ ♠ët ✤÷í♥❣ ❝♦♥❣ ❊❞✇❛r❞s ✤à♥❤ ♥❣❤➽❛ tr➯♥ Q✳ ❑❤✐ ✤â✱ tø t➼♥❤ t♦→♥ ð ▼➺♥❤ ✤➲ ✸✳✶✱ t❛ t❤➜② ✤÷í♥❣ √ ❝♦♥❣ E ❝❤➾ ❝â ❝→❝ ✤✐➸♠ ❝➜♣ ✷ ❧➔ ((0 : 1), (−1 : 1)), ((1 : 0), (1 : ± d))✱ ❝→❝ √ ✤✐➸♠ ❝➜♣ ✹ ❧➔ ((±1 : 1), (0 : 1)), ((1 : ± d), (1 : 0))✱ ✈➔ ❝→❝ ✤✐➸♠ ❝➜♣ ✽ ❧➔ √ √ ((±x8 : 1), (±x8 : 1)), ((1 : ±x8 d), (1 : ±x8 d))✱ ❤ì♥ ♥ú❛ ❝→❝ ✤✐➸♠ ❝➜♣ ✽ ♥❤➙♥ ✤ỉ✐ ❜➡♥❣ ((±1 : 1), (0 : 1)) õ tứ ỵ ▼❛③✉r t❛ s✉② r❛ Et♦r (Q) ∼ = Z/2Z × Z/8Z ự tữỡ tỹ ữ ố ợ ❚❛ s➩ ✤✐ ❝❤ù♥❣ ♠✐♥❤ ❤❛✐ ❤å ✤÷í♥❣ ❝♦♥❣ tr♦♥❣ ỵ tữỡ ữỡ s ỳ t ợ rữợ t sỷ ởt số ữỡ tr♦♥❣ Q✳ ❑❤✐ ✤â ✤÷í♥❣ ❝♦♥❣ ❧➔ ✤÷í♥❣ ❝♦♥❣ t❤✉ë❝ ❤å t❤ù ❤❛✐✳ ✣➦t ❞♦ ✤â x8 ∈ Q \ {0, ±1} ✈➔ d = 1/(x28 (2 − x28 )) √ x8 = x8 d✳ ❚❛ ❝â x82 = x28 d = 1/(2 − x28 ) (2x82 − 1)/x84 = (2/(2 − x28 ) − 1)(2 − x28 )2 = x28 (2 − x28 ) = 1/d✱ ❝ô♥❣ ❧➔ ♠ët sè ❝❤➼♥❤ ♣❤÷ì♥❣✳ ▼➦t ❦❤→❝✱ s✉② r❛ E : x2 + y = + dx2 y x8 = 0, ±1✳ ❉♦ ✈➟② ✤÷í♥❣ ❝♦♥❣ x8 ∈ Q \ {0, ±1} ♥➯♥ x2 + y = + (1/d)x2 y ❣✐→ trà ♥➔② − x28 = 0, 1✱ t❤✉ë❝ ❤å ✤÷í♥❣ ❝♦♥❣ t❤ù ♥❤➜t✳ ◗✉❛♥ ❤➺ t÷ì♥❣ ✤÷ì♥❣ s♦♥❣ ❤ú✉ t➾ ❝õ❛ ❤❛✐ ❤å ✤÷í♥❣ ❝♦♥❣ ♥➔② ✤÷đ❝ ❝❤ù♥❣ ♠✐♥❤ q✉❛ ❜ê ✤➲ s❛✉✳ ❇ê ✤➲ ✸✳✾✳ ❈❤♦ d ❧➔ ♠ët sè ❝❤➼♥❤ ♣❤÷ì♥❣✳ ❍❛✐ ✤÷í♥❣ ❝♦♥❣ ❊❞✇❛r❞s x√+ y 2 = ✈➔ u2 + v2 = + (1/d)u2v2 q✉❛ →♥❤ ①↕ (x, y) → (u, v) = (x d, 1/y) √ ✈ỵ✐ →♥❤ ①↕ ♥❣÷đ❝ (u, v) → (x, y) = (u/ d, 1/v) ❧➔ t÷ì♥❣ ✤÷ì♥❣ s♦♥❣ ❤ú✉ t➾✳ ⑩♥❤ ①↕ ♥➔② ❜↔♦ t♦➔♥ ❝→❝ ✤✐➸♠ (0, ±1)✳ + dx2 y √ ❈❤ù♥❣ ♠✐♥❤✳ ❚❤❛② u = x t❛ ✤÷đ❝ d, v = 1/y u2 /d + 1/v = + u2 /v ✳ u2 + v = + (1/d)u2 v ✳ ⑩♥❤ ①↕ ✈➔♦ ♣❤÷ì♥❣ tr➻♥❤ x2 + y = + dx2 y ữỡ tr ợ ✈ỵ✐ √ (x, y) → (u, v) = (x d, 1/y) v2 ❝❤♦ t❛ ❝❤➾ ❝â ❤ú✉ ❤↕♥ ❈❤÷ì♥❣ ✸✳ ▼ët sè ù♥❣ ❞ư♥❣ ❝õ❛ ✤÷í♥❣ ❝♦♥❣ ❞↕♥❣ ❝❤✉➞♥ ❊❞✇❛r❞s ✤✐➸♠ ❝→ ❜✐➺t ❧➔ (±1, 0)✳ ✺✹ (0, ±1)✳ ❉➵ t❤➜② →♥❤ ①↕ ♥➔② ❜↔♦ t♦➔♥ ❝→❝ ✤✐➸♠ ❈❤✐➲✉ ♥❣÷đ❝ ❧↕✐ ✤÷đ❝ ❝❤ù♥❣ ♠✐♥❤ t÷ì♥❣ tü✳ ❉♦ ❤❛✐ ❤å ✤÷í♥❣ tr ỵ tữỡ ữỡ s ỳ t➾ q✉❛ ❇ê ✤➲ ✸✳✾ ♥➯♥ t❛ ❝â t❤➸ ❤↕♥ ❝❤➳ ✈✐➺❝ ❦❤↔♦ s→t ❤❛✐ ❤å ✤÷í♥❣ ❝♦♥❣ ♥➔② ❝❤➾ ợ ữớ tự t tự ữớ rs tr ợ Z/2Z ì Z/8Z d t t số õ số ỵ Q ✈➔ ❝→❝ ✤✐➸♠ ❝➜♣ ✽ ♥❤➙♥ ✤æ✐ t❤➔♥❤ ❝â ♥❤â♠ ①♦➢♥ (±1, 0)✳ ❇➙② ❣✐í ❝õ❛ ❤å ✤÷í♥❣ ❝♦♥❣ ♥➔②✳ ✣÷í♥❣ ❝♦♥❣ ❊❞✇❛r❞s E : x2 + y2 = + dx2y2 tr➯♥ Q ❝â ♥❤â♠ ①♦➢♥ Et♦r(Q) ✤➥♥❣ ợ Z/2Z ì Z/8Z ♥❤➙♥ ✤æ✐ ❜➡♥❣ (±1, 0) ❦❤✐ ✈➔ ❝❤➾ ❦❤✐ d= ỵ (t2 2)2 (t2 + 4t + 2)2 , (t2 + 2t + 2)4 ✈ỵ✐ t ∈ Q \ {−2, −1, 0} ❈❤ù♥❣ ♠✐♥❤✳ ✣✐➲✉ ❦✐➺♥ ❝➛♥✿ ●✐↔ sû ✤÷í♥❣ ❝♦♥❣ ❊❞✇❛r❞s E ❝â Et♦r(Q) ∼ = Z/2Z×Z/8Z ✈➔ ❝→❝ ✤✐➸♠ ❝➜♣ ✽ ♥❤➙♥ ✤ỉ✐ ❜➡♥❣ (1, 0) õ t ỵ t õ d = (2x28 − 1)/x48 tù❝ ❧➔ 2x28 − = r2 r ởt số ữỡ tr ợ 2u2 − v = 1✳ r ∈ Q ❘ã r➔♥❣ ♥➔♦ ✤â✳ ✣➸ t❤❛♠ sè ❤â❛ (1, −1) ✈➔ (x8 , r) t = (r + 1)/(x8 − 1)✳ ❑❤✐ ✤â✱ r = t(x8 − 1) − 1✳ ♣❤÷ì♥❣ tr➻♥❤ ❤②♣❡r❜♦❧ t❛ t❤✉ ✤÷đ❝ ♥➔② ❞➝♥ ✤➳♥ x8 = ±1 x8 ✱ t❛ ①➨t ✤÷í♥❣ (1, −1) ❚❤❛② ✈➔ (x8 , r) ❧➔ t✱ (x8 , t(x8 − 1) − 1) 2x28 − = (t(x8 − 1) − 1)2 ✳ x8 (t2 − 2) = + 2t + t2 ✳ 2(x8 + 1) = t(t(x8 − 1) − 2)✱ d = (2x28 − 1)/x48 ❉♦ ❤❛② t÷ì♥❣ ữỡ ợ (t2 + 2t + 2) (t2 2) ♥➯♥ ❞➝♥ ✤➳♥ t = −2, −1, 0✳ ❚❤❛② x8 ✈➔♦ ♣❤÷ì♥❣ tr➻♥❤ t❛ ❝â d= ✣✐➲✉ ❦✐➺♥ ✤õ✿ ✈➔♦ ❚ø ✤➙② s✉② r❛ x8 = x8 = 0, ±1 t❛ P❤÷ì♥❣ tr➻♥❤ 2(x28 − 1) = (t(x8 − 1) − 1)2 − = t(x8 − 1)(t(x8 − 1) − 2)✳ ♥➯♥ t❛ ♥❤➟♥ ✤÷đ❝ ❱➻ ✤✐➲✉ ❦✐➺♥ x8 ∈ Q \ {0, ±1}❀ ❧➔ ❝→❝ ✤✐➸♠ t❤✉ë❝ ✤÷í♥❣ ❤②♣❡r❜♦❧ ♥➔②✳ ●å✐ ❤➺ sè ❣â❝ ❝õ❛ ✤÷í♥❣ t❤➥♥❣ ✤✐ q✉❛ ❝â ✈ỵ✐ Q ●✐↔ sû d (t2 − 2)2 (t2 + 4t + 2)2 (t2 + 2t + 2)4 ữủ ữ ổ tự tr ợ t ∈ Q\ ❈❤÷ì♥❣ ✸✳ ▼ët sè ù♥❣ ❞ư♥❣ ❝õ❛ ✤÷í♥❣ ❝♦♥❣ ❞↕♥❣ ❝❤✉➞♥ ❊❞✇❛r❞s {−2, −1, 0}✳ d = 0, ❑❤✐ ✤â ✈➔ ❧➔ ♠ët sè ❝❤➼♥❤ ♣❤÷ì♥❣ tr♦♥❣ x8 = ❚❛ ❝â x8 = t = 0, −1✳ ✈➻ t2 + 2t + = ✺✺ Q✳ ✣➦t t2 + 2t + t2 − ✈ỵ✐ ♠å✐ t ∈ Q✱ x8 = ✈➻ t = −2✱ ✈➔ x8 = −1 ✈➻ ❚❛ t tt ỵ ữủ tọ ♠➣♥✱ ❞♦ ✈➟② t❛ ❝â ✤✐➲✉ ♣❤↔✐ ❝❤ù♥❣ ♠✐♥❤✳ ✣à♥❤ ỵ ữớ rs E : x tr➯♥ Q ❝â ♥❤â♠ ①♦➢♥ Et♦r(Q) ✤➥♥❣ ❝➜✉ ✈ỵ✐ Z/2Z ì Z/4Z tỗ t s Q t❤ä❛ ♠➣♥ d = s2 ✈➔ ♣❤÷ì♥❣ tr➻♥❤ (dx4 − 2x2 + 1)(dx4 − 2dx2 + 1) = ❦❤æ♥❣ ❝â ♥❣❤✐➺♠ tr♦♥❣ Q✳ + y = + dx2 y ❈❤ù♥❣ ♠✐♥❤✳ ✣✐➲✉ ❦✐➺♥ ❝➛♥✿ ●✐↔ sû ✤÷í♥❣ ❝♦♥❣ ❊❞✇❛r❞s E ❝â Et♦r(Q) ∼ = Z/2Z × Z/4Z✳ ❑❤✐ ✤â tr➯♥ E ❝â ✤ó♥❣ ❜❛ ✤✐➸♠ ❝➜♣ ✷✱ ❜è♥ ✤✐➸♠ ❝➜♣ ✹ ✈➔ ❦❤æ♥❣ ❝â ✤✐➸♠ ❝➜♣ ✽✳ ❚❤❡♦ ▼➺♥❤ ✤➲ ✸✳✶✱ ✤✐➲✉ ♥➔② ❝❤➾ ①↔② r❛ ❦❤✐ ✈➔ ❝❤➾ ❦❤✐ ❝❤➼♥❤ ♣❤÷ì♥❣ tr♦♥❣ Q✱ tù❝ ❧➔ d = s2 s ∈ Q✳ ✈ỵ✐ ❇➯♥ ❝↕♥❤ ✤â✱ ❞♦ ✤✐➸♠ ❝➜♣ ✽ ♥➯♥ tø ▼➺♥❤ ✤➲ ✸✳✶✱ s✉② r❛ ❝→❝ ♣❤÷ì♥❣ tr➻♥❤ dx4 − 2dx2 + = ✣✐➲✉ ❦✐➺♥ ✤õ✿ E d ❧➔ sè ❦❤æ♥❣ ❝â dx4 − 2x2 + = ✈➔ ❧➔ ✈æ ♥❣❤✐➺♠✳ ●✐↔ sû d t❤ä❛ ♠➣♥ ❝→❝ ✤✐➲✉ ❦✐➺♥ ✤➣ ♥➯✉✳ ❑❤✐ ✤â✱ tø t➼♥❤ t♦→♥ tr♦♥❣ ▼➺♥❤ ỵ r t õ ❝❤ù♥❣ ♠✐♥❤✳ ❚ê♥❣ ❤ñ♣ t➜t ❝↔ ❦➳t q✉↔ ✤➣ tr➻♥❤ ❜➔② ð tr➯♥✱ t❛ ❝â✿ ❍➺ q✉↔ ✸✳✶✷✳ ●✐↔ sû d ∈ Q \ {0, 1}✳ ❑❤✐ ✤â ✤÷í♥❣ ❝♦♥❣ ❡❧❧✐♣t✐❝ ①→❝ ✤à♥❤ tr➯♥ Q E : Y = X3 + ❝â ♥❤â♠ ①♦➢♥ Et♦r(Q) ✤➥♥❣ ❝➜✉ ✈ỵ✐ Z/12Z, Z/2Z × Z/8Z, Z/2Z × Z/4Z, Z/8Z, Z/4Z, + d (1 − d)2 X + X 16 ) ♥➳✉ d = (1+t(1−t)) (1−4t+t ✈ỵ✐ t ∈ Q \ {0, ±1}; (1+t) (t +4t+2) ♥➳✉ d = (t −2) ✈ỵ✐ t ∈ Q \ {−2, −1, 0}; (t +2t+2) ♥➳✉ d ∈ Q2 ✈➔ (dx4 − 2x2 + 1)(dx4 − 2dx2 + 1) = 0, ∀x ∈ Q; ♥➳✉ d ∈/ Q2 ✈➔ d = 2tt −1 ✈ỵ✐ t ∈ Q \ {0, ±1}; tr♦♥❣ ❝→❝ tr÷í♥❣ ❤đ♣ ❝á♥ ❧↕✐✱ 2 ð ✤➙② ỵ Q2 = {a2 | a Q} 2 4 ❈❤÷ì♥❣ ✸✳ ▼ët sè ù♥❣ ❞ư♥❣ ❝õ❛ ✤÷í♥❣ ❝♦♥❣ ❞↕♥❣ ❝❤✉➞♥ ❊❞✇❛r❞s ✺✻ ❈❤ù♥❣ ♠✐♥❤✳ ❉♦ d = 0, ♥➯♥ t❛ ❝â ✤÷í♥❣ ❝♦♥❣ ❊❞✇❛r❞s ✤à♥❤ ♥❣❤➽❛ tr➯♥ Q ❧➔ E : x2 + y = + dx2 y Et♦r (Q) ❚ø ❍➺ q✉↔ ✷✳✷✵ t❛ ❝â ✤➥♥❣ ❝➜✉ ✈ỵ✐ ♥❤â♠ ①♦➢♥ ❝õ❛ ✤÷í♥❣ ❝♦♥❣ ▼♦♥t✲ ❣♦♠❡r② t÷ì♥❣ ù♥❣ ❧➔ EM : 2(1 + d) v = u3 + u + u 1−d 1−d 1−d ❈❤✐❛ ❝↔ ❤❛✐ ✈➳ ♣❤÷ì♥❣ tr➻♥❤ ✤÷í♥❣ ❝♦♥❣ tr➯♥ ❝❤♦ t ữủ ữỡ tr ợ 1d v2 = 1−d ❚❤ü❝ ❤✐➺♥ ♣❤➨♣ ✤ê✐ ❜✐➳♥ u3 + 1+d 1−d u2 + (1 − d)2 − d u 16 1−d u, 1−d v 4 (u, v) → (X, Y ) = t❛ ♥❤➟♥ ✤÷đ❝ ✤÷í♥❣ ❝♦♥❣ + d (1 − d)2 E :Y =X + X + X 16 Et♦r (Q) ∼ = Et♦r (Q)✳ ❑❤✐ ✤â✱ tø ỵ ứ õ s r ✸✳✼✱ ✸✳✽✱ ✸✳✶✵✱ ✸✳✶✶ t❛ ♥❤➟♥ ✤÷đ❝ ✤✐➲✉ ♣❤↔✐ ❝❤ù♥❣ ♠✐♥❤✳ ❚❛ ①➨t ♠ët ✈➔✐ ✈➼ ❞ö ❝ö t❤➸ s❛✉✿ ❱➼ ❞ư ✸✳✶✸✳ ❈❤♦ ✤÷í♥❣ ❝♦♥❣ ❡❧❧✐♣t✐❝ tr➯♥ E : Y = X3 + ✈ỵ✐ d= (1+t2 )3 (1−4t+t2 ) ✱ (1−t)6 (1+t)2 Q + d (1 − d)2 X + , 16 t ∈ Q \ {0, ±1}✳ E : Y = X3 − ▲➜② t = 2✱ s✉② r❛ d = − 125 ✱ ✈➔ 61 1024 X + X ❚❛ sû ❞ö♥❣ ♣❤➛♥ ♠➲♠ ❙❛❣❡ ❬✶✻❪ ✤➸ ❦✐➸♠ tr❛ ❧↕✐ ❦➳t q✉↔ ✈ỵ✐ ❝→❝ ❧➺♥❤ ❝ư t❤➸ ❊❂❊❧❧✐♣t✐❝❈✉r✈❡✭◗◗✱❬✵✱✲✻✶✴✸✱✵✱✶✵✷✹✴✾✱✵❪✮❀ ❊ ❊✳t♦rs✐♦♥− s✉❜❣r♦✉♣✭✮ ❊✳t♦rs✐♦♥− ♣♦✐♥ts✭✮ ✈➔ ♥❤➟♥ ✤÷đ❝ Et♦r (Q) ∼ = Z/12Z, ❈❤÷ì♥❣ ✸✳ ▼ët sè ù♥❣ ❞ư♥❣ ❝õ❛ ✤÷í♥❣ ❝♦♥❣ ❞↕♥❣ ❝❤✉➞♥ ❊❞✇❛r❞s ✈➔ ❝→❝ ✤✐➸♠ ①♦➢♥ ❝õ❛ E ❧➔ ✭✤✐➸♠ ✤÷đ❝ ✈✐➳t ð ❞↕♥❣ ①↕ ↔♥❤✮✿ ✺✼ (0 : : 1), (0 : : 0), (8/3 : −40/3 : 1), (8/3 : 40/3 : 1), (64/9 : −320/27 : 1), (64/9 : 320/27 : 1), (32/3 : −32/3 : 1), (32/3 : 32/3 : 1), (16 : −80/3 : 1), (16 : 80/3 : 1), (128/3 : −640/3 : 1), (128/3 : 640/3 : 1)✳ ❱➼ ❞ư ✸✳✶✹✳ s✉② r❛ d= ❚r÷í♥❣ ❤đ♣ d= (t2 −2)2 (t2 +4t+2)2 ✈ỵ✐ (t2 +2t+2)4 25921 ✳ ❑❤✐ ✤â t❛ ❝â ✤÷í♥❣ ❝♦♥❣ tr➯♥ 83521 E : Y = X3 + t ∈ Q \ {−2, −1, 0}✳ ▲➜② t = 3✱ Q 54721 207360000 X + X 83521 6975757441 ❙û ❞ö♥❣ ♣❤➛♥ ♠➲♠ ❙❛❣❡ t❛ t➼♥❤ ữủ ữủ Etr (Q) = Z/2Z ì Z/8Z ❝→❝ ✤✐➸♠ ①♦➢♥ ❧➔ (−50625/83521 : : 1)✱ (−34560/83521 : −241920/1419857 : 1)✱ (−34560/83521 : 241920/1419857 : 1)✱ (−14400/83521 : −2318400/24137569 : 1)✱ (−14400/83521 : 2318400/24137569 : 1)✱ (−6000/83521 : −42000/1419857 : 1)✱ (−6000/83521 : 42000/1419857 : 1)✱ (−4096/83521 : : 1)✱ (0 : : 1)✱ (0 : : 0)✱ (2160/83521 : −49680/1419857 : 1)✱ (2160/83521 : 49680/1419857 : 1)✱ (14400/83521 : −14400/83521 : 1)✱ (14400/83521 : 14400/83521 : 1)✱ (96000/83521 : −2208000/1419857 : 1)✱ (96000/83521 : 2208000/1419857 : 1)✳ ❱➼ ❞ö ✸✳✶✺✳ ▲➜② d = 9✱ ❚r÷í♥❣ ❤đ♣ d ∈ Q2 ✈➔ (dx4 −2x2 +1)(dx4 −2dx2 +1) = 0, ∀x ∈ Q✳ t❛ ❝â ✤÷í♥❣ ❝♦♥❣ E : Y = X + 5X + 4X ❑❤✐ ✤â t❛ t➼♥❤ ✤÷đ❝ Et♦r (Q) ∼ = Z/2Z × Z/4Z, ✈➔ ❝→❝ ✤✐➸♠ ①♦➢♥ ❝õ❛ ✤÷í♥❣ ❝♦♥❣ ❧➔ (−4 : : 1), (−2 : −2 : 1), (−2 : : 1), (−1 : : 1), (0 : : 1), (0 : : 0), (2 : −6 : 1), (2 : : 1)✳ ❱➼ ❞ư ✸✳✶✻✳ s✉② r❛ d= ❚r÷í♥❣ ❤đ♣ d∈ / Q2 ✈➔ d= 2t2 −1 ✈ỵ✐ t4 t ∈ Q \ {0, ±1}✳ 17 ✱ ✈➔ ✤÷í♥❣ ❝♦♥❣ 81 E : Y = X3 + 49 256 X + 81 6561 ▲➜② t = 3✱ ❈❤÷ì♥❣ ✸✳ ▼ët sè ù♥❣ ❞ư♥❣ ❝õ❛ ✤÷í♥❣ ❝♦♥❣ ❞↕♥❣ ❝❤✉➞♥ ❊❞✇❛r❞s ✺✽ ❑❤✐ ✤â t❛ t➼♥❤ ✤÷đ❝ Et♦r (Q) ∼ = Z/8Z, ✈➔ ❝→❝ ✤✐➸♠ ①♦➢♥ ❧➔ (−32/81 : −32/243 : 1), (−32/81 : 32/243 : 1), (−8/81 : −8/243 : 1), (−8/81 : 8/243 : 1), (0 : : 1), (0 : : 0), (16/81 : −16/81 : 1), (16/81 : 16/81 : 1)✳ ❱➼ ❞ư ✸✳✶✼✳ ❚r÷í♥❣ ❤đ♣ ❝á♥ ❧↕✐✳ ▲➜② d = 3✱ t❛ ❝â d ❦❤ỉ♥❣ t❤✉ë❝ tr÷í♥❣ ❤đ♣ ♥➔♦ ð tr➯♥ ✈➔ ❦❤✐ ✤â ✤÷í♥❣ ❝♦♥❣ ❧➔ E : Y = X + 2X + X ❙û ❞ư♥❣ ♣❤➛♥ ♠➲♠ ❙❛❣❡ t❛ t➼♥❤ ✤÷đ❝ Et♦r (Q) ∼ = Z/4Z, ✈➔ ❝→❝ ✤✐➸♠ ①♦➢♥ ❧➔ (−1/2 : −1/2 : 1), (−1/2 : 1/2 : 1), (0 : : 1), (0 : : 0)✳ ✸✳✸ Ù♥❣ ❞ư♥❣ ❝õ❛ ✤÷í♥❣ ❝♦♥❣ ❊❞✇❛r❞s tr♦♥❣ ♠➟t ♠➣ ❚r♦♥❣ ♥❤ú♥❣ ởt ợ t ổ ợ ữủ t❤→❝ ✤➸ ❦❤ỉ✐ ♣❤ư❝ t❤ỉ♥❣ t✐♥ ❜➼ ♠➟t ✤÷đ❝ ♥❤ó♥❣ tr♦♥❣ ♠ët t❤✐➳t ❜à ♠➟t ♠➣✱ ❣å✐ ❧➔ t➜♥ ❝æ♥❣ ❦➯♥❤ ❦➲✳ ❇➡♥❣ ❝→❝❤ ❣✐→♠ s→t t❤æ♥❣ t✐♥ ❦➯♥❤ ❦➲ ✭❝❤➥♥❣ ❤↕♥ sü t✐➯✉ t❤ö ✤✐➺♥ ♥➠♥❣✮✱ tr♦♥❣ ♠ët sè tr÷í♥❣ ❤đ♣ t❛ ❝â t❤➸ s✉② r❛ ✤÷đ❝ ♥❤ú♥❣ ❤♦↕t ✤ë♥❣ ❜➯♥ tr♦♥❣ ❝õ❛ ♠ët t❤✉➟t t♦→♥ ♠➟t ♠➣ ✭❦❤æ♥❣ ❛♥ t♦➔♥✮ ✈➔ tø ✤â t➻♠ ✤÷đ❝ t❤ỉ♥❣ t✐♥ ❜➼ ♠➟t✳ ❈â ❤❛✐ ❧♦↕✐ t➜♥ ❝ỉ♥❣ ❦➯♥❤ ❦➲ • P❤➙♥ t➼❝❤ ♥➠♥❣ ❧÷đ♥❣ ✤ì♥ ❣✐↔♥ ✭❙P❆✮ ❧➔ ♣❤➙♥ t➼❝❤ ❦➯♥❤ ❦➲ tø ✈✐➺❝ t❤ü❝ ❤✐➺♥ ✤ì♥ ❣✐↔♥ ❝õ❛ ♠ët t❤✉➟t t t ã P t ữủ s ✭❉P❆✮ ❧➔ t❤ü❝ ❤✐➺♥ t❤✉➟t t♦→♥ ♠ët ✈➔✐ ❧➛♥ ✈➔ s✉② r❛ ✤÷đ❝ ❦➳t q✉↔ ♥❤í ❝→❝ ❝ỉ♥❣ ❝ư t❤è♥❣ ❦➯✳ ❚❤❡♦ ❬✻❪ t❤➻ ❧♦↕✐ t➜♥ ❝æ♥❣ t❤ù ❤❛✐ ✭t➜♥ ❝ỉ♥❣ ❉P❆✮ ❦❤ỉ♥❣ ♣❤↔✐ ❧➔ ♠è✐ ✤❡ ❞å❛ ✤è✐ ✈ỵ✐ ♠➟t ♠➣ ✤÷í♥❣ ❝♦♥❣ ❡❧❧✐♣t✐❝ ✈➻ ❞➵ ❞➔♥❣ tr→♥❤ ✤÷đ❝ ❜➡♥❣ ❝→❝❤ ♥❣➝✉ ♥❤✐➯♥ ❈❤÷ì♥❣ ✸✳ ▼ët sè ù♥❣ ❞ư♥❣ ❝õ❛ ✤÷í♥❣ ❝♦♥❣ ❞↕♥❣ ❝❤✉➞♥ ❊❞✇❛r❞s ✺✾ ❤â❛ ✤➛✉ ✈➔♦ ❝õ❛ ❝→❝ t❤✉➟t t♦→♥✳ ❚r♦♥❣ ❦❤✐ ✤â✱ ♣❤➙♥ t➼❝❤ ❦➯♥❤ ỡ ữủ tỹ ỡ ợ ❝→❝ t❤✉➟t t♦→♥ tr➯♥ ✤÷í♥❣ ❝♦♥❣ ❡❧❧✐♣t✐❝ ❞↕♥❣ ❲❡✐❡rstr❛ss✱ ❜ð✐ ố ợ ữớ t ♥❤➙♥ ✤ỉ✐ ✈➔ ❝ë♥❣ ✤✐➸♠ ❧➔ ❦❤→❝ ♥❤❛✉✳ ▼ët ♣❤÷ì♥❣ ♣❤→♣ ❝❤è♥❣ ❧↕✐ ❦✐➸✉ t➜♥ ❝æ♥❣ ♥➔② ♠ët ❝→❝❤ ❤✐➺✉ q✉↔ ✤➣ ✤÷đ❝ ❜✐➳t ✤➳♥ ♥❤÷♥❣ ❝❤➾ →♣ ❞ư♥❣ ❝❤♦ ❝→❝ ✤÷í♥❣ ❝♦♥❣ ❡❧❧✐♣t✐❝ ❝ư t❤➸✳ ▼➦❝ ❞ị t❛ ❝â t ữớ t ợ t t t ②➯✉ ❝➛✉✱ ♥❤÷♥❣ t❤ỉ♥❣ t❤÷í♥❣ ❝→❝ ✤÷í♥❣ ❝♦♥❣ ❡❧❧✐♣t✐❝ ✤÷đ❝ ❝❤å♥ ❦✐➳♥ ♥❣❤à t❤❡♦ ❝❤✉➞♥✳ ❱➼ ❞ư✱ tr➯♥ ♠ët tr÷í♥❣ ❝â ✤➦❝ sè ♥❣✉②➯♥ tè ❧ỵ♥✱ ◆■❙❚ ❦✐➳♥ ♥❣❤à sû ữớ ợ õ tố ♥❛②✱ t❤✉➟t t♦→♥ ✤÷đ❝ sû ❞ư♥❣ ♣❤ê ❜✐➳♥ ♥❤➜t ✤➸ t➼♥❤ ✤÷í♥❣ ❝♦♥❣ ❡❧❧✐♣t✐❝ ❞↕♥❣ ❲❡✐❡rstr❛ss ❧➔ Q = kP tr➯♥ t❤✉➟t t♦→♥ ♥❤➙♥ ✤ỉ✐✲✈➔✲❝ë♥❣✱ ♠➔ ✤÷đ❝ ✈✐➳t ❧➔ t❤✉➟t t♦→♥ ❜➻♥❤ ♣❤÷ì♥❣✲✈➔✲♥❤➙♥ ✭①❡♠ ❬✶✶✱ ▼ư❝ ✹✳✻✳✸❪✮✳ ●✐↔ sû r➡♥❣ ♣❤➨♣ ♥❤➙♥ ✤ỉ✐ ✤✐➸♠ ✈➔ ♣❤➨♣ ❝ë♥❣ ✤✐➸♠ tr➯♥ ✤÷í♥❣ t ữủ t ợ ổ tự ❦❤✐ ✤â ❤❛✐ ❝æ♥❣ t❤ù❝ ♥➔② ❝â t❤➸ ♣❤➙♥ ❜✐➺t ❜ð✐ ♣❤➙♥ t➼❝❤ ❦➯♥❤ ❦➲✱ ✈➼ ❞ư ♥❤÷ ♣❤➙♥ t➼❝❤ ♥➠♥❣ ❧÷đ♥❣ ✤ì♥ ❣✐↔♥✳ ❑❤✐ ❞➜✉ ❤✐➺✉ ♥➠♥❣ ❧÷đ♥❣ ❝❤➾ r❛ ♠ët ♣❤➨♣ ♥❤➙♥ ✤æ✐ ✈➔ t❤❡♦ s❛✉ ♠ët ♣❤➨♣ ❝ë♥❣ ✤✐➸♠ t❤➻ ❜✐t ❤✐➺♥ t↕✐ ✤÷đ❝ ❣→♥ ❜➡♥❣ ✶ ✈➔ ♥❣÷đ❝ ❧↕✐ ❧➔ ❜➡♥❣ ✵✳ ▼ët ❝→❝❤ t❤ỉ♥❣ t❤÷í♥❣ ✤➸ ❝❤è♥❣ ❧↕✐ ❙P❆ ❧➔ ❧➦♣ ❧↕✐ ❝ò♥❣ ♠ët ♠➝✉ t❤❡♦ ❝❤➾ ❞➝♥ ❜➜t ❦➸ ❞ú ❧✐➺✉ ✤➣ ✤÷đ❝ ①û ỵ ữủ ữ s ã ỹ ❤✐➺♥ ♠ët ✈➔✐ ♣❤➨♣ t♦→♥ ❣✐↔ ✭①❡♠ ❬✻❪✮✳ • ❙û ❞ư♥❣ ♣❤➨♣ ❜✐➸✉ ❞✐➵♥ t❤❛♠ sè t❤❛② t❤➳ ❝❤♦ ✤÷í♥❣ ❝♦♥❣ ❡❧❧✐♣t✐❝✳ ❈❤➥♥❣ ❤↕♥ ♥❤÷ tr♦♥❣ ❬✶✵❪✱ ❝→❝ t→❝ ❣✐↔ ❏♦②❡ ✈➔ ◗✉✐sq✉❛t❡r ✤➲ ①✉➜t sû ❞ö♥❣ ❞↕♥❣ ❍❡ss✐❛♥✳ ✣è✐ ợ ữớ t rs ✤ỉ✐ ✤✐➸♠ sû ❞ư♥❣ ♠ët ❝ỉ♥❣ t❤ù❝ ❞✉② ♥❤➜t ❞♦ ✤â tr→♥❤ ❜à rá r➾ t❤æ♥❣ t✐♥ ❦➯♥❤ ❦➲ tø sü ❦❤→❝ ♥❤❛✉ ❣✐ú❛ ✈✐➺❝ t➼♥❤ t♦→♥ ❝ë♥❣ ✤✐➸♠ ✈➔ ♥❤➙♥ ✤æ✐ ✤✐➸♠✳ ▼➦t ❦❤→❝✱ q✉❛ ❝→❝ t➼♥❤ t♦→♥ ❝õ❛ ❝→❝ t→❝ ❣✐↔ ❇❡r♥st❡✐♥✱ ▲❛♥❣ ✈➔ ❝ë♥❣ sü tr♦♥❣ ❝→❝ t➔✐ ❧✐➺✉ ❬✶✱ ✷✱ ✹❪ ✤➣ ❝❤➾ rã t➼♥❤ ❤✐➺✉ q✉↔ ✈➲ ♠➦t t❤ü❝ ❤➔♥❤ ❦❤✐ sû ❞ö♥❣ ❞↕♥❣ ❝❤✉➞♥ ❊❞✇❛r❞s t❤❛② t❤➳ ❝❤♦ ❞↕♥❣ ❝❤✉➞♥ ❲❡✐❡rstr❛ss tr♦♥❣ ❝→❝ ù♥❣ t ợ ỳ ỵ ữ tr tr ✈å♥❣ ù♥❣ ❞ö♥❣ ❞↕♥❣ ❝❤✉➞♥ ❊❞✇❛r❞s tr♦♥❣ ♠➟t ♠➣ ❤✐➺♥ ♥❛② ❧➔ r➜t ❈❤÷ì♥❣ ✸✳ ▼ët sè ù♥❣ ❞ư♥❣ ❝õ❛ ữớ rs ợ ởt ữợ ự ữủ q t rở r tr ỗ t r õ ÷✉ t❤➳ ✈➲ ♠➦t t➼♥❤ t♦→♥✱ ❞↕♥❣ ❝❤✉➞♥ ❊❞✇❛r❞s ❝ô♥❣ ✤÷đ❝ →♣ ❞ư♥❣ tr♦♥❣ t❤✉➟t t♦→♥ ♣❤➙♥ t➼❝❤ sè ❝õ❛ ▲❡♥str❛ t❤❛② t❤➳ ❝❤♦ ❞↕♥❣ ❝❤✉➞♥ ❲❡✐❡rstr❛ss ✭①❡♠ ❝❤✐ t✐➳t ✈➲ ✈✐➺❝ ❝➔✐ ✤➦t ❞↕♥❣ ❝❤✉➞♥ ❊❞✇❛r❞s ❝❤♦ t❤✉➟t t♦→♥ ♣❤➙♥ t➼❝❤ sè ❊❈▼ tr♦♥❣ ❬✶✷✱ ✷❪✮✳ ❑➳t ❧✉➟♥ ▲✉➟♥ ✈➠♥ ✤➣ t➻♠ ❤✐➸✉ ✈➲ ❞↕♥❣ ❝❤✉➞♥ ❊❞✇❛r❞s ❝❤♦ ✤÷í♥❣ ❝♦♥❣ ❡❧❧✐♣t✐❝✱ ♠è✐ q✉❛♥ ❤➺ ❣✐ú❛ ❞↕♥❣ ♥➔② ✈ỵ✐ ❞↕♥❣ tr rstrss ỗ tớ ❝ô♥❣ tr➻♥❤ ❜➔② ❝❤✐ t✐➳t ✈➲ ♣❤➨♣ ❝ë♥❣ ✤✐➸♠ tr➯♥ ✤÷í♥❣ ❝♦♥❣ ❊❞✇❛r❞s ❝✉ë♥ ✭♥â✐ r✐➯♥❣ ❧➔ ✤÷í♥❣ ❝♦♥❣ ❊❞✇❛r❞s✮ sỹ ợ ữớ tr tữỡ ù♥❣✳ ▲✉➟♥ ✈➠♥ ❝ô♥❣ ✤➣ tr➻♥❤ ❜➔② ❝❤✐ t✐➳t ✈➲ ❝→❝ ♥❤â♠ ①♦➢♥ ❝õ❛ ✤÷í♥❣ ❝♦♥❣ ❊❞✇❛r❞s tr➯♥ Q✱ ✈➔ tứ õ ỹ ởt ợ ữớ t rstrss ợ õ trữợ tr q tr➻♥❤ ❧➔♠ ❧✉➟♥ ✈➠♥✱ ❝❤ó♥❣ tỉ✐ t❤➜② ✈➝♥ ❝á♥ ♠ët sè ✈➜♥ ✤➲ ❝❤÷❛ ❣✐↔✐ q✉②➳t ✤÷đ❝✿ ✶✳ ✣✐➲✉ ❦✐➺♥ ữớ rs tố ữ ố ợ t❤✉➟t t♦→♥ ♣❤➙♥ t➼❝❤ sè sû ❞ư♥❣ ✤÷í♥❣ ❝♦♥❣ ❡❧❧✐♣t✐❝ str tt t ợ ởt ữớ ❊❞✇❛r❞s ❜➜t ❦ý✱ ❧✐➺✉ t❛ ❝â t❤➸ ①➙② ❞ü♥❣ ✤÷đ❝ ♠ët ❤➺ ♠➟t ♠➣ ❛♥ t♦➔♥ ❤❛② ❦❤æ♥❣❄ ✸✳ ❇➯♥ õ ợ ữớ t rstrss ú tổ ①➙② ❞ü♥❣ ♠ỵ✐ ❝❤➾ t❤➸ ❤✐➺♥ ✤✐➲✉ ❦✐➺♥ ✤õ✳ ▲✐➺✉ ❝â t❤➸ ✤÷❛ r❛ ✤✐➲✉ ❦✐➺♥ ❝➛♥ ❝❤♦ ❝→❝❤ ①➙② ❞ü♥❣ ✤â ❤❛② ❦❤ỉ♥❣❄ ◆❣♦➔✐ r❛✱ tr♦♥❣ ❧✉➟♥ ✈➠♥ ❝❤ó♥❣ tỉ✐ ❝ơ♥❣ ❝❤÷❛ ❦❤↔♦ s→t ②➯✉ ❝➛✉ ❝➛♥ t❤✐➳t ✤➸ ♠ët ✤÷í♥❣ ❝♦♥❣ ❊❞✇❛r❞s ❝✉ë♥ ❝â ♥❤â♠ ①♦➢♥ tr➯♥ Q trữợ t ỵ r tự ❜✐➳t ✈➲ ✤÷í♥❣ ❝♦♥❣ ❡❧❧✐♣t✐❝✱ ✤÷í♥❣ ❝♦♥❣ ✤↕✐ sè ✈➔ ❝→❝ ❧➽♥❤ ✈ü❝ ❧✐➯♥ q✉❛♥ ❝á♥ ❤↕♥ ❝❤➳ ♥➯♥ ❧✉➟♥ ợ ữợ t ❝❤✉➞♥ ❊❞✇❛r❞s ❝❤♦ ✤÷í♥❣ ❝♦♥❣ ❡❧❧✐♣t✐❝✳ ❍✐ ✈å♥❣ tr♦♥❣ t❤í✐ ❣✐❛♥ tỵ✐✱ ❝❤ó♥❣ tỉ✐ s➩ ❝â ✤✐➲✉ ❦✐➺♥ ✤✐ s➙✉ ỡ t ữợ ự ❧✐➺✉ t❤❛♠ ❦❤↔♦ ❬✶❪ ❉✳❏✳ ❇❡r♥st❡✐♥✱ P✳ ❇✐r❦♥❡r✱ ▼✳ ❏♦②❡✱ ❚✳ ▲❛♥❣❡✱ ❈✳ P❡t❡rs✱ ❚✇✐st❡❞ ❊❞✇❛r❞s ❝✉r✈❡s✱ ■♥ ❆❢r✐❝❛❝r②♣t ✷✵✵✽✱ ✈♦❧✳ ✺✵✷✸ ♦❢ ▲❡❝t✉r❡ ◆♦t❡s ✐♥ ❈♦♠♣✉t❡r ❙❝✐✲ ❡♥❝❡✱ ♣❛❣❡s ✸✽✾✲✹✵✺✱ ✷✵✵✽✳ ❬✷❪ ❉✳❏✳ ❇❡r♥st❡✐♥✱ P✳ ❇✐r❦♥❡r✱ ❚✳ ▲❛♥❣❡✱ ❈✳ P❡t❡rs✱ ❊❈▼ ✉s✐♥❣ ❊❞✇❛r❞s ❝✉r✈❡s✱ ▼❛t❤❡♠❛t✐❝s ♦❢ ❈♦♠♣✉t❛t✐♦♥✱ ❱♦❧✳ ✽✷✱ ♣❛❣❡s ✶✶✸✾✕✶✶✼✾✱ ❆▼❙✱ ✷✵✶✸✳ ❬✸❪ ❖✳ ❇✐❧❧❡t ❛♥❞ ▼✳ ❏♦②❡✱ ❚❤❡ ❏❛❝♦❜✐ ♠♦❞❡❧ ♦❢ ❛♥ ❡❧❧✐♣t✐❝ ❝✉r✈❡ ❛♥❞ s✐❞❡✲ ❝❤❛♥♥❡❧ ❛♥❛❧②s✐s✱ ■♥ ❆❆❊❈❈✲✶✺✱ ✈♦❧✳ ✷✻✹✸ ♦❢ ▲❡❝t✉r❡ ◆♦t❡s ✐♥ ❈♦♠♣✉t❡r ❙❝✐❡♥❝❡✱ ♣❛❣❡s ✸✹✲✹✷✱ ❙♣r✐♥❣❡r✱ ✷✵✵✸✳ ❬✹❪ ❉✳❏✳ ❇❡r♥st❡✐♥✱ ❚✳ ▲❛♥❣❡✱ ❋❛st❡r ❛❞❞✐t✐♦♥ ❛♥❞ ❞♦✉❜❧✐♥❣ ♦♥ ❡❧❧✐♣t✐❝ ❝✉r✈❡s✱ ■♥ ❆s✐❛❝r②♣t ✷✵✵✼✱ ✈♦❧✳ ✹✽✸✸ ♦❢ ▲❡❝t✉r❡ ◆♦t❡s ✐♥ ❈♦♠♣✉t❡r ❙❝✐❡♥❝❡✱ ♣❛❣❡s ✷✾✲✺✵✱ ❙♣r✐♥❣❡r✱ ✷✵✵✼✳ ❬✺❪ ❉✳❏✳ ❇❡r♥st❡✐♥✱ ❚✳ ▲❛♥❣❡✱ ❆ ❝♦♠♣❧❡t❡ s❡t ♦❢ ❛❞❞✐t✐♦♥ ❧❛✇s ❢♦r ✐♥❝♦♠♣❧❡t❡ ❊❞✇❛r❞s ❝✉r✈❡s✱ ❏♦✉r♥❛❧ ♦❢ ◆✉♠❜❡r ❚❤❡♦r②✱ ✈♦❧✳ ✶✸✶✱ ♣❛❣❡s ✽✺✽✲✽✼✷✱ ✷✵✶✶✳ ❬✻❪ ❏✲❙✳ ❈♦r♦♥✱ ❘❡s✐st❛♥❝❡ ❛❣❛✐♥st ❞✐❢❢❡r❡♥t✐❛❧ ♣♦✇❡r ❛♥❛❧②s✐s ❢♦r ❡❧❧✐♣t✐❝ ❝✉r✈❡ ❝r②♣t♦s②st❡♠s✱ ✬✾✾✱ ❈r②♣t♦❣r❛♣❤✐❝ ❍❛r❞✇❛r❡ ❛♥❞ ❊♠❜❡❞❞❡❞ ❙②st❡♠s ✲ ❈❍❊❙ ✈♦❧✳ ✶✼✶✼ ♦❢ ▲❡❝t✉r❡ ◆♦t❡s ✐♥ ❈♦♠♣✉t❡r ❙❝✐❡♥❝❡✱ ♣❛❣❡s ✷✾✷✕✸✵✷✳ ❙♣r✐♥❣❡r✲❱❡r❧❛❣✱ ✶✾✾✾✳ ❬✼❪ ❍✳▼✳ ❊❞✇❛r❞s✱ ❆ ♥♦r♠❛❧ ❢♦r♠ ♦❢ ❡❧❧✐♣t✐❝ ❝✉r✈❡s✱ ❇✉❧❧❡♥t✐♥ ♦❢ t❤❡ ❆♠❡r✐❝❛♥ ▼❛t❤❡♠❛t✐❝❛❧ ❙♦❝✐❡t②✱ ✈♦❧✳ ✹✹✱ ♣❛❣❡s ✸✾✸✲✹✷✷✱ ✷✵✵✼✳ ❬✽❪ ❉✳ ❍❛♥❦❡rs♦♥✱ ❆✳ ▼❡♥❡③❡s✱ ❙✳ ❱❛♥st♦♥❡✱ ♣❤②✱ ❙♣r✐♥❣❡r✲❱❡r❧❛❣✱ ◆❡✇ ❨♦r❦✱ ✷✵✵✹✳ ✻✷ ●✉✐❞❡ t♦ ❡❧❧✐♣t✐❝ ❝✉r✈❡ ❝r②♣t♦❣r❛✲ ❚➔✐ ❧✐➺✉ t❤❛♠ ❦❤↔♦ ✻✸ ❬✾❪ ❍✳ ❍✐s✐❧✱ ❑✳❑✲❍✳ ❲♦♥❣✱ ●✳ ❈❛rt❡r✱ ❊✳ ❉❛✇s♦♥✱ ❚✇✐st❡❞ ❊❞✇❛r❞s ❝✉r✈❡s r❡✲ ✈✐s✐t❡❞✱ ■♥ ❆s✐❛❝r②♣t ✷✵✵✽✱ ✈♦❧✳ ✺✸✺✵ ♦❢ ▲❡❝t✉r❡ ◆♦t❡s ✐♥ ❈♦♠♣✉t❡r ❙❝✐❡♥❝❡✱ ♣❛❣❡s ✸✷✻✲✸✹✸✱ ❙♣r✐♥❣❡r✱ ❍❡✐❞❡❧❜❡r❣✱ ✷✵✵✽✳ ❬✶✵❪ ▼✳ ❏♦②❡✱ ❏✲❏✳ ◗✉✐sq✉❛t❡r✱ ❍❡ss✐❛♥ ❡❧❧✐♣t✐❝ ❝✉r✈❡s ❛♥❞ s✐❞❡✲❝❤❛♥♥❡❧ ❛tt❛❝❦s✳ ❈r②♣t♦❣r❛♣❤✐❝ ❍❛r❞✇❛r❡ ❛♥❞ ❊♠❜❡❞❞❡❞ ❙②st❡♠s ✲ ❈❍❊❙ ✷✵✵✶✱ ✈♦❧✳ ✷✶✻✷ ♦❢ ▲❡❝t✉r❡ ◆♦t❡s ✐♥ ❈♦♠♣✉t❡r ❙❝✐❡♥❝❡✱ ♣❛❣❡s ✹✶✷✕✹✷✵✳ ❙♣r✐♥❣❡r✲❱❡r❧❛❣✱ ✷✵✵✶✳ ❬✶✶❪ ❉✳❊✳ ❑♥✉t❤✱ ❚❤❡ ❛rt ♦❢ ❝♦♠♣✉t❡r ♣r♦❣r❛♠♠✐♥❣✱ ✈♦❧✳ ✷✿ ❙❡♠✐♥✉♠❡r✐❝❛❧ ❛❧✲ ❣♦r✐t❤♠s✱ ❆❞❞✐s♦♥✲❲❡❧s❧❡②✱ ✶✾✽✶✳ ❬✶✷❪ ❍✳❲✳ ▲❡♥str❛✱ ❋❛❝t♦r✐♥❣ ✐♥t❡❣❡rs ✇✐t❤ ❡❧❧✐♣t✐❝ ❝✉r✈❡s✱ ♠❛t✐❝s✱ ✈♦❧✳ ✶✷✻✱ ♣❛❣❡s ✻✹✾✕✻✼✸✱ ✶✾✽✼✳ ❬✶✸❪ ❑✳ ❖❦❡②❛✱ ❍✳ ❑✉r✉♠❛t❛♥✐✱ ❑✳ ❙❛❦✉r❛✐✱ ❆♥♥❛❧s ♦❢ ▼❛t❤❡✲ ❊❧❧✐♣t✐❝ ▼♦♥t❣♦♠❡r②✲❢♦r♠ ❛♥❞ t❤❡✐r ❝r②♣t♦❣r❛♣❤✐❝ ❛♣♣❧✐❝❛t✐♦♥s✱ ❝✉r✈❡s ✇✐t❤ t❤❡ ■♥ Pr♦❝❡❡❞✐♥❣s ♦❢ P❑❈✬✷✵✵✵✱ ✈♦❧✳ ✶✼✺✶ ♦❢ ▲❡❝t✉r❡ ◆♦t❡s ✐♥ ❈♦♠♣✉t❡r ❙❝✐❡♥❝❡✱ ♣❛❣❡s ✷✸✽✲✷✺✼✱ ❙♣r✐♥❣❡r✲❱❡r❧❛❣✱ ✷✵✵✵✳ ❬✶✹❪ ❏✳ ❍✳ ❙✐❧✈❡r♠❛♥✱ ❚❤❡ ❛r✐t❤♠❡t✐❝ ♦❢ ❡❧❧✐♣t✐❝ ❝✉r✈❡s✱ ✈♦❧✳ ✶✵✻ ♦❢ ●r❛❞✉❛t❡ ❚❡①ts ✐♥ ▼❛t❤❡♠❛t✐❝s✱ ❙♣r✐♥❣❡r✲❱❡r❧❛❣✱ ◆❡✇ ❨♦r❦✱ ✶✾✽✻✳ ❬✶✺❪ ▲✳❈✳ ❲❛s❤✐♥❣t♦♥✱ ❊❧❧✐♣t✐❝ ❈✉r✈❡✿ ◆✉♠❜❡r ❚❤❡♦r② ❛♥❞ ❈r②♣t♦❣r❛♣❤②✱ ❈❘❈ Pr❡ss✱ ❇♦❝❛ ❘❛t♦♥✱ ✷✵✵✽✳ ❬✶✻❪ ❤tt♣✿✴✴✇✇✇✳s❛❣❡♠❛t❤✳♦r❣✳