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▼♦❞✉❧❛r ❋♦r♠s ❢♦r ❖(✷, ✺)

▼♦❞✉❧❛r ❋♦r♠s ❢♦r t❤❡ ❖rt❤♦❣♦♥❛❧ ●r♦✉♣ ❖(✷, ✺)

■♥❣♦ ❑❧ö❝❦❡r

▲❡❤rst✉❤❧ ❆ ❢ür ▼❛t❤❡♠❛t✐❦ ❘❲❚❍ ❆❛❝❤❡♥

❏❛♣❛♥❡s❡✲●❡r♠❛♥ ◆✉♠❜❡r ❚❤❡♦r② ❲♦r❦s❤♦♣ ▼P■▼ ❇♦♥♥ ✷✵✵✺✲✵✼✲✷✼

■♥❣♦ ❑❧ö❝❦❡r ✭❘❲❚❍ ❆❛❝❤❡♥✮ ❏❛♣❛♥❡s❡✲●❡r♠❛♥ ◆✉♠❜❡r ❚❤❡♦r② ❲♦r❦s❤♦♣ ✶ ✴ ✷✾

▼♦❞✉❧❛r ❋♦r♠s ❢♦r ❖(✷, ✺)

❖✉t❧✐♥❡

■♥tr♦❞✉❝t✐♦♥ ❖rt❤♦❣♦♥❛❧ ▼♦❞✉❧❛r ❋♦r♠s ❱❡❝t♦r✲✈❛❧✉❡❞ ▼♦❞✉❧❛r ❋♦r♠s ❇♦r❝❤❡r❞s Pr♦❞✉❝ts ❚❤❡ ●r❛❞❡❞ ❘✐♥❣ ♦❢ ▼♦❞✉❧❛r ❋♦r♠s

■♥❣♦ ❑❧ö❝❦❡r ✭❘❲❚❍ ❆❛❝❤❡♥✮ ❏❛♣❛♥❡s❡✲●❡r♠❛♥ ◆✉♠❜❡r ❚❤❡♦r② ❲♦r❦s❤♦♣ ✷ ✴ ✷✾

▼♦❞✉❧❛r ❋♦r♠s ❢♦r ❖(✷, ✺) ■♥tr♦❞✉❝t✐♦♥

❖✉t❧✐♥❡

■♥tr♦❞✉❝t✐♦♥ ❖rt❤♦❣♦♥❛❧ ▼♦❞✉❧❛r ❋♦r♠s ❱❡❝t♦r✲✈❛❧✉❡❞ ▼♦❞✉❧❛r ❋♦r♠s ❇♦r❝❤❡r❞s Pr♦❞✉❝ts ❚❤❡ ●r❛❞❡❞ ❘✐♥❣ ♦❢ ▼♦❞✉❧❛r ❋♦r♠s

■♥❣♦ ❑❧ö❝❦❡r ✭❘❲❚❍ ❆❛❝❤❡♥✮ ❏❛♣❛♥❡s❡✲●❡r♠❛♥ ◆✉♠❜❡r ❚❤❡♦r② ❲♦r❦s❤♦♣ ✸ ✴ ✷✾

▼♦❞✉❧❛r ❋♦r♠s ❢♦r ❖(✷, ✺) ■♥tr♦❞✉❝t✐♦♥

❍✐st♦r②

◮ ❖(✷, ✶)✿ ❊❧❧✐♣t✐❝ ♠♦❞✉❧❛r ❢♦r♠s✳

■t ✐s ✇❡❧❧✲❦♥♦✇♥ t❤❛t t❤❡ ❣r❛❞❡❞ r✐♥❣ A(❙▲✷(Z)) ♦❢ ❡❧❧✐♣t✐❝ ♠♦❞✉❧❛r ❢♦r♠s ✐s ❛ ♣♦❧②♥♦♠✐❛❧ r✐♥❣ ✐♥ t❤❡ ❡❧❧✐♣t✐❝ ❊✐s❡♥st❡✐♥ s❡r✐❡s ❣✷ ❛♥❞ ❣✸ ✭♦❢ ✇❡✐❣❤t ✹ ❛♥❞ ✻✮✳

◮ ❖(✷, ✷)✿ ❍✐❧❜❡rt ♠♦❞✉❧❛r ❢♦r♠s✳

❈❢✳ ❙✳ ▼❛②❡r✬s t❛❧❦✳

◮ ❖(✷, ✸)✿ ❙✐❡❣❡❧ ♠♦❞✉❧❛r ❢♦r♠s ♦❢ ❞❡❣r❡❡ ✷✳

❏✳✲■✳ ■❣✉s❛ ✭✶✾✻✷✮✿ ❚❤❡ ❣r❛❞❡❞ r✐♥❣ A(❙♣✷(Z)) ✐s ❛ ♣♦❧②♥♦♠✐❛❧ r✐♥❣ ✐♥ t❤❡ ❙✐❡❣❡❧ ❊✐s❡♥st❡✐♥ s❡r✐❡s ❊✹✱ ❊✻✱ ❊✶✵ ❛♥❞ ❊✶✷✳

■♥❣♦ ❑❧ö❝❦❡r ✭❘❲❚❍ ❆❛❝❤❡♥✮ ❏❛♣❛♥❡s❡✲●❡r♠❛♥ ◆✉♠❜❡r ❚❤❡♦r② ❲♦r❦s❤♦♣ ✹ ✴ ✷✾

▼♦❞✉❧❛r ❋♦r♠s ❢♦r ❖(✷, ✺) ■♥tr♦❞✉❝t✐♦♥

❍✐st♦r②

◮ ❖(✷, ✶)✿ ❊❧❧✐♣t✐❝ ♠♦❞✉❧❛r ❢♦r♠s✳

■t ✐s ✇❡❧❧✲❦♥♦✇♥ t❤❛t t❤❡ ❣r❛❞❡❞ r✐♥❣ A(❙▲✷(Z)) ♦❢ ❡❧❧✐♣t✐❝ ♠♦❞✉❧❛r ❢♦r♠s ✐s ❛ ♣♦❧②♥♦♠✐❛❧ r✐♥❣ ✐♥ t❤❡ ❡❧❧✐♣t✐❝ ❊✐s❡♥st❡✐♥ s❡r✐❡s ❣✷ ❛♥❞ ❣✸ ✭♦❢ ✇❡✐❣❤t ✹ ❛♥❞ ✻✮✳

◮ ❖(✷, ✷)✿ ❍✐❧❜❡rt ♠♦❞✉❧❛r ❢♦r♠s✳

❈❢✳ ❙✳ ▼❛②❡r✬s t❛❧❦✳

◮ ❖(✷, ✸)✿ ❙✐❡❣❡❧ ♠♦❞✉❧❛r ❢♦r♠s ♦❢ ❞❡❣r❡❡ ✷✳

❏✳✲■✳ ■❣✉s❛ ✭✶✾✻✷✮✿ ❚❤❡ ❣r❛❞❡❞ r✐♥❣ A(❙♣✷(Z)) ✐s ❛ ♣♦❧②♥♦♠✐❛❧ r✐♥❣ ✐♥ t❤❡ ❙✐❡❣❡❧ ❊✐s❡♥st❡✐♥ s❡r✐❡s ❊✹✱ ❊✻✱ ❊✶✵ ❛♥❞ ❊✶✷✳

■♥❣♦ ❑❧ö❝❦❡r ✭❘❲❚❍ ❆❛❝❤❡♥✮ ❏❛♣❛♥❡s❡✲●❡r♠❛♥ ◆✉♠❜❡r ❚❤❡♦r② ❲♦r❦s❤♦♣ ✹ ✴ ✷✾

▼♦❞✉❧❛r ❋♦r♠s ❢♦r ❖(✷, ✺) ■♥tr♦❞✉❝t✐♦♥

❍✐st♦r②

◮ ❖(✷, ✶)✿ ❊❧❧✐♣t✐❝ ♠♦❞✉❧❛r ❢♦r♠s✳

■t ✐s ✇❡❧❧✲❦♥♦✇♥ t❤❛t t❤❡ ❣r❛❞❡❞ r✐♥❣ A(❙▲✷(Z)) ♦❢ ❡❧❧✐♣t✐❝ ♠♦❞✉❧❛r ❢♦r♠s ✐s ❛ ♣♦❧②♥♦♠✐❛❧ r✐♥❣ ✐♥ t❤❡ ❡❧❧✐♣t✐❝ ❊✐s❡♥st❡✐♥ s❡r✐❡s ❣✷ ❛♥❞ ❣✸ ✭♦❢ ✇❡✐❣❤t ✹ ❛♥❞ ✻✮✳

◮ ❖(✷, ✷)✿ ❍✐❧❜❡rt ♠♦❞✉❧❛r ❢♦r♠s✳

❈❢✳ ❙✳ ▼❛②❡r✬s t❛❧❦✳

◮ ❖(✷, ✸)✿ ❙✐❡❣❡❧ ♠♦❞✉❧❛r ❢♦r♠s ♦❢ ❞❡❣r❡❡ ✷✳

❏✳✲■✳ ■❣✉s❛ ✭✶✾✻✷✮✿ ❚❤❡ ❣r❛❞❡❞ r✐♥❣ A(❙♣✷(Z)) ✐s ❛ ♣♦❧②♥♦♠✐❛❧ r✐♥❣ ✐♥ t❤❡ ❙✐❡❣❡❧ ❊✐s❡♥st❡✐♥ s❡r✐❡s ❊✹✱ ❊✻✱ ❊✶✵ ❛♥❞ ❊✶✷✳

■♥❣♦ ❑❧ö❝❦❡r ✭❘❲❚❍ ❆❛❝❤❡♥✮ ❏❛♣❛♥❡s❡✲●❡r♠❛♥ ◆✉♠❜❡r ❚❤❡♦r② ❲♦r❦s❤♦♣ ✹ ✴ ✷✾

▼♦❞✉❧❛r ❋♦r♠s ❢♦r ❖(✷, ✺) ■♥tr♦❞✉❝t✐♦♥

❍✐st♦r② ✭❝♦♥t✐♥✉❡❞✮

◮ ❖(✷, ✹)✿ ❍❡r♠✐t✐❛♥ ♠♦❞✉❧❛r ❢♦r♠s ♦❢ ❞❡❣r❡❡ ✷✳

❊✳ ❋r❡✐t❛❣ ✭✶✾✻✼✮✿ ❚❤❡ ❣r❛❞❡❞ r✐♥❣ ❢♦r Q(√−✶)✱ ❚✳ ❉❡r♥ ✭✷✵✵✶✮✿ ❚❤❡ ❣r❛❞❡❞ r✐♥❣ ❢♦r Q(√−✸) ❛♥❞ Q(√−✷) ✭✇✐t❤ ❆✳ ❑r✐❡❣✮✳

◮ ❖(✷, ✺)✿ ❚❤✐s ✐s t❤❡ ❝❛s❡ ✇❡ ✇✐❧❧ ❝♦♥s✐❞❡r✳ ◮ ❖(✷, ✻)✿ ◗✉❛t❡r♥✐♦♥✐❝ ♠♦❞✉❧❛r ❢♦r♠s ♦❢ ❞❡❣r❡❡ ✷✳

❆✳ ❑r✐❡❣ ✭✷✵✵✺✮

■♥❣♦ ❑❧ö❝❦❡r ✭❘❲❚❍ ❆❛❝❤❡♥✮ ❏❛♣❛♥❡s❡✲●❡r♠❛♥ ◆✉♠❜❡r ❚❤❡♦r② ❲♦r❦s❤♦♣ ✺ ✴ ✷✾

▼♦❞✉❧❛r ❋♦r♠s ❢♦r ❖(✷, ✺) ■♥tr♦❞✉❝t✐♦♥

❍✐st♦r② ✭❝♦♥t✐♥✉❡❞✮

◮ ❖(✷, ✹)✿ ❍❡r♠✐t✐❛♥ ♠♦❞✉❧❛r ❢♦r♠s ♦❢ ❞❡❣r❡❡ ✷✳

❊✳ ❋r❡✐t❛❣ ✭✶✾✻✼✮✿ ❚❤❡ ❣r❛❞❡❞ r✐♥❣ ❢♦r Q(√−✶)✱ ❚✳ ❉❡r♥ ✭✷✵✵✶✮✿ ❚❤❡ ❣r❛❞❡❞ r✐♥❣ ❢♦r Q(√−✸) ❛♥❞ Q(√−✷) ✭✇✐t❤ ❆✳ ❑r✐❡❣✮✳

◮ ❖(✷, ✺)✿ ❚❤✐s ✐s t❤❡ ❝❛s❡ ✇❡ ✇✐❧❧ ❝♦♥s✐❞❡r✳ ◮ ❖(✷, ✻)✿ ◗✉❛t❡r♥✐♦♥✐❝ ♠♦❞✉❧❛r ❢♦r♠s ♦❢ ❞❡❣r❡❡ ✷✳

❆✳ ❑r✐❡❣ ✭✷✵✵✺✮

■♥❣♦ ❑❧ö❝❦❡r ✭❘❲❚❍ ❆❛❝❤❡♥✮ ❏❛♣❛♥❡s❡✲●❡r♠❛♥ ◆✉♠❜❡r ❚❤❡♦r② ❲♦r❦s❤♦♣ ✺ ✴ ✷✾

▼♦❞✉❧❛r ❋♦r♠s ❢♦r ❖(✷, ✺) ■♥tr♦❞✉❝t✐♦♥

❍✐st♦r② ✭❝♦♥t✐♥✉❡❞✮

◮ ❖(✷, ✹)✿ ❍❡r♠✐t✐❛♥ ♠♦❞✉❧❛r ❢♦r♠s ♦❢ ❞❡❣r❡❡ ✷✳

❊✳ ❋r❡✐t❛❣ ✭✶✾✻✼✮✿ ❚❤❡ ❣r❛❞❡❞ r✐♥❣ ❢♦r Q(√−✶)✱ ❚✳ ❉❡r♥ ✭✷✵✵✶✮✿ ❚❤❡ ❣r❛❞❡❞ r✐♥❣ ❢♦r Q(√−✸) ❛♥❞ Q(√−✷) ✭✇✐t❤ ❆✳ ❑r✐❡❣✮✳

◮ ❖(✷, ✺)✿ ❚❤✐s ✐s t❤❡ ❝❛s❡ ✇❡ ✇✐❧❧ ❝♦♥s✐❞❡r✳ ◮ ❖(✷, ✻)✿ ◗✉❛t❡r♥✐♦♥✐❝ ♠♦❞✉❧❛r ❢♦r♠s ♦❢ ❞❡❣r❡❡ ✷✳

❆✳ ❑r✐❡❣ ✭✷✵✵✺✮

■♥❣♦ ❑❧ö❝❦❡r ✭❘❲❚❍ ❆❛❝❤❡♥✮ ❏❛♣❛♥❡s❡✲●❡r♠❛♥ ◆✉♠❜❡r ❚❤❡♦r② ❲♦r❦s❤♦♣ ✺ ✴ ✷✾

▼♦❞✉❧❛r ❋♦r♠s ❢♦r ❖(✷, ✺) ❖rt❤♦❣♦♥❛❧ ▼♦❞✉❧❛r ❋♦r♠s

❖✉t❧✐♥❡

■♥tr♦❞✉❝t✐♦♥ ❖rt❤♦❣♦♥❛❧ ▼♦❞✉❧❛r ❋♦r♠s ❱❡❝t♦r✲✈❛❧✉❡❞ ▼♦❞✉❧❛r ❋♦r♠s ❇♦r❝❤❡r❞s Pr♦❞✉❝ts ❚❤❡ ●r❛❞❡❞ ❘✐♥❣ ♦❢ ▼♦❞✉❧❛r ❋♦r♠s

■♥❣♦ ❑❧ö❝❦❡r ✭❘❲❚❍ ❆❛❝❤❡♥✮ ❏❛♣❛♥❡s❡✲●❡r♠❛♥ ◆✉♠❜❡r ❚❤❡♦r② ❲♦r❦s❤♦♣ ✻ ✴ ✷✾

▼♦❞✉❧❛r ❋♦r♠s ❢♦r ❖(✷, ✺) ❖rt❤♦❣♦♥❛❧ ▼♦❞✉❧❛r ❋♦r♠s

❙②♠♠❡tr✐❝ ▼❛tr✐❝❡s ❛♥❞ ◗✉❛❞r❛t✐❝ ❋♦r♠s

◮ ❙✿ ❛ s②♠♠❡tr✐❝✱ ♣♦s✐t✐✈❡ ❞❡✜♥✐t❡✱ ❡✈❡♥ ℓ × ℓ ♠❛tr✐① ◮ ❙✵ :=

  ✵ ✵ ✶ ✵ −❙ ✵ ✶ ✵ ✵   , ❙✶ :=   ✵ ✵ ✶ ✵ ❙✵ ✵ ✶ ✵ ✵  ✱ s✐❣♥❛t✉r❡ ♦❢ ❙✶ ✐s (✷, ℓ + ✷)

◮ (①, ②)❚ = t①❚②

❛♥❞ q❚(①) = ✶

✷(①, ①)❚ = ✶ ✷ t①❚① = ✶ ✷❚[①]

❆❜❜r❡✈✐❛t✐♦♥s✿

◮ (·, ·) = (·, ·)❙✱ q = q❙✱ ◮ (·, ·)✵ = (·, ·)❙✵✱ q✵ = q❙✵✱ ◮ (·, ·)✶ = (·, ·)❙✶✱ q✶ = q❙✶✳

◮ ▼♦st❧② ❙ = ❆✸ =

  ✷ ✶ ✵ ✶ ✷ ✶ ✵ ✶ ✷  ✱ q❙(①✶, ①✷, ①✸) = ①✷

✶ + ①✶①✷ + ①✷ ✷ + ①✷①✸ + ①✷ ✸✳

■♥❣♦ ❑❧ö❝❦❡r ✭❘❲❚❍ ❆❛❝❤❡♥✮ ❏❛♣❛♥❡s❡✲●❡r♠❛♥ ◆✉♠❜❡r ❚❤❡♦r② ❲♦r❦s❤♦♣ ✼ ✴ ✷✾

▼♦❞✉❧❛r ❋♦r♠s ❢♦r ❖(✷, ✺) ❖rt❤♦❣♦♥❛❧ ▼♦❞✉❧❛r ❋♦r♠s

❙②♠♠❡tr✐❝ ▼❛tr✐❝❡s ❛♥❞ ◗✉❛❞r❛t✐❝ ❋♦r♠s

◮ ❙✿ ❛ s②♠♠❡tr✐❝✱ ♣♦s✐t✐✈❡ ❞❡✜♥✐t❡✱ ❡✈❡♥ ℓ × ℓ ♠❛tr✐① ◮ ❙✵ :=

  ✵ ✵ ✶ ✵ −❙ ✵ ✶ ✵ ✵   , ❙✶ :=   ✵ ✵ ✶ ✵ ❙✵ ✵ ✶ ✵ ✵  ✱ s✐❣♥❛t✉r❡ ♦❢ ❙✶ ✐s (✷, ℓ + ✷)

◮ (①, ②)❚ = t①❚②

❛♥❞ q❚(①) = ✶

✷(①, ①)❚ = ✶ ✷ t①❚① = ✶ ✷❚[①]

❆❜❜r❡✈✐❛t✐♦♥s✿

◮ (·, ·) = (·, ·)❙✱ q = q❙✱ ◮ (·, ·)✵ = (·, ·)❙✵✱ q✵ = q❙✵✱ ◮ (·, ·)✶ = (·, ·)❙✶✱ q✶ = q❙✶✳

◮ ▼♦st❧② ❙ = ❆✸ =

  ✷ ✶ ✵ ✶ ✷ ✶ ✵ ✶ ✷  ✱ q❙(①✶, ①✷, ①✸) = ①✷

✶ + ①✶①✷ + ①✷ ✷ + ①✷①✸ + ①✷ ✸✳

■♥❣♦ ❑❧ö❝❦❡r ✭❘❲❚❍ ❆❛❝❤❡♥✮ ❏❛♣❛♥❡s❡✲●❡r♠❛♥ ◆✉♠❜❡r ❚❤❡♦r② ❲♦r❦s❤♦♣ ✼ ✴ ✷✾

▼♦❞✉❧❛r ❋♦r♠s ❢♦r ❖(✷, ✺) ❖rt❤♦❣♦♥❛❧ ▼♦❞✉❧❛r ❋♦r♠s

❙②♠♠❡tr✐❝ ▼❛tr✐❝❡s ❛♥❞ ◗✉❛❞r❛t✐❝ ❋♦r♠s

◮ ❙✿ ❛ s②♠♠❡tr✐❝✱ ♣♦s✐t✐✈❡ ❞❡✜♥✐t❡✱ ❡✈❡♥ ℓ × ℓ ♠❛tr✐① ◮ ❙✵ :=

  ✵ ✵ ✶ ✵ −❙ ✵ ✶ ✵ ✵   , ❙✶ :=   ✵ ✵ ✶ ✵ ❙✵ ✵ ✶ ✵ ✵  ✱ s✐❣♥❛t✉r❡ ♦❢ ❙✶ ✐s (✷, ℓ + ✷)

◮ (①, ②)❚ = t①❚②

❛♥❞ q❚(①) = ✶

✷(①, ①)❚ = ✶ ✷ t①❚① = ✶ ✷❚[①]

❆❜❜r❡✈✐❛t✐♦♥s✿

◮ (·, ·) = (·, ·)❙✱ q = q❙✱ ◮ (·, ·)✵ = (·, ·)❙✵✱ q✵ = q❙✵✱ ◮ (·, ·)✶ = (·, ·)❙✶✱ q✶ = q❙✶✳

◮ ▼♦st❧② ❙ = ❆✸ =

  ✷ ✶ ✵ ✶ ✷ ✶ ✵ ✶ ✷  ✱ q❙(①✶, ①✷, ①✸) = ①✷

✶ + ①✶①✷ + ①✷ ✷ + ①✷①✸ + ①✷ ✸✳

■♥❣♦ ❑❧ö❝❦❡r ✭❘❲❚❍ ❆❛❝❤❡♥✮ ❏❛♣❛♥❡s❡✲●❡r♠❛♥ ◆✉♠❜❡r ❚❤❡♦r② ❲♦r❦s❤♦♣ ✼ ✴ ✷✾

▼♦❞✉❧❛r ❋♦r♠s ❢♦r ❖(✷, ✺) ❖rt❤♦❣♦♥❛❧ ▼♦❞✉❧❛r ❋♦r♠s

❙②♠♠❡tr✐❝ ▼❛tr✐❝❡s ❛♥❞ ◗✉❛❞r❛t✐❝ ❋♦r♠s

◮ ❙✿ ❛ s②♠♠❡tr✐❝✱ ♣♦s✐t✐✈❡ ❞❡✜♥✐t❡✱ ❡✈❡♥ ℓ × ℓ ♠❛tr✐① ◮ ❙✵ :=

  ✵ ✵ ✶ ✵ −❙ ✵ ✶ ✵ ✵   , ❙✶ :=   ✵ ✵ ✶ ✵ ❙✵ ✵ ✶ ✵ ✵  ✱ s✐❣♥❛t✉r❡ ♦❢ ❙✶ ✐s (✷, ℓ + ✷)

◮ (①, ②)❚ = t①❚②

❛♥❞ q❚(①) = ✶

✷(①, ①)❚ = ✶ ✷ t①❚① = ✶ ✷❚[①]

❆❜❜r❡✈✐❛t✐♦♥s✿

◮ (·, ·) = (·, ·)❙✱ q = q❙✱ ◮ (·, ·)✵ = (·, ·)❙✵✱ q✵ = q❙✵✱ ◮ (·, ·)✶ = (·, ·)❙✶✱ q✶ = q❙✶✳

◮ ▼♦st❧② ❙ = ❆✸ =

  ✷ ✶ ✵ ✶ ✷ ✶ ✵ ✶ ✷  ✱ q❙(①✶, ①✷, ①✸) = ①✷

✶ + ①✶①✷ + ①✷ ✷ + ①✷①✸ + ①✷ ✸✳

■♥❣♦ ❑❧ö❝❦❡r ✭❘❲❚❍ ❆❛❝❤❡♥✮ ❏❛♣❛♥❡s❡✲●❡r♠❛♥ ◆✉♠❜❡r ❚❤❡♦r② ❲♦r❦s❤♦♣ ✼ ✴ ✷✾

▼♦❞✉❧❛r ❋♦r♠s ❢♦r ❖(✷, ✺) ❖rt❤♦❣♦♥❛❧ ▼♦❞✉❧❛r ❋♦r♠s

▲❛tt✐❝❡s ✐♥ ◗✉❛❞r❛t✐❝ ❙♣❛❝❡s

◮ Λ = Zℓ✱ Λ✵ = Zℓ+✷✱ Λ✶ = Zℓ+✹ ✭❧❛tt✐❝❡s ✐♥ q✉❛❞r❛t✐❝ s♣❛❝❡s

(Λ ⊗ R, (·, ·))✱ ✳✳✳✮

◮ ❉✉❛❧ ❧❛tt✐❝❡s✿ Λ♯ ❚ = {µ ∈ Λ ⊗ R; (λ, µ)❚ ∈ Z ❢♦r ❛❧❧ λ ∈ Λ} = ❚ −✶Λ ◮ ❲❡ ❤❛✈❡✿

◮ Λ♯ = ❙−✶Zℓ✱ Λ♯

✵ = Z × Λ♯ × Z✱ Λ♯ ✶ = Z × Λ♯ ✵ × Z✱

◮ Λ♯/Λ ∼

= Λ♯

✵/Λ✵ ∼

= Λ♯

✶/Λ✶✱

◮ |Λ♯/Λ| = ❞❡t ❙✳

◮ q❚ : Λ♯ ❚/Λ❚ → Q/Z, µ + Λ❚ → q❚(µ) + Z ◮ ❙ = ❆✸✿ Λ♯/Λ = ❆−✶ ✸ Z✸/Z✸ ✐s r❡♣r❡s❡♥t❡❞ ❜② (✵, ✵, ✵)✱ (✶ ✹, ✶ ✷, −✶ ✹)✱

(✶

✷, ✵, ✶ ✷)✱ (−✶ ✹, ✶ ✷, ✶ ✹) ✇✐t❤ ♥♦r♠ ✵✱ ✸ ✽✱ ✶ ✷✱ ✸ ✽✱ r❡s♣❡❝t✐✈❡❧②✳

■♥❣♦ ❑❧ö❝❦❡r ✭❘❲❚❍ ❆❛❝❤❡♥✮ ❏❛♣❛♥❡s❡✲●❡r♠❛♥ ◆✉♠❜❡r ❚❤❡♦r② ❲♦r❦s❤♦♣ ✽ ✴ ✷✾

▼♦❞✉❧❛r ❋♦r♠s ❢♦r ❖(✷, ✺) ❖rt❤♦❣♦♥❛❧ ▼♦❞✉❧❛r ❋♦r♠s

▲❛tt✐❝❡s ✐♥ ◗✉❛❞r❛t✐❝ ❙♣❛❝❡s

◮ Λ = Zℓ✱ Λ✵ = Zℓ+✷✱ Λ✶ = Zℓ+✹ ✭❧❛tt✐❝❡s ✐♥ q✉❛❞r❛t✐❝ s♣❛❝❡s

(Λ ⊗ R, (·, ·))✱ ✳✳✳✮

◮ ❉✉❛❧ ❧❛tt✐❝❡s✿ Λ♯ ❚ = {µ ∈ Λ ⊗ R; (λ, µ)❚ ∈ Z ❢♦r ❛❧❧ λ ∈ Λ} = ❚ −✶Λ ◮ ❲❡ ❤❛✈❡✿

◮ Λ♯ = ❙−✶Zℓ✱ Λ♯

✵ = Z × Λ♯ × Z✱ Λ♯ ✶ = Z × Λ♯ ✵ × Z✱

◮ Λ♯/Λ ∼

= Λ♯

✵/Λ✵ ∼

= Λ♯

✶/Λ✶✱

◮ |Λ♯/Λ| = ❞❡t ❙✳

◮ q❚ : Λ♯ ❚/Λ❚ → Q/Z, µ + Λ❚ → q❚(µ) + Z ◮ ❙ = ❆✸✿ Λ♯/Λ = ❆−✶ ✸ Z✸/Z✸ ✐s r❡♣r❡s❡♥t❡❞ ❜② (✵, ✵, ✵)✱ (✶ ✹, ✶ ✷, −✶ ✹)✱

(✶

✷, ✵, ✶ ✷)✱ (−✶ ✹, ✶ ✷, ✶ ✹) ✇✐t❤ ♥♦r♠ ✵✱ ✸ ✽✱ ✶ ✷✱ ✸ ✽✱ r❡s♣❡❝t✐✈❡❧②✳

■♥❣♦ ❑❧ö❝❦❡r ✭❘❲❚❍ ❆❛❝❤❡♥✮ ❏❛♣❛♥❡s❡✲●❡r♠❛♥ ◆✉♠❜❡r ❚❤❡♦r② ❲♦r❦s❤♦♣ ✽ ✴ ✷✾

▼♦❞✉❧❛r ❋♦r♠s ❢♦r ❖(✷, ✺) ❖rt❤♦❣♦♥❛❧ ▼♦❞✉❧❛r ❋♦r♠s

▲❛tt✐❝❡s ✐♥ ◗✉❛❞r❛t✐❝ ❙♣❛❝❡s

◮ Λ = Zℓ✱ Λ✵ = Zℓ+✷✱ Λ✶ = Zℓ+✹ ✭❧❛tt✐❝❡s ✐♥ q✉❛❞r❛t✐❝ s♣❛❝❡s

(Λ ⊗ R, (·, ·))✱ ✳✳✳✮

◮ ❉✉❛❧ ❧❛tt✐❝❡s✿ Λ♯ ❚ = {µ ∈ Λ ⊗ R; (λ, µ)❚ ∈ Z ❢♦r ❛❧❧ λ ∈ Λ} = ❚ −✶Λ ◮ ❲❡ ❤❛✈❡✿

◮ Λ♯ = ❙−✶Zℓ✱ Λ♯

✵ = Z × Λ♯ × Z✱ Λ♯ ✶ = Z × Λ♯ ✵ × Z✱

◮ Λ♯/Λ ∼

= Λ♯

✵/Λ✵ ∼

= Λ♯

✶/Λ✶✱

◮ |Λ♯/Λ| = ❞❡t ❙✳

◮ q❚ : Λ♯ ❚/Λ❚ → Q/Z, µ + Λ❚ → q❚(µ) + Z ◮ ❙ = ❆✸✿ Λ♯/Λ = ❆−✶ ✸ Z✸/Z✸ ✐s r❡♣r❡s❡♥t❡❞ ❜② (✵, ✵, ✵)✱ (✶ ✹, ✶ ✷, −✶ ✹)✱

(✶

✷, ✵, ✶ ✷)✱ (−✶ ✹, ✶ ✷, ✶ ✹) ✇✐t❤ ♥♦r♠ ✵✱ ✸ ✽✱ ✶ ✷✱ ✸ ✽✱ r❡s♣❡❝t✐✈❡❧②✳

■♥❣♦ ❑❧ö❝❦❡r ✭❘❲❚❍ ❆❛❝❤❡♥✮ ❏❛♣❛♥❡s❡✲●❡r♠❛♥ ◆✉♠❜❡r ❚❤❡♦r② ❲♦r❦s❤♦♣ ✽ ✴ ✷✾

▼♦❞✉❧❛r ❋♦r♠s ❢♦r ❖(✷, ✺) ❖rt❤♦❣♦♥❛❧ ▼♦❞✉❧❛r ❋♦r♠s

▲❛tt✐❝❡s ✐♥ ◗✉❛❞r❛t✐❝ ❙♣❛❝❡s

◮ Λ = Zℓ✱ Λ✵ = Zℓ+✷✱ Λ✶ = Zℓ+✹ ✭❧❛tt✐❝❡s ✐♥ q✉❛❞r❛t✐❝ s♣❛❝❡s

(Λ ⊗ R, (·, ·))✱ ✳✳✳✮

◮ ❉✉❛❧ ❧❛tt✐❝❡s✿ Λ♯ ❚ = {µ ∈ Λ ⊗ R; (λ, µ)❚ ∈ Z ❢♦r ❛❧❧ λ ∈ Λ} = ❚ −✶Λ ◮ ❲❡ ❤❛✈❡✿

◮ Λ♯ = ❙−✶Zℓ✱ Λ♯

✵ = Z × Λ♯ × Z✱ Λ♯ ✶ = Z × Λ♯ ✵ × Z✱

◮ Λ♯/Λ ∼

= Λ♯

✵/Λ✵ ∼

= Λ♯

✶/Λ✶✱

◮ |Λ♯/Λ| = ❞❡t ❙✳

◮ q❚ : Λ♯ ❚/Λ❚ → Q/Z, µ + Λ❚ → q❚(µ) + Z ◮ ❙ = ❆✸✿ Λ♯/Λ = ❆−✶ ✸ Z✸/Z✸ ✐s r❡♣r❡s❡♥t❡❞ ❜② (✵, ✵, ✵)✱ (✶ ✹, ✶ ✷, −✶ ✹)✱

(✶

✷, ✵, ✶ ✷)✱ (−✶ ✹, ✶ ✷, ✶ ✹) ✇✐t❤ ♥♦r♠ ✵✱ ✸ ✽✱ ✶ ✷✱ ✸ ✽✱ r❡s♣❡❝t✐✈❡❧②✳

■♥❣♦ ❑❧ö❝❦❡r ✭❘❲❚❍ ❆❛❝❤❡♥✮ ❏❛♣❛♥❡s❡✲●❡r♠❛♥ ◆✉♠❜❡r ❚❤❡♦r② ❲♦r❦s❤♦♣ ✽ ✴ ✷✾

▼♦❞✉❧❛r ❋♦r♠s ❢♦r ❖(✷, ✺) ❖rt❤♦❣♦♥❛❧ ▼♦❞✉❧❛r ❋♦r♠s

▲❛tt✐❝❡s ✐♥ ◗✉❛❞r❛t✐❝ ❙♣❛❝❡s

◮ Λ = Zℓ✱ Λ✵ = Zℓ+✷✱ Λ✶ = Zℓ+✹ ✭❧❛tt✐❝❡s ✐♥ q✉❛❞r❛t✐❝ s♣❛❝❡s

(Λ ⊗ R, (·, ·))✱ ✳✳✳✮

◮ ❉✉❛❧ ❧❛tt✐❝❡s✿ Λ♯ ❚ = {µ ∈ Λ ⊗ R; (λ, µ)❚ ∈ Z ❢♦r ❛❧❧ λ ∈ Λ} = ❚ −✶Λ ◮ ❲❡ ❤❛✈❡✿

◮ Λ♯ = ❙−✶Zℓ✱ Λ♯

✵ = Z × Λ♯ × Z✱ Λ♯ ✶ = Z × Λ♯ ✵ × Z✱

◮ Λ♯/Λ ∼

= Λ♯

✵/Λ✵ ∼

= Λ♯

✶/Λ✶✱

◮ |Λ♯/Λ| = ❞❡t ❙✳

◮ q❚ : Λ♯ ❚/Λ❚ → Q/Z, µ + Λ❚ → q❚(µ) + Z ◮ ❙ = ❆✸✿ Λ♯/Λ = ❆−✶ ✸ Z✸/Z✸ ✐s r❡♣r❡s❡♥t❡❞ ❜② (✵, ✵, ✵)✱ (✶ ✹, ✶ ✷, −✶ ✹)✱

(✶

✷, ✵, ✶ ✷)✱ (−✶ ✹, ✶ ✷, ✶ ✹) ✇✐t❤ ♥♦r♠ ✵✱ ✸ ✽✱ ✶ ✷✱ ✸ ✽✱ r❡s♣❡❝t✐✈❡❧②✳

■♥❣♦ ❑❧ö❝❦❡r ✭❘❲❚❍ ❆❛❝❤❡♥✮ ❏❛♣❛♥❡s❡✲●❡r♠❛♥ ◆✉♠❜❡r ❚❤❡♦r② ❲♦r❦s❤♦♣ ✽ ✴ ✷✾

▼♦❞✉❧❛r ❋♦r♠s ❢♦r ❖(✷, ✺) ❖rt❤♦❣♦♥❛❧ ▼♦❞✉❧❛r ❋♦r♠s

▲❛tt✐❝❡s ✐♥ ◗✉❛❞r❛t✐❝ ❙♣❛❝❡s

◮ Λ = Zℓ✱ Λ✵ = Zℓ+✷✱ Λ✶ = Zℓ+✹ ✭❧❛tt✐❝❡s ✐♥ q✉❛❞r❛t✐❝ s♣❛❝❡s

(Λ ⊗ R, (·, ·))✱ ✳✳✳✮

◮ ❉✉❛❧ ❧❛tt✐❝❡s✿ Λ♯ ❚ = {µ ∈ Λ ⊗ R; (λ, µ)❚ ∈ Z ❢♦r ❛❧❧ λ ∈ Λ} = ❚ −✶Λ ◮ ❲❡ ❤❛✈❡✿

◮ Λ♯ = ❙−✶Zℓ✱ Λ♯

✵ = Z × Λ♯ × Z✱ Λ♯ ✶ = Z × Λ♯ ✵ × Z✱

◮ Λ♯/Λ ∼

= Λ♯

✵/Λ✵ ∼

= Λ♯

✶/Λ✶✱

◮ |Λ♯/Λ| = ❞❡t ❙✳

◮ q❚ : Λ♯ ❚/Λ❚ → Q/Z, µ + Λ❚ → q❚(µ) + Z ◮ ❙ = ❆✸✿ Λ♯/Λ = ❆−✶ ✸ Z✸/Z✸ ✐s r❡♣r❡s❡♥t❡❞ ❜② (✵, ✵, ✵)✱ (✶ ✹, ✶ ✷, −✶ ✹)✱

(✶

✷, ✵, ✶ ✷)✱ (−✶ ✹, ✶ ✷, ✶ ✹) ✇✐t❤ ♥♦r♠ ✵✱ ✸ ✽✱ ✶ ✷✱ ✸ ✽✱ r❡s♣❡❝t✐✈❡❧②✳

■♥❣♦ ❑❧ö❝❦❡r ✭❘❲❚❍ ❆❛❝❤❡♥✮ ❏❛♣❛♥❡s❡✲●❡r♠❛♥ ◆✉♠❜❡r ❚❤❡♦r② ❲♦r❦s❤♦♣ ✽ ✴ ✷✾

▼♦❞✉❧❛r ❋♦r♠s ❢♦r ❖(✷, ✺) ❖rt❤♦❣♦♥❛❧ ▼♦❞✉❧❛r ❋♦r♠s

▲❛tt✐❝❡s ✐♥ ◗✉❛❞r❛t✐❝ ❙♣❛❝❡s

◮ Λ = Zℓ✱ Λ✵ = Zℓ+✷✱ Λ✶ = Zℓ+✹ ✭❧❛tt✐❝❡s ✐♥ q✉❛❞r❛t✐❝ s♣❛❝❡s

(Λ ⊗ R, (·, ·))✱ ✳✳✳✮

◮ ❉✉❛❧ ❧❛tt✐❝❡s✿ Λ♯ ❚ = {µ ∈ Λ ⊗ R; (λ, µ)❚ ∈ Z ❢♦r ❛❧❧ λ ∈ Λ} = ❚ −✶Λ ◮ ❲❡ ❤❛✈❡✿

◮ Λ♯ = ❙−✶Zℓ✱ Λ♯

✵ = Z × Λ♯ × Z✱ Λ♯ ✶ = Z × Λ♯ ✵ × Z✱

◮ Λ♯/Λ ∼

= Λ♯

✵/Λ✵ ∼

= Λ♯

✶/Λ✶✱

◮ |Λ♯/Λ| = ❞❡t ❙✳

◮ q❚ : Λ♯ ❚/Λ❚ → Q/Z, µ + Λ❚ → q❚(µ) + Z ◮ ❙ = ❆✸✿ Λ♯/Λ = ❆−✶ ✸ Z✸/Z✸ ✐s r❡♣r❡s❡♥t❡❞ ❜② (✵, ✵, ✵)✱ (✶ ✹, ✶ ✷, −✶ ✹)✱

(✶

✷, ✵, ✶ ✷)✱ (−✶ ✹, ✶ ✷, ✶ ✹) ✇✐t❤ ♥♦r♠ ✵✱ ✸ ✽✱ ✶ ✷✱ ✸ ✽✱ r❡s♣❡❝t✐✈❡❧②✳

■♥❣♦ ❑❧ö❝❦❡r ✭❘❲❚❍ ❆❛❝❤❡♥✮ ❏❛♣❛♥❡s❡✲●❡r♠❛♥ ◆✉♠❜❡r ❚❤❡♦r② ❲♦r❦s❤♦♣ ✽ ✴ ✷✾

▼♦❞✉❧❛r ❋♦r♠s ❢♦r ❖(✷, ✺) ❖rt❤♦❣♦♥❛❧ ▼♦❞✉❧❛r ❋♦r♠s

❖rt❤♦❣♦♥❛❧ ●r♦✉♣s ❛♥❞ t❤❡ ❍❛❧❢✲s♣❛❝❡

◮ ❖(❚; R) = {▼ ∈ ▼❛t(ℓ; R); ❚[▼] := t▼❚▼ = ❚}

= {▼ ∈ ▼❛t(ℓ; R); q❚(▼①) = q❚(①) ❢♦r ❛❧❧ ① ∈ Rℓ}.

◮ ❖(Λ) = {▼ ∈ ❖(❚; R); ▼Λ = Λ} ◮ P❙ = {✈ ∈ Rℓ+✷; q✵(✈) > ✵, t✈❙✵ ❡ > ✵} ✇❤❡r❡ ❡ = (✶, ✵, . . . , ✵, ✶) ◮ ❍❛❧❢ s♣❛❝❡✿ H❙ = {✇ = ✉ + ✐✈ ∈ Cℓ+✷; ✈ = ■♠(✇) ∈ P❙} ◮ ❖(❙✶; R) ❛❝ts ♦♥ H❙ ∪ (−H❙)✿

▼✇ = ❥(▼, ✇)−✶ · (−q✵(✇)❜ + ❆✇ + ❝) ❥(▼, ✇) = −γq✵(✇) + t❞✇ + δ ▼ =   α

t❛

β ❜ ❆ ❝ γ

t❞

δ  

◮ ❖+(❙✶; R) = {▼ ∈ ❖(❙✶; R); ▼H❙ = H❙} ◮ Γ❙ = ❖(Λ✶) ∩ ❖+(❙✶; R)

■♥❣♦ ❑❧ö❝❦❡r ✭❘❲❚❍ ❆❛❝❤❡♥✮ ❏❛♣❛♥❡s❡✲●❡r♠❛♥ ◆✉♠❜❡r ❚❤❡♦r② ❲♦r❦s❤♦♣ ✾ ✴ ✷✾

▼♦❞✉❧❛r ❋♦r♠s ❢♦r ❖(✷, ✺) ❖rt❤♦❣♦♥❛❧ ▼♦❞✉❧❛r ❋♦r♠s

❖rt❤♦❣♦♥❛❧ ●r♦✉♣s ❛♥❞ t❤❡ ❍❛❧❢✲s♣❛❝❡

◮ ❖(❚; R) = {▼ ∈ ▼❛t(ℓ; R); ❚[▼] := t▼❚▼ = ❚}

= {▼ ∈ ▼❛t(ℓ; R); q❚(▼①) = q❚(①) ❢♦r ❛❧❧ ① ∈ Rℓ}.

◮ ❖(Λ) = {▼ ∈ ❖(❚; R); ▼Λ = Λ} ◮ P❙ = {✈ ∈ Rℓ+✷; q✵(✈) > ✵, t✈❙✵ ❡ > ✵} ✇❤❡r❡ ❡ = (✶, ✵, . . . , ✵, ✶) ◮ ❍❛❧❢ s♣❛❝❡✿ H❙ = {✇ = ✉ + ✐✈ ∈ Cℓ+✷; ✈ = ■♠(✇) ∈ P❙} ◮ ❖(❙✶; R) ❛❝ts ♦♥ H❙ ∪ (−H❙)✿

▼✇ = ❥(▼, ✇)−✶ · (−q✵(✇)❜ + ❆✇ + ❝) ❥(▼, ✇) = −γq✵(✇) + t❞✇ + δ ▼ =   α

t❛

β ❜ ❆ ❝ γ

t❞

δ  

◮ ❖+(❙✶; R) = {▼ ∈ ❖(❙✶; R); ▼H❙ = H❙} ◮ Γ❙ = ❖(Λ✶) ∩ ❖+(❙✶; R)

■♥❣♦ ❑❧ö❝❦❡r ✭❘❲❚❍ ❆❛❝❤❡♥✮ ❏❛♣❛♥❡s❡✲●❡r♠❛♥ ◆✉♠❜❡r ❚❤❡♦r② ❲♦r❦s❤♦♣ ✾ ✴ ✷✾

▼♦❞✉❧❛r ❋♦r♠s ❢♦r ❖(✷, ✺) ❖rt❤♦❣♦♥❛❧ ▼♦❞✉❧❛r ❋♦r♠s

❖rt❤♦❣♦♥❛❧ ●r♦✉♣s ❛♥❞ t❤❡ ❍❛❧❢✲s♣❛❝❡

◮ ❖(❚; R) = {▼ ∈ ▼❛t(ℓ; R); ❚[▼] := t▼❚▼ = ❚}

= {▼ ∈ ▼❛t(ℓ; R); q❚(▼①) = q❚(①) ❢♦r ❛❧❧ ① ∈ Rℓ}.

◮ ❖(Λ) = {▼ ∈ ❖(❚; R); ▼Λ = Λ} ◮ P❙ = {✈ ∈ Rℓ+✷; q✵(✈) > ✵, t✈❙✵ ❡ > ✵} ✇❤❡r❡ ❡ = (✶, ✵, . . . , ✵, ✶) ◮ ❍❛❧❢ s♣❛❝❡✿ H❙ = {✇ = ✉ + ✐✈ ∈ Cℓ+✷; ✈ = ■♠(✇) ∈ P❙} ◮ ❖(❙✶; R) ❛❝ts ♦♥ H❙ ∪ (−H❙)✿

▼✇ = ❥(▼, ✇)−✶ · (−q✵(✇)❜ + ❆✇ + ❝) ❥(▼, ✇) = −γq✵(✇) + t❞✇ + δ ▼ =   α

t❛

β ❜ ❆ ❝ γ

t❞

δ  

◮ ❖+(❙✶; R) = {▼ ∈ ❖(❙✶; R); ▼H❙ = H❙} ◮ Γ❙ = ❖(Λ✶) ∩ ❖+(❙✶; R)

■♥❣♦ ❑❧ö❝❦❡r ✭❘❲❚❍ ❆❛❝❤❡♥✮ ❏❛♣❛♥❡s❡✲●❡r♠❛♥ ◆✉♠❜❡r ❚❤❡♦r② ❲♦r❦s❤♦♣ ✾ ✴ ✷✾

▼♦❞✉❧❛r ❋♦r♠s ❢♦r ❖(✷, ✺) ❖rt❤♦❣♦♥❛❧ ▼♦❞✉❧❛r ❋♦r♠s

❖rt❤♦❣♦♥❛❧ ●r♦✉♣s ❛♥❞ t❤❡ ❍❛❧❢✲s♣❛❝❡

◮ ❖(❚; R) = {▼ ∈ ▼❛t(ℓ; R); ❚[▼] := t▼❚▼ = ❚}

= {▼ ∈ ▼❛t(ℓ; R); q❚(▼①) = q❚(①) ❢♦r ❛❧❧ ① ∈ Rℓ}.

◮ ❖(Λ) = {▼ ∈ ❖(❚; R); ▼Λ = Λ} ◮ P❙ = {✈ ∈ Rℓ+✷; q✵(✈) > ✵, t✈❙✵ ❡ > ✵} ✇❤❡r❡ ❡ = (✶, ✵, . . . , ✵, ✶) ◮ ❍❛❧❢ s♣❛❝❡✿ H❙ = {✇ = ✉ + ✐✈ ∈ Cℓ+✷; ✈ = ■♠(✇) ∈ P❙} ◮ ❖(❙✶; R) ❛❝ts ♦♥ H❙ ∪ (−H❙)✿

▼✇ = ❥(▼, ✇)−✶ · (−q✵(✇)❜ + ❆✇ + ❝) ❥(▼, ✇) = −γq✵(✇) + t❞✇ + δ ▼ =   α

t❛

β ❜ ❆ ❝ γ

t❞

δ  

◮ ❖+(❙✶; R) = {▼ ∈ ❖(❙✶; R); ▼H❙ = H❙} ◮ Γ❙ = ❖(Λ✶) ∩ ❖+(❙✶; R)

■♥❣♦ ❑❧ö❝❦❡r ✭❘❲❚❍ ❆❛❝❤❡♥✮ ❏❛♣❛♥❡s❡✲●❡r♠❛♥ ◆✉♠❜❡r ❚❤❡♦r② ❲♦r❦s❤♦♣ ✾ ✴ ✷✾

▼♦❞✉❧❛r ❋♦r♠s ❢♦r ❖(✷, ✺) ❖rt❤♦❣♦♥❛❧ ▼♦❞✉❧❛r ❋♦r♠s

❖rt❤♦❣♦♥❛❧ ●r♦✉♣s ❛♥❞ t❤❡ ❍❛❧❢✲s♣❛❝❡

◮ ❖(❚; R) = {▼ ∈ ▼❛t(ℓ; R); ❚[▼] := t▼❚▼ = ❚}

= {▼ ∈ ▼❛t(ℓ; R); q❚(▼①) = q❚(①) ❢♦r ❛❧❧ ① ∈ Rℓ}.

◮ ❖(Λ) = {▼ ∈ ❖(❚; R); ▼Λ = Λ} ◮ P❙ = {✈ ∈ Rℓ+✷; q✵(✈) > ✵, t✈❙✵ ❡ > ✵} ✇❤❡r❡ ❡ = (✶, ✵, . . . , ✵, ✶) ◮ ❍❛❧❢ s♣❛❝❡✿ H❙ = {✇ = ✉ + ✐✈ ∈ Cℓ+✷; ✈ = ■♠(✇) ∈ P❙} ◮ ❖(❙✶; R) ❛❝ts ♦♥ H❙ ∪ (−H❙)✿

▼✇ = ❥(▼, ✇)−✶ · (−q✵(✇)❜ + ❆✇ + ❝) ❥(▼, ✇) = −γq✵(✇) + t❞✇ + δ ▼ =   α

t❛

β ❜ ❆ ❝ γ

t❞

δ  

◮ ❖+(❙✶; R) = {▼ ∈ ❖(❙✶; R); ▼H❙ = H❙} ◮ Γ❙ = ❖(Λ✶) ∩ ❖+(❙✶; R)

■♥❣♦ ❑❧ö❝❦❡r ✭❘❲❚❍ ❆❛❝❤❡♥✮ ❏❛♣❛♥❡s❡✲●❡r♠❛♥ ◆✉♠❜❡r ❚❤❡♦r② ❲♦r❦s❤♦♣ ✾ ✴ ✷✾

▼♦❞✉❧❛r ❋♦r♠s ❢♦r ❖(✷, ✺) ❖rt❤♦❣♦♥❛❧ ▼♦❞✉❧❛r ❋♦r♠s

❖rt❤♦❣♦♥❛❧ ●r♦✉♣s ❛♥❞ t❤❡ ❍❛❧❢✲s♣❛❝❡

◮ ❖(❚; R) = {▼ ∈ ▼❛t(ℓ; R); ❚[▼] := t▼❚▼ = ❚}

= {▼ ∈ ▼❛t(ℓ; R); q❚(▼①) = q❚(①) ❢♦r ❛❧❧ ① ∈ Rℓ}.

◮ ❖(Λ) = {▼ ∈ ❖(❚; R); ▼Λ = Λ} ◮ P❙ = {✈ ∈ Rℓ+✷; q✵(✈) > ✵, t✈❙✵ ❡ > ✵} ✇❤❡r❡ ❡ = (✶, ✵, . . . , ✵, ✶) ◮ ❍❛❧❢ s♣❛❝❡✿ H❙ = {✇ = ✉ + ✐✈ ∈ Cℓ+✷; ✈ = ■♠(✇) ∈ P❙} ◮ ❖(❙✶; R) ❛❝ts ♦♥ H❙ ∪ (−H❙)✿

▼✇ = ❥(▼, ✇)−✶ · (−q✵(✇)❜ + ❆✇ + ❝) ❥(▼, ✇) = −γq✵(✇) + t❞✇ + δ ▼ =   α

t❛

β ❜ ❆ ❝ γ

t❞

δ  

◮ ❖+(❙✶; R) = {▼ ∈ ❖(❙✶; R); ▼H❙ = H❙} ◮ Γ❙ = ❖(Λ✶) ∩ ❖+(❙✶; R)

■♥❣♦ ❑❧ö❝❦❡r ✭❘❲❚❍ ❆❛❝❤❡♥✮ ❏❛♣❛♥❡s❡✲●❡r♠❛♥ ◆✉♠❜❡r ❚❤❡♦r② ❲♦r❦s❤♦♣ ✾ ✴ ✷✾

▼♦❞✉❧❛r ❋♦r♠s ❢♦r ❖(✷, ✺) ❖rt❤♦❣♦♥❛❧ ▼♦❞✉❧❛r ❋♦r♠s

❖rt❤♦❣♦♥❛❧ ●r♦✉♣s ❛♥❞ t❤❡ ❍❛❧❢✲s♣❛❝❡

◮ ❖(❚; R) = {▼ ∈ ▼❛t(ℓ; R); ❚[▼] := t▼❚▼ = ❚}

= {▼ ∈ ▼❛t(ℓ; R); q❚(▼①) = q❚(①) ❢♦r ❛❧❧ ① ∈ Rℓ}.

◮ ❖(Λ) = {▼ ∈ ❖(❚; R); ▼Λ = Λ} ◮ P❙ = {✈ ∈ Rℓ+✷; q✵(✈) > ✵, t✈❙✵ ❡ > ✵} ✇❤❡r❡ ❡ = (✶, ✵, . . . , ✵, ✶) ◮ ❍❛❧❢ s♣❛❝❡✿ H❙ = {✇ = ✉ + ✐✈ ∈ Cℓ+✷; ✈ = ■♠(✇) ∈ P❙} ◮ ❖(❙✶; R) ❛❝ts ♦♥ H❙ ∪ (−H❙)✿

▼✇ = ❥(▼, ✇)−✶ · (−q✵(✇)❜ + ❆✇ + ❝) ❥(▼, ✇) = −γq✵(✇) + t❞✇ + δ ▼ =   α

t❛

β ❜ ❆ ❝ γ

t❞

δ  

◮ ❖+(❙✶; R) = {▼ ∈ ❖(❙✶; R); ▼H❙ = H❙} ◮ Γ❙ = ❖(Λ✶) ∩ ❖+(❙✶; R)

■♥❣♦ ❑❧ö❝❦❡r ✭❘❲❚❍ ❆❛❝❤❡♥✮ ❏❛♣❛♥❡s❡✲●❡r♠❛♥ ◆✉♠❜❡r ❚❤❡♦r② ❲♦r❦s❤♦♣ ✾ ✴ ✷✾

▼♦❞✉❧❛r ❋♦r♠s ❢♦r ❖(✷, ✺) ❖rt❤♦❣♦♥❛❧ ▼♦❞✉❧❛r ❋♦r♠s

Pr♦♣❡rt✐❡s ♦❢ t❤❡ ❖rt❤♦❣♦♥❛❧ ▼♦❞✉❧❛r ●r♦✉♣

◮ Γ❙ ✐s ✭✐♥ ♦✉r ❝❛s❡✮ ❣❡♥❡r❛t❡❞ ❜② ❏✱ ❚λ✱ λ ∈ Λ✵✱ ❛♥❞ ❘❆✱ ❆ ∈ ❖(Λ)✱

✇❤❡r❡ ❏✇ = −q✵(✇)−✶ · (τ✷, −③, τ✶) (✐♥✈❡rs✐♦♥), ❚λ✇ = ✇ + λ (tr❛♥s❧❛t✐♦♥), ❘❆✇ = (τ✶, ❆③, τ✷) (r♦t❛t✐♦♥).

◮ Γ❙ ❛❝ts ♦♥ Λ♯ ✶/Λ✶ ❜② ♠✉❧t✐♣❧✐❝❛t✐♦♥✳

■t ♣❡r♠✉t❡s ❡❧❡♠❡♥ts ♦❢ Λ♯

✶/Λ✶ ✇✐t❤ t❤❡ s❛♠❡ ♥♦r♠ ✭♠♦❞✉❧♦ Z✮✳

❚❤❡ s✐❣♥s ♦❢ t❤♦s❡ ♣❡r♠✉t❛t✐♦♥s ❛r❡ ❛❜❡❧✐❛♥ ❝❤❛r❛❝t❡rs ♦❢ Γ❙✳

◮ ❙ = ❆✸✿ ❆❜❡❧✐❛♥ ❝❤❛r❛❝t❡rs ♦❢ Γ❆✸✿

Γ❛❜

❆✸ = νπ, ❞❡t ,

✇❤❡r❡ νπ ✐s t❤❡ s✐❣♥ ♦❢ t❤❡ ♣❡r♠✉t❛t✐♦♥ ♦❢ t❤❡ t✇♦ ❡❧❡♠❡♥ts ♦❢ Λ♯/Λ ♦❢ ♥♦r♠ ✸

✽✳

■♥❣♦ ❑❧ö❝❦❡r ✭❘❲❚❍ ❆❛❝❤❡♥✮ ❏❛♣❛♥❡s❡✲●❡r♠❛♥ ◆✉♠❜❡r ❚❤❡♦r② ❲♦r❦s❤♦♣ ✶✵ ✴ ✷✾

▼♦❞✉❧❛r ❋♦r♠s ❢♦r ❖(✷, ✺) ❖rt❤♦❣♦♥❛❧ ▼♦❞✉❧❛r ❋♦r♠s

Pr♦♣❡rt✐❡s ♦❢ t❤❡ ❖rt❤♦❣♦♥❛❧ ▼♦❞✉❧❛r ●r♦✉♣

◮ Γ❙ ✐s ✭✐♥ ♦✉r ❝❛s❡✮ ❣❡♥❡r❛t❡❞ ❜② ❏✱ ❚λ✱ λ ∈ Λ✵✱ ❛♥❞ ❘❆✱ ❆ ∈ ❖(Λ)✱

✇❤❡r❡ ❏✇ = −q✵(✇)−✶ · (τ✷, −③, τ✶) (✐♥✈❡rs✐♦♥), ❚λ✇ = ✇ + λ (tr❛♥s❧❛t✐♦♥), ❘❆✇ = (τ✶, ❆③, τ✷) (r♦t❛t✐♦♥).

◮ Γ❙ ❛❝ts ♦♥ Λ♯ ✶/Λ✶ ❜② ♠✉❧t✐♣❧✐❝❛t✐♦♥✳

■t ♣❡r♠✉t❡s ❡❧❡♠❡♥ts ♦❢ Λ♯

✶/Λ✶ ✇✐t❤ t❤❡ s❛♠❡ ♥♦r♠ ✭♠♦❞✉❧♦ Z✮✳

❚❤❡ s✐❣♥s ♦❢ t❤♦s❡ ♣❡r♠✉t❛t✐♦♥s ❛r❡ ❛❜❡❧✐❛♥ ❝❤❛r❛❝t❡rs ♦❢ Γ❙✳

◮ ❙ = ❆✸✿ ❆❜❡❧✐❛♥ ❝❤❛r❛❝t❡rs ♦❢ Γ❆✸✿

Γ❛❜

❆✸ = νπ, ❞❡t ,

✇❤❡r❡ νπ ✐s t❤❡ s✐❣♥ ♦❢ t❤❡ ♣❡r♠✉t❛t✐♦♥ ♦❢ t❤❡ t✇♦ ❡❧❡♠❡♥ts ♦❢ Λ♯/Λ ♦❢ ♥♦r♠ ✸

✽✳

■♥❣♦ ❑❧ö❝❦❡r ✭❘❲❚❍ ❆❛❝❤❡♥✮ ❏❛♣❛♥❡s❡✲●❡r♠❛♥ ◆✉♠❜❡r ❚❤❡♦r② ❲♦r❦s❤♦♣ ✶✵ ✴ ✷✾

▼♦❞✉❧❛r ❋♦r♠s ❢♦r ❖(✷, ✺) ❖rt❤♦❣♦♥❛❧ ▼♦❞✉❧❛r ❋♦r♠s

Pr♦♣❡rt✐❡s ♦❢ t❤❡ ❖rt❤♦❣♦♥❛❧ ▼♦❞✉❧❛r ●r♦✉♣

◮ Γ❙ ✐s ✭✐♥ ♦✉r ❝❛s❡✮ ❣❡♥❡r❛t❡❞ ❜② ❏✱ ❚λ✱ λ ∈ Λ✵✱ ❛♥❞ ❘❆✱ ❆ ∈ ❖(Λ)✱

✇❤❡r❡ ❏✇ = −q✵(✇)−✶ · (τ✷, −③, τ✶) (✐♥✈❡rs✐♦♥), ❚λ✇ = ✇ + λ (tr❛♥s❧❛t✐♦♥), ❘❆✇ = (τ✶, ❆③, τ✷) (r♦t❛t✐♦♥).

◮ Γ❙ ❛❝ts ♦♥ Λ♯ ✶/Λ✶ ❜② ♠✉❧t✐♣❧✐❝❛t✐♦♥✳

■t ♣❡r♠✉t❡s ❡❧❡♠❡♥ts ♦❢ Λ♯

✶/Λ✶ ✇✐t❤ t❤❡ s❛♠❡ ♥♦r♠ ✭♠♦❞✉❧♦ Z✮✳

❚❤❡ s✐❣♥s ♦❢ t❤♦s❡ ♣❡r♠✉t❛t✐♦♥s ❛r❡ ❛❜❡❧✐❛♥ ❝❤❛r❛❝t❡rs ♦❢ Γ❙✳

◮ ❙ = ❆✸✿ ❆❜❡❧✐❛♥ ❝❤❛r❛❝t❡rs ♦❢ Γ❆✸✿

Γ❛❜

❆✸ = νπ, ❞❡t ,

✇❤❡r❡ νπ ✐s t❤❡ s✐❣♥ ♦❢ t❤❡ ♣❡r♠✉t❛t✐♦♥ ♦❢ t❤❡ t✇♦ ❡❧❡♠❡♥ts ♦❢ Λ♯/Λ ♦❢ ♥♦r♠ ✸

✽✳

■♥❣♦ ❑❧ö❝❦❡r ✭❘❲❚❍ ❆❛❝❤❡♥✮ ❏❛♣❛♥❡s❡✲●❡r♠❛♥ ◆✉♠❜❡r ❚❤❡♦r② ❲♦r❦s❤♦♣ ✶✵ ✴ ✷✾

▼♦❞✉❧❛r ❋♦r♠s ❢♦r ❖(✷, ✺) ❖rt❤♦❣♦♥❛❧ ▼♦❞✉❧❛r ❋♦r♠s

❲❤❛t ✐s ❛♥ ❖rt❤♦❣♦♥❛❧ ▼♦❞✉❧❛r ❋♦r♠❄

❉❡✜♥✐t✐♦♥

❆♥ ✭♦rt❤♦❣♦♥❛❧✮ ♠♦❞✉❧❛r ❢♦r♠ ♦❢ ✇❡✐❣❤t ❦ ∈ Z ✇✐t❤ r❡s♣❡❝t t♦ ❛ s✉❜❣r♦✉♣ Γ ♦❢ Γ❙ ♦❢ ✜♥✐t❡ ✐♥❞❡① ❛♥❞ ❛♥ ❛❜❡❧✐❛♥ ❝❤❛r❛❝t❡r ν : Γ → C× ♦❢ ✜♥✐t❡ ♦r❞❡r ✐s ❛ ❤♦❧♦♠♦r♣❤✐❝ ❢✉♥❝t✐♦♥ ❢ : H❙ → C s❛t✐❢②✐♥❣ ❢ (▼✇) = ν(▼) ❥(▼, ✇)❦ ❢ (✇) ❢♦r ❛❧❧ ✇ ∈ H❙ ❛♥❞ ▼ ∈ Γ. ❲❡ ❞❡♥♦t❡ t❤❡ ✈❡❝t♦r s♣❛❝❡ ♦❢ ❛❧❧ s✉❝❤ ❢✉♥❝t✐♦♥s ❜② [Γ, ❦, ν]✳

◮ ■❢ −■ ∈ Γ ❛♥❞ ν(−■) = (−✶)❦ t❤❡♥ [Γ, ❦, ν] = {✵}✳ ◮ ■❢ ❦ < ✵ t❤❡♥ [Γ, ❦, ν] = {✵}✳

■♥❣♦ ❑❧ö❝❦❡r ✭❘❲❚❍ ❆❛❝❤❡♥✮ ❏❛♣❛♥❡s❡✲●❡r♠❛♥ ◆✉♠❜❡r ❚❤❡♦r② ❲♦r❦s❤♦♣ ✶✶ ✴ ✷✾

▼♦❞✉❧❛r ❋♦r♠s ❢♦r ❖(✷, ✺) ❖rt❤♦❣♦♥❛❧ ▼♦❞✉❧❛r ❋♦r♠s

❲❤❛t ✐s ❛♥ ❖rt❤♦❣♦♥❛❧ ▼♦❞✉❧❛r ❋♦r♠❄

❉❡✜♥✐t✐♦♥

❆♥ ✭♦rt❤♦❣♦♥❛❧✮ ♠♦❞✉❧❛r ❢♦r♠ ♦❢ ✇❡✐❣❤t ❦ ∈ Z ✇✐t❤ r❡s♣❡❝t t♦ ❛ s✉❜❣r♦✉♣ Γ ♦❢ Γ❙ ♦❢ ✜♥✐t❡ ✐♥❞❡① ❛♥❞ ❛♥ ❛❜❡❧✐❛♥ ❝❤❛r❛❝t❡r ν : Γ → C× ♦❢ ✜♥✐t❡ ♦r❞❡r ✐s ❛ ❤♦❧♦♠♦r♣❤✐❝ ❢✉♥❝t✐♦♥ ❢ : H❙ → C s❛t✐❢②✐♥❣ ❢ (▼✇) = ν(▼) ❥(▼, ✇)❦ ❢ (✇) ❢♦r ❛❧❧ ✇ ∈ H❙ ❛♥❞ ▼ ∈ Γ. ❲❡ ❞❡♥♦t❡ t❤❡ ✈❡❝t♦r s♣❛❝❡ ♦❢ ❛❧❧ s✉❝❤ ❢✉♥❝t✐♦♥s ❜② [Γ, ❦, ν]✳

◮ ■❢ −■ ∈ Γ ❛♥❞ ν(−■) = (−✶)❦ t❤❡♥ [Γ, ❦, ν] = {✵}✳ ◮ ■❢ ❦ < ✵ t❤❡♥ [Γ, ❦, ν] = {✵}✳

■♥❣♦ ❑❧ö❝❦❡r ✭❘❲❚❍ ❆❛❝❤❡♥✮ ❏❛♣❛♥❡s❡✲●❡r♠❛♥ ◆✉♠❜❡r ❚❤❡♦r② ❲♦r❦s❤♦♣ ✶✶ ✴ ✷✾

▼♦❞✉❧❛r ❋♦r♠s ❢♦r ❖(✷, ✺) ❖rt❤♦❣♦♥❛❧ ▼♦❞✉❧❛r ❋♦r♠s

❲❤❛t ✐s ❛♥ ❖rt❤♦❣♦♥❛❧ ▼♦❞✉❧❛r ❋♦r♠❄

❉❡✜♥✐t✐♦♥

❆♥ ✭♦rt❤♦❣♦♥❛❧✮ ♠♦❞✉❧❛r ❢♦r♠ ♦❢ ✇❡✐❣❤t ❦ ∈ Z ✇✐t❤ r❡s♣❡❝t t♦ ❛ s✉❜❣r♦✉♣ Γ ♦❢ Γ❙ ♦❢ ✜♥✐t❡ ✐♥❞❡① ❛♥❞ ❛♥ ❛❜❡❧✐❛♥ ❝❤❛r❛❝t❡r ν : Γ → C× ♦❢ ✜♥✐t❡ ♦r❞❡r ✐s ❛ ❤♦❧♦♠♦r♣❤✐❝ ❢✉♥❝t✐♦♥ ❢ : H❙ → C s❛t✐❢②✐♥❣ ❢ (▼✇) = ν(▼) ❥(▼, ✇)❦ ❢ (✇) ❢♦r ❛❧❧ ✇ ∈ H❙ ❛♥❞ ▼ ∈ Γ. ❲❡ ❞❡♥♦t❡ t❤❡ ✈❡❝t♦r s♣❛❝❡ ♦❢ ❛❧❧ s✉❝❤ ❢✉♥❝t✐♦♥s ❜② [Γ, ❦, ν]✳

◮ ■❢ −■ ∈ Γ ❛♥❞ ν(−■) = (−✶)❦ t❤❡♥ [Γ, ❦, ν] = {✵}✳ ◮ ■❢ ❦ < ✵ t❤❡♥ [Γ, ❦, ν] = {✵}✳

■♥❣♦ ❑❧ö❝❦❡r ✭❘❲❚❍ ❆❛❝❤❡♥✮ ❏❛♣❛♥❡s❡✲●❡r♠❛♥ ◆✉♠❜❡r ❚❤❡♦r② ❲♦r❦s❤♦♣ ✶✶ ✴ ✷✾

▼♦❞✉❧❛r ❋♦r♠s ❢♦r ❖(✷, ✺) ❖rt❤♦❣♦♥❛❧ ▼♦❞✉❧❛r ❋♦r♠s

❲❤❛t ✐s ❖✉r ●♦❛❧❄

Pr♦❞✉❝ts ♦❢ ♠♦❞✉❧❛r ❢♦r♠s ❛r❡ ❛❣❛✐♥ ♠♦❞✉❧❛r ❢♦r♠s✳ ❚❤✉s A(Γ❙) =

• ❦∈Z

[Γ❙, ❦, ✶] ❛♥❞ A(Γ′

❙) =

• ❦∈Z

[Γ′

❙, ❦, ✶] =

• ❦∈Z
• ν∈Γ❛❜

❙

[Γ❙, ❦, ν] ❢♦r♠ ❣r❛❞❡❞ r✐♥❣s✳

• ♦❛❧✿ ❉❡t❡r♠✐♥❡ ❣❡♥❡r❛t♦rs ❛♥❞ ❛❧❣❡❜r❛✐❝ str✉❝t✉r❡ ♦❢ A(Γ❆✸) ❛♥❞ A(Γ′

❆✸)✳

❉✉❡ t♦ −■ ∈ Γ❆✸ ❛♥❞ νπ(−■) = ❞❡t(−■) = −✶ ✇❡ ❣❡t ❛ ✜rst r❡s✉❧t✿

◮ ■❢ ❦ ✐s ❡✈❡♥ t❤❡♥ [Γ❆✸, ❦, νπ] = [Γ❆✸, ❦, ❞❡t] = {✵}✳ ◮ ■❢ ❦ ✐s ♦❞❞ t❤❡♥ [Γ❆✸, ❦, ✶] = [Γ❆✸, ❦, νπ ❞❡t] = {✵}✳

■♥❣♦ ❑❧ö❝❦❡r ✭❘❲❚❍ ❆❛❝❤❡♥✮ ❏❛♣❛♥❡s❡✲●❡r♠❛♥ ◆✉♠❜❡r ❚❤❡♦r② ❲♦r❦s❤♦♣ ✶✷ ✴ ✷✾

▼♦❞✉❧❛r ❋♦r♠s ❢♦r ❖(✷, ✺) ❖rt❤♦❣♦♥❛❧ ▼♦❞✉❧❛r ❋♦r♠s

❲❤❛t ✐s ❖✉r ●♦❛❧❄

Pr♦❞✉❝ts ♦❢ ♠♦❞✉❧❛r ❢♦r♠s ❛r❡ ❛❣❛✐♥ ♠♦❞✉❧❛r ❢♦r♠s✳ ❚❤✉s A(Γ❙) =

• ❦∈Z

[Γ❙, ❦, ✶] ❛♥❞ A(Γ′

❙) =

• ❦∈Z

[Γ′

❙, ❦, ✶] =

• ❦∈Z
• ν∈Γ❛❜

❙

[Γ❙, ❦, ν] ❢♦r♠ ❣r❛❞❡❞ r✐♥❣s✳

• ♦❛❧✿ ❉❡t❡r♠✐♥❡ ❣❡♥❡r❛t♦rs ❛♥❞ ❛❧❣❡❜r❛✐❝ str✉❝t✉r❡ ♦❢ A(Γ❆✸) ❛♥❞ A(Γ′

❆✸)✳

❉✉❡ t♦ −■ ∈ Γ❆✸ ❛♥❞ νπ(−■) = ❞❡t(−■) = −✶ ✇❡ ❣❡t ❛ ✜rst r❡s✉❧t✿

◮ ■❢ ❦ ✐s ❡✈❡♥ t❤❡♥ [Γ❆✸, ❦, νπ] = [Γ❆✸, ❦, ❞❡t] = {✵}✳ ◮ ■❢ ❦ ✐s ♦❞❞ t❤❡♥ [Γ❆✸, ❦, ✶] = [Γ❆✸, ❦, νπ ❞❡t] = {✵}✳

■♥❣♦ ❑❧ö❝❦❡r ✭❘❲❚❍ ❆❛❝❤❡♥✮ ❏❛♣❛♥❡s❡✲●❡r♠❛♥ ◆✉♠❜❡r ❚❤❡♦r② ❲♦r❦s❤♦♣ ✶✷ ✴ ✷✾

▼♦❞✉❧❛r ❋♦r♠s ❢♦r ❖(✷, ✺) ❖rt❤♦❣♦♥❛❧ ▼♦❞✉❧❛r ❋♦r♠s

❲❤❛t ✐s ❖✉r ●♦❛❧❄

Pr♦❞✉❝ts ♦❢ ♠♦❞✉❧❛r ❢♦r♠s ❛r❡ ❛❣❛✐♥ ♠♦❞✉❧❛r ❢♦r♠s✳ ❚❤✉s A(Γ❙) =

• ❦∈Z

[Γ❙, ❦, ✶] ❛♥❞ A(Γ′

❙) =

• ❦∈Z

[Γ′

❙, ❦, ✶] =

• ❦∈Z
• ν∈Γ❛❜

❙

[Γ❙, ❦, ν] ❢♦r♠ ❣r❛❞❡❞ r✐♥❣s✳

• ♦❛❧✿ ❉❡t❡r♠✐♥❡ ❣❡♥❡r❛t♦rs ❛♥❞ ❛❧❣❡❜r❛✐❝ str✉❝t✉r❡ ♦❢ A(Γ❆✸) ❛♥❞ A(Γ′

❆✸)✳

❉✉❡ t♦ −■ ∈ Γ❆✸ ❛♥❞ νπ(−■) = ❞❡t(−■) = −✶ ✇❡ ❣❡t ❛ ✜rst r❡s✉❧t✿

◮ ■❢ ❦ ✐s ❡✈❡♥ t❤❡♥ [Γ❆✸, ❦, νπ] = [Γ❆✸, ❦, ❞❡t] = {✵}✳ ◮ ■❢ ❦ ✐s ♦❞❞ t❤❡♥ [Γ❆✸, ❦, ✶] = [Γ❆✸, ❦, νπ ❞❡t] = {✵}✳

■♥❣♦ ❑❧ö❝❦❡r ✭❘❲❚❍ ❆❛❝❤❡♥✮ ❏❛♣❛♥❡s❡✲●❡r♠❛♥ ◆✉♠❜❡r ❚❤❡♦r② ❲♦r❦s❤♦♣ ✶✷ ✴ ✷✾

▼♦❞✉❧❛r ❋♦r♠s ❢♦r ❖(✷, ✺) ❱❡❝t♦r✲✈❛❧✉❡❞ ▼♦❞✉❧❛r ❋♦r♠s

❖✉t❧✐♥❡

■♥tr♦❞✉❝t✐♦♥ ❖rt❤♦❣♦♥❛❧ ▼♦❞✉❧❛r ❋♦r♠s ❱❡❝t♦r✲✈❛❧✉❡❞ ▼♦❞✉❧❛r ❋♦r♠s ❇♦r❝❤❡r❞s Pr♦❞✉❝ts ❚❤❡ ●r❛❞❡❞ ❘✐♥❣ ♦❢ ▼♦❞✉❧❛r ❋♦r♠s

■♥❣♦ ❑❧ö❝❦❡r ✭❘❲❚❍ ❆❛❝❤❡♥✮ ❏❛♣❛♥❡s❡✲●❡r♠❛♥ ◆✉♠❜❡r ❚❤❡♦r② ❲♦r❦s❤♦♣ ✶✸ ✴ ✷✾

▼♦❞✉❧❛r ❋♦r♠s ❢♦r ❖(✷, ✺) ❱❡❝t♦r✲✈❛❧✉❡❞ ▼♦❞✉❧❛r ❋♦r♠s

❚❤❡ ▼❡t❛♣❧❡❝t✐❝ ●r♦✉♣

❚❤❡ ♠❡t❛♣❧❡❝t✐❝ ❣r♦✉♣ ▼♣✷(Z) ✐s ❣✐✈❡♥ ❜②

• (▼, ϕ) ; ▼ =

❛ ❜

❝ ❞

• ∈ ❙▲✷(Z), ϕ : H → C ❤♦❧♦♠✳, ϕ✷(τ) = ❝τ + ❞
• .

■t ♦♣❡r❛t❡s ♦♥ t❤❡ ✉♣♣❡r ❤❛❧❢✲♣❧❛♥❡ H ✈✐❛ (▼, ϕ)τ = ▼τ = ❛τ + ❜ ❝τ + ❞ ❛♥❞ ✐s ❣❡♥❡r❛t❡❞ ❜② ❚ = ✶ ✶ ✵ ✶

• , ✶
• ❛♥❞

❏ = ✵ −✶ ✶ ✵

• , √τ
• .

■♥❣♦ ❑❧ö❝❦❡r ✭❘❲❚❍ ❆❛❝❤❡♥✮ ❏❛♣❛♥❡s❡✲●❡r♠❛♥ ◆✉♠❜❡r ❚❤❡♦r② ❲♦r❦s❤♦♣ ✶✹ ✴ ✷✾

▼♦❞✉❧❛r ❋♦r♠s ❢♦r ❖(✷, ✺) ❱❡❝t♦r✲✈❛❧✉❡❞ ▼♦❞✉❧❛r ❋♦r♠s

❚❤❡ ▼❡t❛♣❧❡❝t✐❝ ●r♦✉♣

❚❤❡ ♠❡t❛♣❧❡❝t✐❝ ❣r♦✉♣ ▼♣✷(Z) ✐s ❣✐✈❡♥ ❜②

• (▼, ϕ) ; ▼ =

❛ ❜

❝ ❞

• ∈ ❙▲✷(Z), ϕ : H → C ❤♦❧♦♠✳, ϕ✷(τ) = ❝τ + ❞
• .

■t ♦♣❡r❛t❡s ♦♥ t❤❡ ✉♣♣❡r ❤❛❧❢✲♣❧❛♥❡ H ✈✐❛ (▼, ϕ)τ = ▼τ = ❛τ + ❜ ❝τ + ❞ ❛♥❞ ✐s ❣❡♥❡r❛t❡❞ ❜② ❚ = ✶ ✶ ✵ ✶

• , ✶
• ❛♥❞

❏ = ✵ −✶ ✶ ✵

• , √τ
• .

■♥❣♦ ❑❧ö❝❦❡r ✭❘❲❚❍ ❆❛❝❤❡♥✮ ❏❛♣❛♥❡s❡✲●❡r♠❛♥ ◆✉♠❜❡r ❚❤❡♦r② ❲♦r❦s❤♦♣ ✶✹ ✴ ✷✾

▼♦❞✉❧❛r ❋♦r♠s ❢♦r ❖(✷, ✺) ❱❡❝t♦r✲✈❛❧✉❡❞ ▼♦❞✉❧❛r ❋♦r♠s

❚❤❡ ▼❡t❛♣❧❡❝t✐❝ ●r♦✉♣

❚❤❡ ♠❡t❛♣❧❡❝t✐❝ ❣r♦✉♣ ▼♣✷(Z) ✐s ❣✐✈❡♥ ❜②

• (▼, ϕ) ; ▼ =

❛ ❜

❝ ❞

• ∈ ❙▲✷(Z), ϕ : H → C ❤♦❧♦♠✳, ϕ✷(τ) = ❝τ + ❞
• .

■t ♦♣❡r❛t❡s ♦♥ t❤❡ ✉♣♣❡r ❤❛❧❢✲♣❧❛♥❡ H ✈✐❛ (▼, ϕ)τ = ▼τ = ❛τ + ❜ ❝τ + ❞ ❛♥❞ ✐s ❣❡♥❡r❛t❡❞ ❜② ❚ = ✶ ✶ ✵ ✶

• , ✶
• ❛♥❞

❏ = ✵ −✶ ✶ ✵

• , √τ
• .

■♥❣♦ ❑❧ö❝❦❡r ✭❘❲❚❍ ❆❛❝❤❡♥✮ ❏❛♣❛♥❡s❡✲●❡r♠❛♥ ◆✉♠❜❡r ❚❤❡♦r② ❲♦r❦s❤♦♣ ✶✹ ✴ ✷✾

▼♦❞✉❧❛r ❋♦r♠s ❢♦r ❖(✷, ✺) ❱❡❝t♦r✲✈❛❧✉❡❞ ▼♦❞✉❧❛r ❋♦r♠s

❚❤❡ ❲❡✐❧ ❘❡♣r❡s❡♥t❛t✐♦♥

▲❡t

◮ ❙ ∈ ❙②♠(ℓ; R) ❜❡ ❛ s②♠♠❡tr✐❝✱ ❡✈❡♥ ♠❛tr✐① ♦❢ s✐❣♥❛t✉r❡ (❜+, ❜−)✱ ◮ Λ = Zℓ✱ ◮ (·, ·) = (·, ·)❙✱ ◮ (❡µ)µ∈Λ♯/Λ ❜❡ t❤❡ st❛♥❞❛r❞ ❜❛s✐s ♦❢ t❤❡ ❣r♦✉♣ r✐♥❣ C[Λ♯/Λ]✳

❚❤❡ ❲❡✐❧ r❡♣r❡s❡♥t❛t✐♦♥ ρ❙ ♦❢ ▼♣✷(Z) ♦♥ C[Λ♯/Λ] ✐s ❞❡✜♥❡❞ ❜② ρ❙(❚) ❡µ = ❡π✐(µ,µ) ❡µ, ρ❙(❏) ❡µ = √ ✐

❜−−❜+

• | ❞❡t ❙|
• ν∈Λ♯/Λ

❡−✷π✐(µ,ν) ❡ν.

■♥❣♦ ❑❧ö❝❦❡r ✭❘❲❚❍ ❆❛❝❤❡♥✮ ❏❛♣❛♥❡s❡✲●❡r♠❛♥ ◆✉♠❜❡r ❚❤❡♦r② ❲♦r❦s❤♦♣ ✶✺ ✴ ✷✾

▼♦❞✉❧❛r ❋♦r♠s ❢♦r ❖(✷, ✺) ❱❡❝t♦r✲✈❛❧✉❡❞ ▼♦❞✉❧❛r ❋♦r♠s

❱❡❝t♦r✲✈❛❧✉❡❞ ▼♦❞✉❧❛r ❋♦r♠s

❉❡✜♥✐t✐♦♥

❆ ❤♦❧♦♠♦r♣❤✐❝ ❢✉♥❝t✐♦♥ ❢ : H → C[Λ♯/Λ] ✐s ❛ ✈❡❝t♦r✲✈❛❧✉❡❞ ♠♦❞✉❧❛r ❢♦r♠ ♦❢ ✇❡✐❣❤t ❦ ∈ ✶

✷Z ✇✐t❤ r❡s♣❡❝t t♦ ρ❙ ✐❢

❢ (▼τ) = ϕ(τ)✷❦ ρ❙(▼, ϕ) ❢ (τ), ❢♦r ❛❧❧ (▼, ϕ) ∈ ▼♣✷(Z) ❛♥❞ ✐❢ ❢ ❤❛s ❛ ❋♦✉r✐❡r ❡①♣❛♥s✐♦♥ ♦❢ t❤❡ ❢♦r♠ ❢ (τ) =

• µ∈Λ♯/Λ
• ♥∈q❙(µ)+Z

♥≥♥✵

❝µ(♥)q♥ ❡µ.

◮ ♥✵ ≥ ✵✿ ❍♦❧♦♠♦r♣❤✐❝ ♠♦❞✉❧❛r ❢♦r♠s✱ [▼♣✷(Z), ❦, ρ❙]✱ ◮ ♥✵ < ✵✿ ◆❡❛r❧② ❤♦❧♦♠♦r♣❤✐❝ ♠♦❞✉❧❛r ❢♦r♠s✱ [▼♣✷(Z), ❦, ρ❙]∞✳

■♥❣♦ ❑❧ö❝❦❡r ✭❘❲❚❍ ❆❛❝❤❡♥✮ ❏❛♣❛♥❡s❡✲●❡r♠❛♥ ◆✉♠❜❡r ❚❤❡♦r② ❲♦r❦s❤♦♣ ✶✻ ✴ ✷✾

▼♦❞✉❧❛r ❋♦r♠s ❢♦r ❖(✷, ✺) ❱❡❝t♦r✲✈❛❧✉❡❞ ▼♦❞✉❧❛r ❋♦r♠s

❲❤❛t t♦ ❑♥♦✇ ❆❜♦✉t ❱❡❝t♦r✲✈❛❧✉❡❞ ▼♦❞✉❧❛r ❋♦r♠s

◮ ◆❡❛r❧② ❤♦❧♦♠♦r♣❤✐❝ ♠♦❞✉❧❛r ❢♦r♠s ❛r❡ ✉♥✐q✉❡❧② ❞❡t❡r♠✐♥❡❞ ❜② t❤❡✐r

♣r✐♥❝✐♣❛❧ ♣❛rt

• µ∈Λ♯/Λ
• ♥∈q❙(µ)+Z

♥≤✵

❝µ(♥)q♥ ❡µ.

◮ ❙❦♦r✉♣♣❛✿ ❋♦r♠✉❧❛ ❢♦r ❞✐♠❡♥s✐♦♥ ♦❢ [▼♣✷(Z), ❦, ρ❙] ❢♦r ❦ ≥ ✷✳ ◮ ❊①❛♠♣❧❡s✿

◮ ❊✐s❡♥st❡✐♥ s❡r✐❡s ✭❦ ∈ ✶

✷Z✱ ❦ > ✷✮

❊❦(τ) = ✶ ✷

• (▼,ϕ)∈❚\ ▼♣✷(Z)

ϕ(τ)−✷❦ ρ❙(▼, ϕ)−✶❡✵. ❇r✉✐♥✐❡r✱ ❑✉ss✿ ❋♦r♠✉❧❛ ❢♦r ❋♦✉r✐❡r ❝♦❡✣❝✐❡♥ts ♦❢ ❊❦✳

◮ ❚❤❡t❛ s❡r✐❡s✳ ■♥❣♦ ❑❧ö❝❦❡r ✭❘❲❚❍ ❆❛❝❤❡♥✮ ❏❛♣❛♥❡s❡✲●❡r♠❛♥ ◆✉♠❜❡r ❚❤❡♦r② ❲♦r❦s❤♦♣ ✶✼ ✴ ✷✾

▼♦❞✉❧❛r ❋♦r♠s ❢♦r ❖(✷, ✺) ❱❡❝t♦r✲✈❛❧✉❡❞ ▼♦❞✉❧❛r ❋♦r♠s

❲❤❛t t♦ ❑♥♦✇ ❆❜♦✉t ❱❡❝t♦r✲✈❛❧✉❡❞ ▼♦❞✉❧❛r ❋♦r♠s

◮ ◆❡❛r❧② ❤♦❧♦♠♦r♣❤✐❝ ♠♦❞✉❧❛r ❢♦r♠s ❛r❡ ✉♥✐q✉❡❧② ❞❡t❡r♠✐♥❡❞ ❜② t❤❡✐r

♣r✐♥❝✐♣❛❧ ♣❛rt

• µ∈Λ♯/Λ
• ♥∈q❙(µ)+Z

♥≤✵

❝µ(♥)q♥ ❡µ.

◮ ❙❦♦r✉♣♣❛✿ ❋♦r♠✉❧❛ ❢♦r ❞✐♠❡♥s✐♦♥ ♦❢ [▼♣✷(Z), ❦, ρ❙] ❢♦r ❦ ≥ ✷✳ ◮ ❊①❛♠♣❧❡s✿

◮ ❊✐s❡♥st❡✐♥ s❡r✐❡s ✭❦ ∈ ✶

✷Z✱ ❦ > ✷✮

❊❦(τ) = ✶ ✷

• (▼,ϕ)∈❚\ ▼♣✷(Z)

ϕ(τ)−✷❦ ρ❙(▼, ϕ)−✶❡✵. ❇r✉✐♥✐❡r✱ ❑✉ss✿ ❋♦r♠✉❧❛ ❢♦r ❋♦✉r✐❡r ❝♦❡✣❝✐❡♥ts ♦❢ ❊❦✳

◮ ❚❤❡t❛ s❡r✐❡s✳ ■♥❣♦ ❑❧ö❝❦❡r ✭❘❲❚❍ ❆❛❝❤❡♥✮ ❏❛♣❛♥❡s❡✲●❡r♠❛♥ ◆✉♠❜❡r ❚❤❡♦r② ❲♦r❦s❤♦♣ ✶✼ ✴ ✷✾

▼♦❞✉❧❛r ❋♦r♠s ❢♦r ❖(✷, ✺) ❱❡❝t♦r✲✈❛❧✉❡❞ ▼♦❞✉❧❛r ❋♦r♠s

❲❤❛t t♦ ❑♥♦✇ ❆❜♦✉t ❱❡❝t♦r✲✈❛❧✉❡❞ ▼♦❞✉❧❛r ❋♦r♠s

◮ ◆❡❛r❧② ❤♦❧♦♠♦r♣❤✐❝ ♠♦❞✉❧❛r ❢♦r♠s ❛r❡ ✉♥✐q✉❡❧② ❞❡t❡r♠✐♥❡❞ ❜② t❤❡✐r

♣r✐♥❝✐♣❛❧ ♣❛rt

• µ∈Λ♯/Λ
• ♥∈q❙(µ)+Z

♥≤✵

❝µ(♥)q♥ ❡µ.

◮ ❙❦♦r✉♣♣❛✿ ❋♦r♠✉❧❛ ❢♦r ❞✐♠❡♥s✐♦♥ ♦❢ [▼♣✷(Z), ❦, ρ❙] ❢♦r ❦ ≥ ✷✳ ◮ ❊①❛♠♣❧❡s✿

◮ ❊✐s❡♥st❡✐♥ s❡r✐❡s ✭❦ ∈ ✶

✷Z✱ ❦ > ✷✮

❊❦(τ) = ✶ ✷

• (▼,ϕ)∈❚\ ▼♣✷(Z)

ϕ(τ)−✷❦ ρ❙(▼, ϕ)−✶❡✵. ❇r✉✐♥✐❡r✱ ❑✉ss✿ ❋♦r♠✉❧❛ ❢♦r ❋♦✉r✐❡r ❝♦❡✣❝✐❡♥ts ♦❢ ❊❦✳

◮ ❚❤❡t❛ s❡r✐❡s✳ ■♥❣♦ ❑❧ö❝❦❡r ✭❘❲❚❍ ❆❛❝❤❡♥✮ ❏❛♣❛♥❡s❡✲●❡r♠❛♥ ◆✉♠❜❡r ❚❤❡♦r② ❲♦r❦s❤♦♣ ✶✼ ✴ ✷✾

▼♦❞✉❧❛r ❋♦r♠s ❢♦r ❖(✷, ✺) ❇♦r❝❤❡r❞s Pr♦❞✉❝ts

❖✉t❧✐♥❡

■♥tr♦❞✉❝t✐♦♥ ❖rt❤♦❣♦♥❛❧ ▼♦❞✉❧❛r ❋♦r♠s ❱❡❝t♦r✲✈❛❧✉❡❞ ▼♦❞✉❧❛r ❋♦r♠s ❇♦r❝❤❡r❞s Pr♦❞✉❝ts ❚❤❡ ●r❛❞❡❞ ❘✐♥❣ ♦❢ ▼♦❞✉❧❛r ❋♦r♠s

■♥❣♦ ❑❧ö❝❦❡r ✭❘❲❚❍ ❆❛❝❤❡♥✮ ❏❛♣❛♥❡s❡✲●❡r♠❛♥ ◆✉♠❜❡r ❚❤❡♦r② ❲♦r❦s❤♦♣ ✶✽ ✴ ✷✾

▼♦❞✉❧❛r ❋♦r♠s ❢♦r ❖(✷, ✺) ❇♦r❝❤❡r❞s Pr♦❞✉❝ts

❇♦r❝❤❡r❞s ❚❤❡♦r❡♠

❚❤❡♦r❡♠

▲❡t ❢ ∈ [▼♣✷(Z), −ℓ/✷, ρ♯

❙]∞ ✇✐t❤ ❋♦✉r✐❡r ❝♦❡✣❝✐❡♥ts ❝µ(♥)✱ s✉❝❤ t❤❛t

❝✵(✵) ∈ ✷Z ❛♥❞ ❝µ(♥) ∈ Z ✇❤❡♥❡✈❡r ♥ < ✵✳ ❚❤❡♥ t❤❡r❡ ❡①✐sts ❛ ❇♦r❝❤❡r❞s ♣r♦❞✉❝t ψ❦ : H❙ → C ✇✐t❤ t❤❡ ❢♦❧❧♦✇✐♥❣ ♣r♦♣❡rt✐❡s✿

◮ ψ❦ ✐s ❛ ♠❡r♦♠♦r♣❤✐❝ ♠♦❞✉❧❛r ❢♦r♠ ♦❢ ✇❡✐❣❤t ❦ = ❝✵(✵)/✷ ✇✐t❤ r❡s♣❡❝t

t♦ Γ❙ ❛♥❞ s♦♠❡ ❛❜❡❧✐❛♥ ❝❤❛r❛❝t❡r χ✳

◮ ❚❤❡ ③❡r♦s ❛♥❞ ♣♦❧❡s ♦❢ ψ❦ ❛r❡ ❡①♣❧✐❝✐t❡❧② ❦♥♦✇♥ ❛♥❞ ❞❡♣❡♥❞ ♦♥❧② ♦♥

t❤❡ ♣r✐♥❝✐♣❛❧ ♣❛rt ♦❢ ❢ ✳

◮ ψ❦ ✐s ❣✐✈❡♥ ❜② t❤❡ ♥♦r♠❛❧❧② ❝♦♥✈❡r❣❡♥t ♣r♦❞✉❝t ❡①♣❛♥s✐♦♥

ψ❦(✇) = ❡✷π✐(̺❢ ,✇)

λ✵∈Λ♯

✵

λ✵>✵

• ✶ − ❡✷π✐(λ✵,✇)❝(✵,λ✵,✵)(q✵(λ✵))

.

■♥❣♦ ❑❧ö❝❦❡r ✭❘❲❚❍ ❆❛❝❤❡♥✮ ❏❛♣❛♥❡s❡✲●❡r♠❛♥ ◆✉♠❜❡r ❚❤❡♦r② ❲♦r❦s❤♦♣ ✶✾ ✴ ✷✾

▼♦❞✉❧❛r ❋♦r♠s ❢♦r ❖(✷, ✺) ❇♦r❝❤❡r❞s Pr♦❞✉❝ts

❇♦r❝❤❡r❞s ❚❤❡♦r❡♠

❚❤❡♦r❡♠

▲❡t ❢ ∈ [▼♣✷(Z), −ℓ/✷, ρ♯

❙]∞ ✇✐t❤ ❋♦✉r✐❡r ❝♦❡✣❝✐❡♥ts ❝µ(♥)✱ s✉❝❤ t❤❛t

❝✵(✵) ∈ ✷Z ❛♥❞ ❝µ(♥) ∈ Z ✇❤❡♥❡✈❡r ♥ < ✵✳ ❚❤❡♥ t❤❡r❡ ❡①✐sts ❛ ❇♦r❝❤❡r❞s ♣r♦❞✉❝t ψ❦ : H❙ → C ✇✐t❤ t❤❡ ❢♦❧❧♦✇✐♥❣ ♣r♦♣❡rt✐❡s✿

◮ ψ❦ ✐s ❛ ♠❡r♦♠♦r♣❤✐❝ ♠♦❞✉❧❛r ❢♦r♠ ♦❢ ✇❡✐❣❤t ❦ = ❝✵(✵)/✷ ✇✐t❤ r❡s♣❡❝t

t♦ Γ❙ ❛♥❞ s♦♠❡ ❛❜❡❧✐❛♥ ❝❤❛r❛❝t❡r χ✳

◮ ❚❤❡ ③❡r♦s ❛♥❞ ♣♦❧❡s ♦❢ ψ❦ ❛r❡ ❡①♣❧✐❝✐t❡❧② ❦♥♦✇♥ ❛♥❞ ❞❡♣❡♥❞ ♦♥❧② ♦♥

t❤❡ ♣r✐♥❝✐♣❛❧ ♣❛rt ♦❢ ❢ ✳

◮ ψ❦ ✐s ❣✐✈❡♥ ❜② t❤❡ ♥♦r♠❛❧❧② ❝♦♥✈❡r❣❡♥t ♣r♦❞✉❝t ❡①♣❛♥s✐♦♥

ψ❦(✇) = ❡✷π✐(̺❢ ,✇)

λ✵∈Λ♯

✵

λ✵>✵

• ✶ − ❡✷π✐(λ✵,✇)❝(✵,λ✵,✵)(q✵(λ✵))

.

■♥❣♦ ❑❧ö❝❦❡r ✭❘❲❚❍ ❆❛❝❤❡♥✮ ❏❛♣❛♥❡s❡✲●❡r♠❛♥ ◆✉♠❜❡r ❚❤❡♦r② ❲♦r❦s❤♦♣ ✶✾ ✴ ✷✾

▼♦❞✉❧❛r ❋♦r♠s ❢♦r ❖(✷, ✺) ❇♦r❝❤❡r❞s Pr♦❞✉❝ts

❇♦r❝❤❡r❞s ❚❤❡♦r❡♠

❚❤❡♦r❡♠

▲❡t ❢ ∈ [▼♣✷(Z), −ℓ/✷, ρ♯

❙]∞ ✇✐t❤ ❋♦✉r✐❡r ❝♦❡✣❝✐❡♥ts ❝µ(♥)✱ s✉❝❤ t❤❛t

❝✵(✵) ∈ ✷Z ❛♥❞ ❝µ(♥) ∈ Z ✇❤❡♥❡✈❡r ♥ < ✵✳ ❚❤❡♥ t❤❡r❡ ❡①✐sts ❛ ❇♦r❝❤❡r❞s ♣r♦❞✉❝t ψ❦ : H❙ → C ✇✐t❤ t❤❡ ❢♦❧❧♦✇✐♥❣ ♣r♦♣❡rt✐❡s✿

◮ ψ❦ ✐s ❛ ♠❡r♦♠♦r♣❤✐❝ ♠♦❞✉❧❛r ❢♦r♠ ♦❢ ✇❡✐❣❤t ❦ = ❝✵(✵)/✷ ✇✐t❤ r❡s♣❡❝t

t♦ Γ❙ ❛♥❞ s♦♠❡ ❛❜❡❧✐❛♥ ❝❤❛r❛❝t❡r χ✳

◮ ❚❤❡ ③❡r♦s ❛♥❞ ♣♦❧❡s ♦❢ ψ❦ ❛r❡ ❡①♣❧✐❝✐t❡❧② ❦♥♦✇♥ ❛♥❞ ❞❡♣❡♥❞ ♦♥❧② ♦♥

t❤❡ ♣r✐♥❝✐♣❛❧ ♣❛rt ♦❢ ❢ ✳

◮ ψ❦ ✐s ❣✐✈❡♥ ❜② t❤❡ ♥♦r♠❛❧❧② ❝♦♥✈❡r❣❡♥t ♣r♦❞✉❝t ❡①♣❛♥s✐♦♥

ψ❦(✇) = ❡✷π✐(̺❢ ,✇)

λ✵∈Λ♯

✵

λ✵>✵

• ✶ − ❡✷π✐(λ✵,✇)❝(✵,λ✵,✵)(q✵(λ✵))

.

■♥❣♦ ❑❧ö❝❦❡r ✭❘❲❚❍ ❆❛❝❤❡♥✮ ❏❛♣❛♥❡s❡✲●❡r♠❛♥ ◆✉♠❜❡r ❚❤❡♦r② ❲♦r❦s❤♦♣ ✶✾ ✴ ✷✾

▼♦❞✉❧❛r ❋♦r♠s ❢♦r ❖(✷, ✺) ❇♦r❝❤❡r❞s Pr♦❞✉❝ts

❇♦r❝❤❡r❞s✬ ❖❜str✉❝t✐♦♥ ❈♦♥❞✐t✐♦♥

❆ ♥❡❝❡ss❛r② ❛♥❞ s✉✣❝✐❡♥t ❝♦♥❞✐t✐♦♥ ❢♦r t❤❡ ❡①✐st❡♥❝❡ ♦❢ ♥❡❛r❧② ❤♦❧♦♠♦r♣❤✐❝ ♠♦❞✉❧❛r ❢♦r♠s

❚❤❡♦r❡♠

❚❤❡r❡ ❡①✐sts ❛ ♥❡❛r❧② ❤♦❧♦♠♦r♣❤✐❝ ♠♦❞✉❧❛r ❢♦r♠ ❢ ∈ [▼♣✷(Z), −ℓ/✷, ρ♯

❙]∞

✇✐t❤ ♣r❡s❝r✐❜❡❞ ♣r✐♥❝✐♣❛❧ ♣❛rt

• µ∈Λ♯

✶/Λ✶

• ♥∈−q❙(µ)+Z

♥≤✵

❝µ(♥)q♥ ❡µ, ✐❢ ❛♥❞ ♦♥❧② ✐❢

• µ∈Λ♯

✶/Λ✶

• ♥∈−q❙(µ)+Z

♥≤✵

❝µ(♥) ❛µ(−♥) = ✵ ❢♦r ❛❧❧ ❤♦❧♦♠♦r♣❤✐❝ ♠♦❞✉❧❛r ❢♦r♠s ❣ ∈ [▼♣✷(Z), ✷ + ℓ/✷, ρ❙] ✇✐t❤ ❋♦✉r✐❡r ❡①♣❛♥s✐♦♥ ❣(τ) =

µ∈Λ♯

✶/Λ✶

• ♥∈q❙(µ)+Z, ♥≥✵ ❛µ(♥)q♥ ❡µ✳

■♥❣♦ ❑❧ö❝❦❡r ✭❘❲❚❍ ❆❛❝❤❡♥✮ ❏❛♣❛♥❡s❡✲●❡r♠❛♥ ◆✉♠❜❡r ❚❤❡♦r② ❲♦r❦s❤♦♣ ✷✵ ✴ ✷✾

▼♦❞✉❧❛r ❋♦r♠s ❢♦r ❖(✷, ✺) ❇♦r❝❤❡r❞s Pr♦❞✉❝ts

■♥♣✉t ❢♦r ❇♦r❝❤❡r❞s ❚❤❡♦r❡♠ ✐♥ t❤❡ ❈❛s❡ ❙ = ❆✸

❚❤❡ ♦❜str✉❝t✐♦♥ s♣❛❝❡ [▼♣✷(Z), ✼/✷, ρ❆✸] ✐s ♦❢ ❞✐♠❡♥s✐♦♥ ✶ ❛♥❞ s♣❛♥♥❡❞ ❜② ❊✼/✷(τ) = ✶ ❡✵ − ✽ q✸/✽ ❡ ✶

✹ + ❡− ✶ ✹

• − ✶✽ q✶/✷ ❡ ✶

✷ − ✶✵✽ q ❡✵ + ❖(q✶✶/✽).

❚❤✉s t❤❡ ❝♦♥❞✐t✐♦♥ ❢♦r t❤❡ ♣r✐♥❝✐♣❛❧ ♣❛rt ♦❢ ❢ ∈ [▼♣✷(Z), −✸/✷, ρ♯

❆✸]∞ ✐s

❝✵(✵) = ✽

• ❝ ✶

✹ (−✸

✽) + ❝− ✶

✹ (−✸

✽)

• + ✶✽ ❝ ✶

✷ (−✶

✷) + ✶✵✽ ❝✵(−✶) + · · · .

P♦ss✐❜❧❡ ♣r✐♥❝✐♣❛❧ ♣❛rts ❛r❡ ❣✐✈❡♥ ❜② q−✸/✽ ❡ ✶

✹ + ❡− ✶ ✹

• + ✶✻ ❡✵,

q−✶/✷ ❡ ✶

✷

+ ✶✽ ❡✵, q−✶ ❡✵ + ✶✵✽ ❡✵.

■♥❣♦ ❑❧ö❝❦❡r ✭❘❲❚❍ ❆❛❝❤❡♥✮ ❏❛♣❛♥❡s❡✲●❡r♠❛♥ ◆✉♠❜❡r ❚❤❡♦r② ❲♦r❦s❤♦♣ ✷✶ ✴ ✷✾

▼♦❞✉❧❛r ❋♦r♠s ❢♦r ❖(✷, ✺) ❇♦r❝❤❡r❞s Pr♦❞✉❝ts

■♥♣✉t ❢♦r ❇♦r❝❤❡r❞s ❚❤❡♦r❡♠ ✐♥ t❤❡ ❈❛s❡ ❙ = ❆✸

❚❤❡ ♦❜str✉❝t✐♦♥ s♣❛❝❡ [▼♣✷(Z), ✼/✷, ρ❆✸] ✐s ♦❢ ❞✐♠❡♥s✐♦♥ ✶ ❛♥❞ s♣❛♥♥❡❞ ❜② ❊✼/✷(τ) = ✶ ❡✵ − ✽ q✸/✽ ❡ ✶

✹ + ❡− ✶ ✹

• − ✶✽ q✶/✷ ❡ ✶

✷ − ✶✵✽ q ❡✵ + ❖(q✶✶/✽).

❚❤✉s t❤❡ ❝♦♥❞✐t✐♦♥ ❢♦r t❤❡ ♣r✐♥❝✐♣❛❧ ♣❛rt ♦❢ ❢ ∈ [▼♣✷(Z), −✸/✷, ρ♯

❆✸]∞ ✐s

❝✵(✵) = ✽

• ❝ ✶

✹ (−✸

✽) + ❝− ✶

✹ (−✸

✽)

• + ✶✽ ❝ ✶

✷ (−✶

✷) + ✶✵✽ ❝✵(−✶) + · · · .

P♦ss✐❜❧❡ ♣r✐♥❝✐♣❛❧ ♣❛rts ❛r❡ ❣✐✈❡♥ ❜② q−✸/✽ ❡ ✶

✹ + ❡− ✶ ✹

• + ✶✻ ❡✵,

q−✶/✷ ❡ ✶

✷

+ ✶✽ ❡✵, q−✶ ❡✵ + ✶✵✽ ❡✵.

■♥❣♦ ❑❧ö❝❦❡r ✭❘❲❚❍ ❆❛❝❤❡♥✮ ❏❛♣❛♥❡s❡✲●❡r♠❛♥ ◆✉♠❜❡r ❚❤❡♦r② ❲♦r❦s❤♦♣ ✷✶ ✴ ✷✾

▼♦❞✉❧❛r ❋♦r♠s ❢♦r ❖(✷, ✺) ❇♦r❝❤❡r❞s Pr♦❞✉❝ts

■♥♣✉t ❢♦r ❇♦r❝❤❡r❞s ❚❤❡♦r❡♠ ✐♥ t❤❡ ❈❛s❡ ❙ = ❆✸

❚❤❡ ♦❜str✉❝t✐♦♥ s♣❛❝❡ [▼♣✷(Z), ✼/✷, ρ❆✸] ✐s ♦❢ ❞✐♠❡♥s✐♦♥ ✶ ❛♥❞ s♣❛♥♥❡❞ ❜② ❊✼/✷(τ) = ✶ ❡✵ − ✽ q✸/✽ ❡ ✶

✹ + ❡− ✶ ✹

• − ✶✽ q✶/✷ ❡ ✶

✷ − ✶✵✽ q ❡✵ + ❖(q✶✶/✽).

❚❤✉s t❤❡ ❝♦♥❞✐t✐♦♥ ❢♦r t❤❡ ♣r✐♥❝✐♣❛❧ ♣❛rt ♦❢ ❢ ∈ [▼♣✷(Z), −✸/✷, ρ♯

❆✸]∞ ✐s

❝✵(✵) = ✽

• ❝ ✶

✹ (−✸

✽) + ❝− ✶

✹ (−✸

✽)

• + ✶✽ ❝ ✶

✷ (−✶

✷) + ✶✵✽ ❝✵(−✶) + · · · .

P♦ss✐❜❧❡ ♣r✐♥❝✐♣❛❧ ♣❛rts ❛r❡ ❣✐✈❡♥ ❜② q−✸/✽ ❡ ✶

✹ + ❡− ✶ ✹

• + ✶✻ ❡✵,

q−✶/✷ ❡ ✶

✷

+ ✶✽ ❡✵, q−✶ ❡✵ + ✶✵✽ ❡✵.

■♥❣♦ ❑❧ö❝❦❡r ✭❘❲❚❍ ❆❛❝❤❡♥✮ ❏❛♣❛♥❡s❡✲●❡r♠❛♥ ◆✉♠❜❡r ❚❤❡♦r② ❲♦r❦s❤♦♣ ✷✶ ✴ ✷✾

▼♦❞✉❧❛r ❋♦r♠s ❢♦r ❖(✷, ✺) ❇♦r❝❤❡r❞s Pr♦❞✉❝ts

❇♦r❝❤❡r❞s Pr♦❞✉❝ts ❢♦r Γ❆✸

❚❤❡♦r❡♠

❚❤❡r❡ ❡①✐st ❇♦r❝❤❡r❞s ♣r♦❞✉❝ts ψ✽ ∈ [Γ❆✸, ✽, ✶], ψ✾ ∈ [Γ❆✸, ✾, νπ] ❛♥❞ ψ✺✹ ∈ [Γ❆✸, ✺✹, νπ ❞❡t]. ❚❤❡ ③❡r♦s ♦❢ t❤❡ ♣r♦❞✉❝ts ❛r❡ ❛❧❧ ♦❢ ✜rst ♦r❞❡r ❛♥❞ ❛r❡ ❣✐✈❡♥ ❜②

• ▼∈Γ❆✸

▼H❆✷,

• ▼∈Γ❆✸

▼H❆✷

✶

❛♥❞

• ▼∈Γ❆✸

▼H❙✷, r❡s♣❡❝t✐✈❡❧②✱ ✇❤❡r❡ H❆✷ = {(τ✶, ③✶, ③✷, ③✸, τ✷) ∈ H❆✸; ③✸ = ✵}, H❆✷

✶ = {(τ✶, ③✶, ③✷, ③✸, τ✷) ∈ H❆✸; ③✷ = ✵},

H❙✷ = {(τ✶, ③✶, ③✷, ③✸, τ✷) ∈ H❆✸; ③✶ + ③✸ = ✵}.

■♥❣♦ ❑❧ö❝❦❡r ✭❘❲❚❍ ❆❛❝❤❡♥✮ ❏❛♣❛♥❡s❡✲●❡r♠❛♥ ◆✉♠❜❡r ❚❤❡♦r② ❲♦r❦s❤♦♣ ✷✷ ✴ ✷✾

▼♦❞✉❧❛r ❋♦r♠s ❢♦r ❖(✷, ✺) ❚❤❡ ●r❛❞❡❞ ❘✐♥❣ ♦❢ ▼♦❞✉❧❛r ❋♦r♠s

❖✉t❧✐♥❡

■♥tr♦❞✉❝t✐♦♥ ❖rt❤♦❣♦♥❛❧ ▼♦❞✉❧❛r ❋♦r♠s ❱❡❝t♦r✲✈❛❧✉❡❞ ▼♦❞✉❧❛r ❋♦r♠s ❇♦r❝❤❡r❞s Pr♦❞✉❝ts ❚❤❡ ●r❛❞❡❞ ❘✐♥❣ ♦❢ ▼♦❞✉❧❛r ❋♦r♠s

■♥❣♦ ❑❧ö❝❦❡r ✭❘❲❚❍ ❆❛❝❤❡♥✮ ❏❛♣❛♥❡s❡✲●❡r♠❛♥ ◆✉♠❜❡r ❚❤❡♦r② ❲♦r❦s❤♦♣ ✷✸ ✴ ✷✾

▼♦❞✉❧❛r ❋♦r♠s ❢♦r ❖(✷, ✺) ❚❤❡ ●r❛❞❡❞ ❘✐♥❣ ♦❢ ▼♦❞✉❧❛r ❋♦r♠s

❚♦♦❧s ❢♦r Pr♦✈✐♥❣ t❤❡ ▼❛✐♥ ❘❡s✉❧t

❈♦r♦❧❧❛r②

▲❡t ❦ ∈ Z ❛♥❞ ♥ ∈ {✵, ✶}✳ ✶✳ ■❢ ❦ ✐s ♦❞❞ ❛♥❞ ❢ ∈ [Γ❆✸, ❦, ν♥+✶

π

❞❡t♥]✱ t❤❡♥ ❢ ✈❛♥✐s❤❡s ♦♥ H❆✷

✶ ❛♥❞

❢ /ψ✾ ∈ [Γ❆✸, ❦ − ✾, ν♥

π ❞❡t♥]✳

✷✳ ■❢ ❢ ∈ [Γ❆✸, ❦, ν❦+✶

π

❞❡t]✱ t❤❡♥ ❢ ✈❛♥✐s❤❡s ♦♥ H❙✷ ❛♥❞ ❢ /ψ✺✹ ∈ [Γ❆✸, ❦ − ✺✹, ν❦

π]✳

❚❤❡♦r❡♠

✭❉❡r♥ ✷✵✵✶✮ ❚❤❡ ❣r❛❞❡❞ r✐♥❣ A(Γ❆✷) =

❦∈Z[Γ❆✷, ✷❦, ✶] ✐s ❛ ♣♦❧②♥♦♠✐❛❧

r✐♥❣ ✐♥ ❊✹|H❆✷, ❊✻|H❆✷, ❊✶✵|H❆✷, ❊✶✷|H❆✷ ❛♥❞ ψ✷

✾|H❆✷.

■♥❣♦ ❑❧ö❝❦❡r ✭❘❲❚❍ ❆❛❝❤❡♥✮ ❏❛♣❛♥❡s❡✲●❡r♠❛♥ ◆✉♠❜❡r ❚❤❡♦r② ❲♦r❦s❤♦♣ ✷✹ ✴ ✷✾

▼♦❞✉❧❛r ❋♦r♠s ❢♦r ❖(✷, ✺) ❚❤❡ ●r❛❞❡❞ ❘✐♥❣ ♦❢ ▼♦❞✉❧❛r ❋♦r♠s

❚♦♦❧s ❢♦r Pr♦✈✐♥❣ t❤❡ ▼❛✐♥ ❘❡s✉❧t

❈♦r♦❧❧❛r②

▲❡t ❦ ∈ Z ❛♥❞ ♥ ∈ {✵, ✶}✳ ✶✳ ■❢ ❦ ✐s ♦❞❞ ❛♥❞ ❢ ∈ [Γ❆✸, ❦, ν♥+✶

π

❞❡t♥]✱ t❤❡♥ ❢ ✈❛♥✐s❤❡s ♦♥ H❆✷

✶ ❛♥❞

❢ /ψ✾ ∈ [Γ❆✸, ❦ − ✾, ν♥

π ❞❡t♥]✳

✷✳ ■❢ ❢ ∈ [Γ❆✸, ❦, ν❦+✶

π

❞❡t]✱ t❤❡♥ ❢ ✈❛♥✐s❤❡s ♦♥ H❙✷ ❛♥❞ ❢ /ψ✺✹ ∈ [Γ❆✸, ❦ − ✺✹, ν❦

π]✳

❚❤❡♦r❡♠

✭❉❡r♥ ✷✵✵✶✮ ❚❤❡ ❣r❛❞❡❞ r✐♥❣ A(Γ❆✷) =

❦∈Z[Γ❆✷, ✷❦, ✶] ✐s ❛ ♣♦❧②♥♦♠✐❛❧

r✐♥❣ ✐♥ ❊✹|H❆✷, ❊✻|H❆✷, ❊✶✵|H❆✷, ❊✶✷|H❆✷ ❛♥❞ ψ✷

✾|H❆✷.

■♥❣♦ ❑❧ö❝❦❡r ✭❘❲❚❍ ❆❛❝❤❡♥✮ ❏❛♣❛♥❡s❡✲●❡r♠❛♥ ◆✉♠❜❡r ❚❤❡♦r② ❲♦r❦s❤♦♣ ✷✹ ✴ ✷✾

▼♦❞✉❧❛r ❋♦r♠s ❢♦r ❖(✷, ✺) ❚❤❡ ●r❛❞❡❞ ❘✐♥❣ ♦❢ ▼♦❞✉❧❛r ❋♦r♠s

❚❤❡ ●r❛❞❡❞ ❘✐♥❣ ♦❢ ▼♦❞✉❧❛r ❋♦r♠s ❢♦r Γ❆✸

❚❤❡♦r❡♠

❚❤❡ ❣r❛❞❡❞ r✐♥❣ A(Γ❆✸) =

❦∈Z[Γ❆✸, ✷❦, ✶] ✐s ❛ ♣♦❧②♥♦♠✐❛❧ r✐♥❣ ✐♥

❊✹, ❊✻, ψ✽, ❊✶✵, ❊✶✷ ❛♥❞ ψ✷

✾.

Pr♦♦❢✳

◮ ▲❡t ❢ ∈ [Γ❆✸, ✷❦, ✶]✳ ◮ ❆❝❝♦r❞✐♥❣ t♦ ❉❡r♥ ❢ |H❆✷ ✐s ❡q✉❛❧ t♦ ❛ ♣♦❧②♥♦♠✐❛❧ ♣ ✐♥ ❊✹|H❆✷✱ ❊✻|H❆✷✱

❊✶✵|H❆✷✱ ❊✶✷|H❆✷✱ ψ✷

✾|H❆✷✳ ◮ ❚❤✉s ❢ − ♣(❊✹, ❊✻, ❊✶✵, ❊✶✷, ψ✷ ✾) ✈❛♥✐s❤❡s ♦♥ H❆✷✳ ◮ ❚❤❡♥ (❢ − ♣(❊✹, ❊✻, ❊✶✵, ❊✶✷, ψ✷ ✾)/ψ✽ ∈ [Γ❆✸, ✷❦ − ✽, ✶]✳ ◮ ❚❤❡ ❛ss❡rt✐♦♥ ❢♦❧❧♦✇s ❜② ✐♥❞✉❝t✐♦♥✳

■♥❣♦ ❑❧ö❝❦❡r ✭❘❲❚❍ ❆❛❝❤❡♥✮ ❏❛♣❛♥❡s❡✲●❡r♠❛♥ ◆✉♠❜❡r ❚❤❡♦r② ❲♦r❦s❤♦♣ ✷✺ ✴ ✷✾

▼♦❞✉❧❛r ❋♦r♠s ❢♦r ❖(✷, ✺) ❚❤❡ ●r❛❞❡❞ ❘✐♥❣ ♦❢ ▼♦❞✉❧❛r ❋♦r♠s

❚❤❡ ●r❛❞❡❞ ❘✐♥❣ ♦❢ ▼♦❞✉❧❛r ❋♦r♠s ❢♦r Γ❆✸

❚❤❡♦r❡♠

❚❤❡ ❣r❛❞❡❞ r✐♥❣ A(Γ❆✸) =

❦∈Z[Γ❆✸, ✷❦, ✶] ✐s ❛ ♣♦❧②♥♦♠✐❛❧ r✐♥❣ ✐♥

❊✹, ❊✻, ψ✽, ❊✶✵, ❊✶✷ ❛♥❞ ψ✷

✾.

Pr♦♦❢✳

◮ ▲❡t ❢ ∈ [Γ❆✸, ✷❦, ✶]✳ ◮ ❆❝❝♦r❞✐♥❣ t♦ ❉❡r♥ ❢ |H❆✷ ✐s ❡q✉❛❧ t♦ ❛ ♣♦❧②♥♦♠✐❛❧ ♣ ✐♥ ❊✹|H❆✷✱ ❊✻|H❆✷✱

❊✶✵|H❆✷✱ ❊✶✷|H❆✷✱ ψ✷

✾|H❆✷✳ ◮ ❚❤✉s ❢ − ♣(❊✹, ❊✻, ❊✶✵, ❊✶✷, ψ✷ ✾) ✈❛♥✐s❤❡s ♦♥ H❆✷✳ ◮ ❚❤❡♥ (❢ − ♣(❊✹, ❊✻, ❊✶✵, ❊✶✷, ψ✷ ✾)/ψ✽ ∈ [Γ❆✸, ✷❦ − ✽, ✶]✳ ◮ ❚❤❡ ❛ss❡rt✐♦♥ ❢♦❧❧♦✇s ❜② ✐♥❞✉❝t✐♦♥✳

■♥❣♦ ❑❧ö❝❦❡r ✭❘❲❚❍ ❆❛❝❤❡♥✮ ❏❛♣❛♥❡s❡✲●❡r♠❛♥ ◆✉♠❜❡r ❚❤❡♦r② ❲♦r❦s❤♦♣ ✷✺ ✴ ✷✾

▼♦❞✉❧❛r ❋♦r♠s ❢♦r ❖(✷, ✺) ❚❤❡ ●r❛❞❡❞ ❘✐♥❣ ♦❢ ▼♦❞✉❧❛r ❋♦r♠s

❚❤❡ ●r❛❞❡❞ ❘✐♥❣ ♦❢ ▼♦❞✉❧❛r ❋♦r♠s ❢♦r Γ❆✸

❚❤❡♦r❡♠

❚❤❡ ❣r❛❞❡❞ r✐♥❣ A(Γ❆✸) =

❦∈Z[Γ❆✸, ✷❦, ✶] ✐s ❛ ♣♦❧②♥♦♠✐❛❧ r✐♥❣ ✐♥

❊✹, ❊✻, ψ✽, ❊✶✵, ❊✶✷ ❛♥❞ ψ✷

✾.

Pr♦♦❢✳

◮ ▲❡t ❢ ∈ [Γ❆✸, ✷❦, ✶]✳ ◮ ❆❝❝♦r❞✐♥❣ t♦ ❉❡r♥ ❢ |H❆✷ ✐s ❡q✉❛❧ t♦ ❛ ♣♦❧②♥♦♠✐❛❧ ♣ ✐♥ ❊✹|H❆✷✱ ❊✻|H❆✷✱

❊✶✵|H❆✷✱ ❊✶✷|H❆✷✱ ψ✷

✾|H❆✷✳ ◮ ❚❤✉s ❢ − ♣(❊✹, ❊✻, ❊✶✵, ❊✶✷, ψ✷ ✾) ✈❛♥✐s❤❡s ♦♥ H❆✷✳ ◮ ❚❤❡♥ (❢ − ♣(❊✹, ❊✻, ❊✶✵, ❊✶✷, ψ✷ ✾)/ψ✽ ∈ [Γ❆✸, ✷❦ − ✽, ✶]✳ ◮ ❚❤❡ ❛ss❡rt✐♦♥ ❢♦❧❧♦✇s ❜② ✐♥❞✉❝t✐♦♥✳

■♥❣♦ ❑❧ö❝❦❡r ✭❘❲❚❍ ❆❛❝❤❡♥✮ ❏❛♣❛♥❡s❡✲●❡r♠❛♥ ◆✉♠❜❡r ❚❤❡♦r② ❲♦r❦s❤♦♣ ✷✺ ✴ ✷✾

▼♦❞✉❧❛r ❋♦r♠s ❢♦r ❖(✷, ✺) ❚❤❡ ●r❛❞❡❞ ❘✐♥❣ ♦❢ ▼♦❞✉❧❛r ❋♦r♠s

❚❤❡ ●r❛❞❡❞ ❘✐♥❣ ♦❢ ▼♦❞✉❧❛r ❋♦r♠s ❢♦r Γ❆✸

❚❤❡♦r❡♠

❚❤❡ ❣r❛❞❡❞ r✐♥❣ A(Γ❆✸) =

❦∈Z[Γ❆✸, ✷❦, ✶] ✐s ❛ ♣♦❧②♥♦♠✐❛❧ r✐♥❣ ✐♥

❊✹, ❊✻, ψ✽, ❊✶✵, ❊✶✷ ❛♥❞ ψ✷

✾.

Pr♦♦❢✳

◮ ▲❡t ❢ ∈ [Γ❆✸, ✷❦, ✶]✳ ◮ ❆❝❝♦r❞✐♥❣ t♦ ❉❡r♥ ❢ |H❆✷ ✐s ❡q✉❛❧ t♦ ❛ ♣♦❧②♥♦♠✐❛❧ ♣ ✐♥ ❊✹|H❆✷✱ ❊✻|H❆✷✱

❊✶✵|H❆✷✱ ❊✶✷|H❆✷✱ ψ✷

✾|H❆✷✳ ◮ ❚❤✉s ❢ − ♣(❊✹, ❊✻, ❊✶✵, ❊✶✷, ψ✷ ✾) ✈❛♥✐s❤❡s ♦♥ H❆✷✳ ◮ ❚❤❡♥ (❢ − ♣(❊✹, ❊✻, ❊✶✵, ❊✶✷, ψ✷ ✾)/ψ✽ ∈ [Γ❆✸, ✷❦ − ✽, ✶]✳ ◮ ❚❤❡ ❛ss❡rt✐♦♥ ❢♦❧❧♦✇s ❜② ✐♥❞✉❝t✐♦♥✳

■♥❣♦ ❑❧ö❝❦❡r ✭❘❲❚❍ ❆❛❝❤❡♥✮ ❏❛♣❛♥❡s❡✲●❡r♠❛♥ ◆✉♠❜❡r ❚❤❡♦r② ❲♦r❦s❤♦♣ ✷✺ ✴ ✷✾

▼♦❞✉❧❛r ❋♦r♠s ❢♦r ❖(✷, ✺) ❚❤❡ ●r❛❞❡❞ ❘✐♥❣ ♦❢ ▼♦❞✉❧❛r ❋♦r♠s

❚❤❡ ●r❛❞❡❞ ❘✐♥❣ ♦❢ ▼♦❞✉❧❛r ❋♦r♠s ❢♦r Γ❆✸

❚❤❡♦r❡♠

❚❤❡ ❣r❛❞❡❞ r✐♥❣ A(Γ❆✸) =

❦∈Z[Γ❆✸, ✷❦, ✶] ✐s ❛ ♣♦❧②♥♦♠✐❛❧ r✐♥❣ ✐♥

❊✹, ❊✻, ψ✽, ❊✶✵, ❊✶✷ ❛♥❞ ψ✷

✾.

Pr♦♦❢✳

◮ ▲❡t ❢ ∈ [Γ❆✸, ✷❦, ✶]✳ ◮ ❆❝❝♦r❞✐♥❣ t♦ ❉❡r♥ ❢ |H❆✷ ✐s ❡q✉❛❧ t♦ ❛ ♣♦❧②♥♦♠✐❛❧ ♣ ✐♥ ❊✹|H❆✷✱ ❊✻|H❆✷✱

❊✶✵|H❆✷✱ ❊✶✷|H❆✷✱ ψ✷

✾|H❆✷✳ ◮ ❚❤✉s ❢ − ♣(❊✹, ❊✻, ❊✶✵, ❊✶✷, ψ✷ ✾) ✈❛♥✐s❤❡s ♦♥ H❆✷✳ ◮ ❚❤❡♥ (❢ − ♣(❊✹, ❊✻, ❊✶✵, ❊✶✷, ψ✷ ✾)/ψ✽ ∈ [Γ❆✸, ✷❦ − ✽, ✶]✳ ◮ ❚❤❡ ❛ss❡rt✐♦♥ ❢♦❧❧♦✇s ❜② ✐♥❞✉❝t✐♦♥✳

■♥❣♦ ❑❧ö❝❦❡r ✭❘❲❚❍ ❆❛❝❤❡♥✮ ❏❛♣❛♥❡s❡✲●❡r♠❛♥ ◆✉♠❜❡r ❚❤❡♦r② ❲♦r❦s❤♦♣ ✷✺ ✴ ✷✾

▼♦❞✉❧❛r ❋♦r♠s ❢♦r ❖(✷, ✺) ❚❤❡ ●r❛❞❡❞ ❘✐♥❣ ♦❢ ▼♦❞✉❧❛r ❋♦r♠s

❚❤❡ ●r❛❞❡❞ ❘✐♥❣ ♦❢ ▼♦❞✉❧❛r ❋♦r♠s ❢♦r Γ❆✸

❚❤❡♦r❡♠

❚❤❡ ❣r❛❞❡❞ r✐♥❣ A(Γ❆✸) =

❦∈Z[Γ❆✸, ✷❦, ✶] ✐s ❛ ♣♦❧②♥♦♠✐❛❧ r✐♥❣ ✐♥

❊✹, ❊✻, ψ✽, ❊✶✵, ❊✶✷ ❛♥❞ ψ✷

✾.

Pr♦♦❢✳

◮ ▲❡t ❢ ∈ [Γ❆✸, ✷❦, ✶]✳ ◮ ❆❝❝♦r❞✐♥❣ t♦ ❉❡r♥ ❢ |H❆✷ ✐s ❡q✉❛❧ t♦ ❛ ♣♦❧②♥♦♠✐❛❧ ♣ ✐♥ ❊✹|H❆✷✱ ❊✻|H❆✷✱

❊✶✵|H❆✷✱ ❊✶✷|H❆✷✱ ψ✷

✾|H❆✷✳ ◮ ❚❤✉s ❢ − ♣(❊✹, ❊✻, ❊✶✵, ❊✶✷, ψ✷ ✾) ✈❛♥✐s❤❡s ♦♥ H❆✷✳ ◮ ❚❤❡♥ (❢ − ♣(❊✹, ❊✻, ❊✶✵, ❊✶✷, ψ✷ ✾)/ψ✽ ∈ [Γ❆✸, ✷❦ − ✽, ✶]✳ ◮ ❚❤❡ ❛ss❡rt✐♦♥ ❢♦❧❧♦✇s ❜② ✐♥❞✉❝t✐♦♥✳

■♥❣♦ ❑❧ö❝❦❡r ✭❘❲❚❍ ❆❛❝❤❡♥✮ ❏❛♣❛♥❡s❡✲●❡r♠❛♥ ◆✉♠❜❡r ❚❤❡♦r② ❲♦r❦s❤♦♣ ✷✺ ✴ ✷✾

▼♦❞✉❧❛r ❋♦r♠s ❢♦r ❖(✷, ✺) ❚❤❡ ●r❛❞❡❞ ❘✐♥❣ ♦❢ ▼♦❞✉❧❛r ❋♦r♠s

❚❤❡ ●r❛❞❡❞ ❘✐♥❣ ♦❢ ▼♦❞✉❧❛r ❋♦r♠s ❢♦r Γ′

❆✸

❚❤❡♦r❡♠

❚❤❡ ❣r❛❞❡❞ r✐♥❣ A(Γ′

❆✸) = ❦∈Z[Γ′ ❆✸, ❦, ✶] ✐s ❣❡♥❡r❛t❡❞ ❜②

❊✹, ❊✻, ψ✽, ψ✾, ❊✶✵, ❊✶✷ ❛♥❞ ψ✺✹ ❛♥❞ ✐s ✐s♦♠♦r♣❤✐❝ t♦ C[❳✶, . . . , ❳✼]/(❳ ✷

✼ − ♣(❳✶, . . . , ❳✻))

✇❤❡r❡ ♣ ∈ C[❳✶, . . . , ❳✻] ✐s t❤❡ ✉♥✐q✉❡❧② ❞❡t❡r♠✐♥❡❞ ♣♦❧②♥♦♠✐❛❧ ✇✐t❤ ψ✷

✺✹ = ♣(❊✹, ❊✻, ψ✽, ψ✾, ❊✶✵, ❊✶✷).

■♥❣♦ ❑❧ö❝❦❡r ✭❘❲❚❍ ❆❛❝❤❡♥✮ ❏❛♣❛♥❡s❡✲●❡r♠❛♥ ◆✉♠❜❡r ❚❤❡♦r② ❲♦r❦s❤♦♣ ✷✻ ✴ ✷✾

▼♦❞✉❧❛r ❋♦r♠s ❢♦r ❖(✷, ✺) ❚❤❡ ●r❛❞❡❞ ❘✐♥❣ ♦❢ ▼♦❞✉❧❛r ❋♦r♠s

Pr♦♦❢ ♦❢ t❤❡ ❙❡❝♦♥❞ ▼❛✐♥ ❘❡s✉❧t

Pr♦♦❢✳

◮ ▲❡t ❢ ∈ [Γ′ ❆✸, ❦, ✶]✳ ◮ ■❢ ❦ ✐s ♦❞❞✱ t❤❡♥ ❢ ✈❛♥✐s❤❡s ♦♥ H❆✷

✶ ❛♥❞ ✇❡ ❤❛✈❡

❢ /ψ✾ ∈ [Γ′

❆✸, ❦ − ✾, ✶]✳ ❙♦ ✇❡ ❝❛♥ ❛ss✉♠❡ t❤❛t ❦ ✐s ❡✈❡♥✳ ◮ ❲❡ ❦♥♦✇ t❤❛t [Γ′ ❆✸, ✷❦, ✶] = [Γ❆✸, ✷❦, ✶] ⊕ [Γ❆✸, ✷❦, νπ ❞❡t]✳ ❚❤✉s

❢ = ❢✶ + ❢νπ ❞❡t ✇✐t❤ ❢ν ∈ [Γ❆✸, ✷❦, ν]✳

◮ ❢νπ ❞❡t ✈❛♥✐s❤❡s ♦♥ H❙✷ ❛♥❞ ✇❡ ❤❛✈❡ ❢νπ ❞❡t/ψ✺✹ ∈ [Γ❆✸, ✷❦ − ✺✹, ✶]✳ ◮ ◆♦✇ ❢✶ ❛♥❞ ❢νπ ❞❡t/ψ✺✹ ❛r❡ ♣♦❧②♥♦♠✐❛❧s ✐♥ ❊✹✱ ❊✻✱ ψ✽✱ ❊✶✵✱ ❊✶✷✱ ψ✷ ✾✳

❚❤✐s ❝♦♠♣❧❡t❡s t❤❡ ♣r♦♦❢ ♦❢ t❤❡ ✜rst r❡s✉❧t✳

■♥❣♦ ❑❧ö❝❦❡r ✭❘❲❚❍ ❆❛❝❤❡♥✮ ❏❛♣❛♥❡s❡✲●❡r♠❛♥ ◆✉♠❜❡r ❚❤❡♦r② ❲♦r❦s❤♦♣ ✷✼ ✴ ✷✾

▼♦❞✉❧❛r ❋♦r♠s ❢♦r ❖(✷, ✺) ❚❤❡ ●r❛❞❡❞ ❘✐♥❣ ♦❢ ▼♦❞✉❧❛r ❋♦r♠s

Pr♦♦❢ ♦❢ t❤❡ ❙❡❝♦♥❞ ▼❛✐♥ ❘❡s✉❧t

Pr♦♦❢✳

◮ ▲❡t ❢ ∈ [Γ′ ❆✸, ❦, ✶]✳ ◮ ■❢ ❦ ✐s ♦❞❞✱ t❤❡♥ ❢ ✈❛♥✐s❤❡s ♦♥ H❆✷

✶ ❛♥❞ ✇❡ ❤❛✈❡

❢ /ψ✾ ∈ [Γ′

❆✸, ❦ − ✾, ✶]✳ ❙♦ ✇❡ ❝❛♥ ❛ss✉♠❡ t❤❛t ❦ ✐s ❡✈❡♥✳ ◮ ❲❡ ❦♥♦✇ t❤❛t [Γ′ ❆✸, ✷❦, ✶] = [Γ❆✸, ✷❦, ✶] ⊕ [Γ❆✸, ✷❦, νπ ❞❡t]✳ ❚❤✉s

❢ = ❢✶ + ❢νπ ❞❡t ✇✐t❤ ❢ν ∈ [Γ❆✸, ✷❦, ν]✳

◮ ❢νπ ❞❡t ✈❛♥✐s❤❡s ♦♥ H❙✷ ❛♥❞ ✇❡ ❤❛✈❡ ❢νπ ❞❡t/ψ✺✹ ∈ [Γ❆✸, ✷❦ − ✺✹, ✶]✳ ◮ ◆♦✇ ❢✶ ❛♥❞ ❢νπ ❞❡t/ψ✺✹ ❛r❡ ♣♦❧②♥♦♠✐❛❧s ✐♥ ❊✹✱ ❊✻✱ ψ✽✱ ❊✶✵✱ ❊✶✷✱ ψ✷ ✾✳

❚❤✐s ❝♦♠♣❧❡t❡s t❤❡ ♣r♦♦❢ ♦❢ t❤❡ ✜rst r❡s✉❧t✳

■♥❣♦ ❑❧ö❝❦❡r ✭❘❲❚❍ ❆❛❝❤❡♥✮ ❏❛♣❛♥❡s❡✲●❡r♠❛♥ ◆✉♠❜❡r ❚❤❡♦r② ❲♦r❦s❤♦♣ ✷✼ ✴ ✷✾

▼♦❞✉❧❛r ❋♦r♠s ❢♦r ❖(✷, ✺) ❚❤❡ ●r❛❞❡❞ ❘✐♥❣ ♦❢ ▼♦❞✉❧❛r ❋♦r♠s

Pr♦♦❢ ♦❢ t❤❡ ❙❡❝♦♥❞ ▼❛✐♥ ❘❡s✉❧t

Pr♦♦❢✳

◮ ▲❡t ❢ ∈ [Γ′ ❆✸, ❦, ✶]✳ ◮ ■❢ ❦ ✐s ♦❞❞✱ t❤❡♥ ❢ ✈❛♥✐s❤❡s ♦♥ H❆✷

✶ ❛♥❞ ✇❡ ❤❛✈❡

❢ /ψ✾ ∈ [Γ′

❆✸, ❦ − ✾, ✶]✳ ❙♦ ✇❡ ❝❛♥ ❛ss✉♠❡ t❤❛t ❦ ✐s ❡✈❡♥✳ ◮ ❲❡ ❦♥♦✇ t❤❛t [Γ′ ❆✸, ✷❦, ✶] = [Γ❆✸, ✷❦, ✶] ⊕ [Γ❆✸, ✷❦, νπ ❞❡t]✳ ❚❤✉s

❢ = ❢✶ + ❢νπ ❞❡t ✇✐t❤ ❢ν ∈ [Γ❆✸, ✷❦, ν]✳

◮ ❢νπ ❞❡t ✈❛♥✐s❤❡s ♦♥ H❙✷ ❛♥❞ ✇❡ ❤❛✈❡ ❢νπ ❞❡t/ψ✺✹ ∈ [Γ❆✸, ✷❦ − ✺✹, ✶]✳ ◮ ◆♦✇ ❢✶ ❛♥❞ ❢νπ ❞❡t/ψ✺✹ ❛r❡ ♣♦❧②♥♦♠✐❛❧s ✐♥ ❊✹✱ ❊✻✱ ψ✽✱ ❊✶✵✱ ❊✶✷✱ ψ✷ ✾✳

❚❤✐s ❝♦♠♣❧❡t❡s t❤❡ ♣r♦♦❢ ♦❢ t❤❡ ✜rst r❡s✉❧t✳

■♥❣♦ ❑❧ö❝❦❡r ✭❘❲❚❍ ❆❛❝❤❡♥✮ ❏❛♣❛♥❡s❡✲●❡r♠❛♥ ◆✉♠❜❡r ❚❤❡♦r② ❲♦r❦s❤♦♣ ✷✼ ✴ ✷✾

▼♦❞✉❧❛r ❋♦r♠s ❢♦r ❖(✷, ✺) ❚❤❡ ●r❛❞❡❞ ❘✐♥❣ ♦❢ ▼♦❞✉❧❛r ❋♦r♠s

Pr♦♦❢ ♦❢ t❤❡ ❙❡❝♦♥❞ ▼❛✐♥ ❘❡s✉❧t

Pr♦♦❢✳

◮ ▲❡t ❢ ∈ [Γ′ ❆✸, ❦, ✶]✳ ◮ ■❢ ❦ ✐s ♦❞❞✱ t❤❡♥ ❢ ✈❛♥✐s❤❡s ♦♥ H❆✷

✶ ❛♥❞ ✇❡ ❤❛✈❡

❢ /ψ✾ ∈ [Γ′

❆✸, ❦ − ✾, ✶]✳ ❙♦ ✇❡ ❝❛♥ ❛ss✉♠❡ t❤❛t ❦ ✐s ❡✈❡♥✳ ◮ ❲❡ ❦♥♦✇ t❤❛t [Γ′ ❆✸, ✷❦, ✶] = [Γ❆✸, ✷❦, ✶] ⊕ [Γ❆✸, ✷❦, νπ ❞❡t]✳ ❚❤✉s

❢ = ❢✶ + ❢νπ ❞❡t ✇✐t❤ ❢ν ∈ [Γ❆✸, ✷❦, ν]✳

◮ ❢νπ ❞❡t ✈❛♥✐s❤❡s ♦♥ H❙✷ ❛♥❞ ✇❡ ❤❛✈❡ ❢νπ ❞❡t/ψ✺✹ ∈ [Γ❆✸, ✷❦ − ✺✹, ✶]✳ ◮ ◆♦✇ ❢✶ ❛♥❞ ❢νπ ❞❡t/ψ✺✹ ❛r❡ ♣♦❧②♥♦♠✐❛❧s ✐♥ ❊✹✱ ❊✻✱ ψ✽✱ ❊✶✵✱ ❊✶✷✱ ψ✷ ✾✳

❚❤✐s ❝♦♠♣❧❡t❡s t❤❡ ♣r♦♦❢ ♦❢ t❤❡ ✜rst r❡s✉❧t✳

■♥❣♦ ❑❧ö❝❦❡r ✭❘❲❚❍ ❆❛❝❤❡♥✮ ❏❛♣❛♥❡s❡✲●❡r♠❛♥ ◆✉♠❜❡r ❚❤❡♦r② ❲♦r❦s❤♦♣ ✷✼ ✴ ✷✾

▼♦❞✉❧❛r ❋♦r♠s ❢♦r ❖(✷, ✺) ❚❤❡ ●r❛❞❡❞ ❘✐♥❣ ♦❢ ▼♦❞✉❧❛r ❋♦r♠s

Pr♦♦❢ ♦❢ t❤❡ ❙❡❝♦♥❞ ▼❛✐♥ ❘❡s✉❧t

Pr♦♦❢✳

◮ ▲❡t ❢ ∈ [Γ′ ❆✸, ❦, ✶]✳ ◮ ■❢ ❦ ✐s ♦❞❞✱ t❤❡♥ ❢ ✈❛♥✐s❤❡s ♦♥ H❆✷

✶ ❛♥❞ ✇❡ ❤❛✈❡

❢ /ψ✾ ∈ [Γ′

❆✸, ❦ − ✾, ✶]✳ ❙♦ ✇❡ ❝❛♥ ❛ss✉♠❡ t❤❛t ❦ ✐s ❡✈❡♥✳ ◮ ❲❡ ❦♥♦✇ t❤❛t [Γ′ ❆✸, ✷❦, ✶] = [Γ❆✸, ✷❦, ✶] ⊕ [Γ❆✸, ✷❦, νπ ❞❡t]✳ ❚❤✉s

❢ = ❢✶ + ❢νπ ❞❡t ✇✐t❤ ❢ν ∈ [Γ❆✸, ✷❦, ν]✳

◮ ❢νπ ❞❡t ✈❛♥✐s❤❡s ♦♥ H❙✷ ❛♥❞ ✇❡ ❤❛✈❡ ❢νπ ❞❡t/ψ✺✹ ∈ [Γ❆✸, ✷❦ − ✺✹, ✶]✳ ◮ ◆♦✇ ❢✶ ❛♥❞ ❢νπ ❞❡t/ψ✺✹ ❛r❡ ♣♦❧②♥♦♠✐❛❧s ✐♥ ❊✹✱ ❊✻✱ ψ✽✱ ❊✶✵✱ ❊✶✷✱ ψ✷ ✾✳

❚❤✐s ❝♦♠♣❧❡t❡s t❤❡ ♣r♦♦❢ ♦❢ t❤❡ ✜rst r❡s✉❧t✳

■♥❣♦ ❑❧ö❝❦❡r ✭❘❲❚❍ ❆❛❝❤❡♥✮ ❏❛♣❛♥❡s❡✲●❡r♠❛♥ ◆✉♠❜❡r ❚❤❡♦r② ❲♦r❦s❤♦♣ ✷✼ ✴ ✷✾

▼♦❞✉❧❛r ❋♦r♠s ❢♦r ❖(✷, ✺) ❚❤❡ ●r❛❞❡❞ ❘✐♥❣ ♦❢ ▼♦❞✉❧❛r ❋♦r♠s

Pr♦♦❢ ♦❢ t❤❡ ❙❡❝♦♥❞ ▼❛✐♥ ❘❡s✉❧t ✭❝♦♥t✬❞✮

Pr♦♦❢✳

◮ ❲❡ ❤❛✈❡ ψ✷ ✺✹ ∈ [Γ❆✸, ✶✵✽, ✶]✳ ❚❤✉s ψ✷ ✺✹ = ♣(❊✹, ❊✻, ψ✽, ψ✾, ❊✶✵, ❊✶✷)✳ ◮ ❲❡ ✇❛♥t t♦ s❤♦✇ t❤❛t A(Γ′ ❙) ∼

= C[❳✶, . . . , ❳✼]/(❳ ✷

✼ − ♣(❳✶, . . . , ❳✻))✳

❙♦ ❧❡t ◗ ∈ C[❳✶, . . . , ❳✼] s✉❝❤ t❤❛t ◗(❊✹, . . . , ❊✶✷, ψ✺✹) = ✵✳

◮ ❚❤❡r❡ ❡①✐st ◗✵, ◗✶ ∈ C[❳✶, . . . , ❳✻] s✉❝❤ t❤❛t

◗ ∈ ◗✵ + ❳✼◗✶ + (❳ ✷

✼ − ♣(❳✶, . . . , ❳✻))✳ ◮ ❚❤✉s ◗✵(❊✹, . . . , ❊✶✷) + ψ✺✹ · ◗✶(❊✹, . . . , ❊✶✷) = ✵✳ ◮ ❚❤❡r❡ ❡①✐sts ❛ ♠♦❞✉❧❛r s✉❜st✐t✉t✐♦♥ ♠❛♣♣✐♥❣ ψ✺✹ t♦ −ψ✺✹ ❛♥❞ ❧❡❛✈✐♥❣

❊✹, . . . , ❊✶✷ ✐♥✈❛r✐❛♥t❀ ❤❡♥❝❡ ◗✵(❊✹, . . . , ❊✶✷) − ψ✺✹ · ◗✶(❊✹, . . . , ❊✶✷) = ✵✳

◮ ❲❡ ❝♦♥❝❧✉❞❡ ◗✵(❊✹, . . . , ❊✶✷) = ✵ ❛♥❞ ◗✶(❊✹, . . . , ❊✶✷) = ✵✳ ◮ ❊✹, . . . , ❊✶✷ ❛r❡ ❛❧❣❡❜r❛✐❝❛❧❧② ✐♥❞❡♣❡♥❞❡♥t✳ ❚❤❡r❡❢♦r❡ ◗✵ = ◗✶ = ✵✱

✇❤✐❝❤ ❝♦♠♣❧❡t❡s t❤❡ ♣r♦♦❢✳

■♥❣♦ ❑❧ö❝❦❡r ✭❘❲❚❍ ❆❛❝❤❡♥✮ ❏❛♣❛♥❡s❡✲●❡r♠❛♥ ◆✉♠❜❡r ❚❤❡♦r② ❲♦r❦s❤♦♣ ✷✽ ✴ ✷✾

▼♦❞✉❧❛r ❋♦r♠s ❢♦r ❖(✷, ✺) ❚❤❡ ●r❛❞❡❞ ❘✐♥❣ ♦❢ ▼♦❞✉❧❛r ❋♦r♠s

Pr♦♦❢ ♦❢ t❤❡ ❙❡❝♦♥❞ ▼❛✐♥ ❘❡s✉❧t ✭❝♦♥t✬❞✮

Pr♦♦❢✳

◮ ❲❡ ❤❛✈❡ ψ✷ ✺✹ ∈ [Γ❆✸, ✶✵✽, ✶]✳ ❚❤✉s ψ✷ ✺✹ = ♣(❊✹, ❊✻, ψ✽, ψ✾, ❊✶✵, ❊✶✷)✳ ◮ ❲❡ ✇❛♥t t♦ s❤♦✇ t❤❛t A(Γ′ ❙) ∼

= C[❳✶, . . . , ❳✼]/(❳ ✷

✼ − ♣(❳✶, . . . , ❳✻))✳

❙♦ ❧❡t ◗ ∈ C[❳✶, . . . , ❳✼] s✉❝❤ t❤❛t ◗(❊✹, . . . , ❊✶✷, ψ✺✹) = ✵✳

◮ ❚❤❡r❡ ❡①✐st ◗✵, ◗✶ ∈ C[❳✶, . . . , ❳✻] s✉❝❤ t❤❛t

◗ ∈ ◗✵ + ❳✼◗✶ + (❳ ✷

✼ − ♣(❳✶, . . . , ❳✻))✳ ◮ ❚❤✉s ◗✵(❊✹, . . . , ❊✶✷) + ψ✺✹ · ◗✶(❊✹, . . . , ❊✶✷) = ✵✳ ◮ ❚❤❡r❡ ❡①✐sts ❛ ♠♦❞✉❧❛r s✉❜st✐t✉t✐♦♥ ♠❛♣♣✐♥❣ ψ✺✹ t♦ −ψ✺✹ ❛♥❞ ❧❡❛✈✐♥❣

❊✹, . . . , ❊✶✷ ✐♥✈❛r✐❛♥t❀ ❤❡♥❝❡ ◗✵(❊✹, . . . , ❊✶✷) − ψ✺✹ · ◗✶(❊✹, . . . , ❊✶✷) = ✵✳

◮ ❲❡ ❝♦♥❝❧✉❞❡ ◗✵(❊✹, . . . , ❊✶✷) = ✵ ❛♥❞ ◗✶(❊✹, . . . , ❊✶✷) = ✵✳ ◮ ❊✹, . . . , ❊✶✷ ❛r❡ ❛❧❣❡❜r❛✐❝❛❧❧② ✐♥❞❡♣❡♥❞❡♥t✳ ❚❤❡r❡❢♦r❡ ◗✵ = ◗✶ = ✵✱

✇❤✐❝❤ ❝♦♠♣❧❡t❡s t❤❡ ♣r♦♦❢✳

■♥❣♦ ❑❧ö❝❦❡r ✭❘❲❚❍ ❆❛❝❤❡♥✮ ❏❛♣❛♥❡s❡✲●❡r♠❛♥ ◆✉♠❜❡r ❚❤❡♦r② ❲♦r❦s❤♦♣ ✷✽ ✴ ✷✾

▼♦❞✉❧❛r ❋♦r♠s ❢♦r ❖(✷, ✺) ❚❤❡ ●r❛❞❡❞ ❘✐♥❣ ♦❢ ▼♦❞✉❧❛r ❋♦r♠s

Pr♦♦❢ ♦❢ t❤❡ ❙❡❝♦♥❞ ▼❛✐♥ ❘❡s✉❧t ✭❝♦♥t✬❞✮

Pr♦♦❢✳

◮ ❲❡ ❤❛✈❡ ψ✷ ✺✹ ∈ [Γ❆✸, ✶✵✽, ✶]✳ ❚❤✉s ψ✷ ✺✹ = ♣(❊✹, ❊✻, ψ✽, ψ✾, ❊✶✵, ❊✶✷)✳ ◮ ❲❡ ✇❛♥t t♦ s❤♦✇ t❤❛t A(Γ′ ❙) ∼

= C[❳✶, . . . , ❳✼]/(❳ ✷

✼ − ♣(❳✶, . . . , ❳✻))✳

❙♦ ❧❡t ◗ ∈ C[❳✶, . . . , ❳✼] s✉❝❤ t❤❛t ◗(❊✹, . . . , ❊✶✷, ψ✺✹) = ✵✳

◮ ❚❤❡r❡ ❡①✐st ◗✵, ◗✶ ∈ C[❳✶, . . . , ❳✻] s✉❝❤ t❤❛t

◗ ∈ ◗✵ + ❳✼◗✶ + (❳ ✷

✼ − ♣(❳✶, . . . , ❳✻))✳ ◮ ❚❤✉s ◗✵(❊✹, . . . , ❊✶✷) + ψ✺✹ · ◗✶(❊✹, . . . , ❊✶✷) = ✵✳ ◮ ❚❤❡r❡ ❡①✐sts ❛ ♠♦❞✉❧❛r s✉❜st✐t✉t✐♦♥ ♠❛♣♣✐♥❣ ψ✺✹ t♦ −ψ✺✹ ❛♥❞ ❧❡❛✈✐♥❣

❊✹, . . . , ❊✶✷ ✐♥✈❛r✐❛♥t❀ ❤❡♥❝❡ ◗✵(❊✹, . . . , ❊✶✷) − ψ✺✹ · ◗✶(❊✹, . . . , ❊✶✷) = ✵✳

◮ ❲❡ ❝♦♥❝❧✉❞❡ ◗✵(❊✹, . . . , ❊✶✷) = ✵ ❛♥❞ ◗✶(❊✹, . . . , ❊✶✷) = ✵✳ ◮ ❊✹, . . . , ❊✶✷ ❛r❡ ❛❧❣❡❜r❛✐❝❛❧❧② ✐♥❞❡♣❡♥❞❡♥t✳ ❚❤❡r❡❢♦r❡ ◗✵ = ◗✶ = ✵✱

✇❤✐❝❤ ❝♦♠♣❧❡t❡s t❤❡ ♣r♦♦❢✳

■♥❣♦ ❑❧ö❝❦❡r ✭❘❲❚❍ ❆❛❝❤❡♥✮ ❏❛♣❛♥❡s❡✲●❡r♠❛♥ ◆✉♠❜❡r ❚❤❡♦r② ❲♦r❦s❤♦♣ ✷✽ ✴ ✷✾

▼♦❞✉❧❛r ❋♦r♠s ❢♦r ❖(✷, ✺) ❚❤❡ ●r❛❞❡❞ ❘✐♥❣ ♦❢ ▼♦❞✉❧❛r ❋♦r♠s

Pr♦♦❢ ♦❢ t❤❡ ❙❡❝♦♥❞ ▼❛✐♥ ❘❡s✉❧t ✭❝♦♥t✬❞✮

Pr♦♦❢✳

◮ ❲❡ ❤❛✈❡ ψ✷ ✺✹ ∈ [Γ❆✸, ✶✵✽, ✶]✳ ❚❤✉s ψ✷ ✺✹ = ♣(❊✹, ❊✻, ψ✽, ψ✾, ❊✶✵, ❊✶✷)✳ ◮ ❲❡ ✇❛♥t t♦ s❤♦✇ t❤❛t A(Γ′ ❙) ∼

= C[❳✶, . . . , ❳✼]/(❳ ✷

✼ − ♣(❳✶, . . . , ❳✻))✳

❙♦ ❧❡t ◗ ∈ C[❳✶, . . . , ❳✼] s✉❝❤ t❤❛t ◗(❊✹, . . . , ❊✶✷, ψ✺✹) = ✵✳

◮ ❚❤❡r❡ ❡①✐st ◗✵, ◗✶ ∈ C[❳✶, . . . , ❳✻] s✉❝❤ t❤❛t

◗ ∈ ◗✵ + ❳✼◗✶ + (❳ ✷

✼ − ♣(❳✶, . . . , ❳✻))✳ ◮ ❚❤✉s ◗✵(❊✹, . . . , ❊✶✷) + ψ✺✹ · ◗✶(❊✹, . . . , ❊✶✷) = ✵✳ ◮ ❚❤❡r❡ ❡①✐sts ❛ ♠♦❞✉❧❛r s✉❜st✐t✉t✐♦♥ ♠❛♣♣✐♥❣ ψ✺✹ t♦ −ψ✺✹ ❛♥❞ ❧❡❛✈✐♥❣

❊✹, . . . , ❊✶✷ ✐♥✈❛r✐❛♥t❀ ❤❡♥❝❡ ◗✵(❊✹, . . . , ❊✶✷) − ψ✺✹ · ◗✶(❊✹, . . . , ❊✶✷) = ✵✳

◮ ❲❡ ❝♦♥❝❧✉❞❡ ◗✵(❊✹, . . . , ❊✶✷) = ✵ ❛♥❞ ◗✶(❊✹, . . . , ❊✶✷) = ✵✳ ◮ ❊✹, . . . , ❊✶✷ ❛r❡ ❛❧❣❡❜r❛✐❝❛❧❧② ✐♥❞❡♣❡♥❞❡♥t✳ ❚❤❡r❡❢♦r❡ ◗✵ = ◗✶ = ✵✱

✇❤✐❝❤ ❝♦♠♣❧❡t❡s t❤❡ ♣r♦♦❢✳

■♥❣♦ ❑❧ö❝❦❡r ✭❘❲❚❍ ❆❛❝❤❡♥✮ ❏❛♣❛♥❡s❡✲●❡r♠❛♥ ◆✉♠❜❡r ❚❤❡♦r② ❲♦r❦s❤♦♣ ✷✽ ✴ ✷✾

▼♦❞✉❧❛r ❋♦r♠s ❢♦r ❖(✷, ✺) ❚❤❡ ●r❛❞❡❞ ❘✐♥❣ ♦❢ ▼♦❞✉❧❛r ❋♦r♠s

Pr♦♦❢ ♦❢ t❤❡ ❙❡❝♦♥❞ ▼❛✐♥ ❘❡s✉❧t ✭❝♦♥t✬❞✮

Pr♦♦❢✳

◮ ❲❡ ❤❛✈❡ ψ✷ ✺✹ ∈ [Γ❆✸, ✶✵✽, ✶]✳ ❚❤✉s ψ✷ ✺✹ = ♣(❊✹, ❊✻, ψ✽, ψ✾, ❊✶✵, ❊✶✷)✳ ◮ ❲❡ ✇❛♥t t♦ s❤♦✇ t❤❛t A(Γ′ ❙) ∼

= C[❳✶, . . . , ❳✼]/(❳ ✷

✼ − ♣(❳✶, . . . , ❳✻))✳

❙♦ ❧❡t ◗ ∈ C[❳✶, . . . , ❳✼] s✉❝❤ t❤❛t ◗(❊✹, . . . , ❊✶✷, ψ✺✹) = ✵✳

◮ ❚❤❡r❡ ❡①✐st ◗✵, ◗✶ ∈ C[❳✶, . . . , ❳✻] s✉❝❤ t❤❛t

◗ ∈ ◗✵ + ❳✼◗✶ + (❳ ✷

✼ − ♣(❳✶, . . . , ❳✻))✳ ◮ ❚❤✉s ◗✵(❊✹, . . . , ❊✶✷) + ψ✺✹ · ◗✶(❊✹, . . . , ❊✶✷) = ✵✳ ◮ ❚❤❡r❡ ❡①✐sts ❛ ♠♦❞✉❧❛r s✉❜st✐t✉t✐♦♥ ♠❛♣♣✐♥❣ ψ✺✹ t♦ −ψ✺✹ ❛♥❞ ❧❡❛✈✐♥❣

❊✹, . . . , ❊✶✷ ✐♥✈❛r✐❛♥t❀ ❤❡♥❝❡ ◗✵(❊✹, . . . , ❊✶✷) − ψ✺✹ · ◗✶(❊✹, . . . , ❊✶✷) = ✵✳

◮ ❲❡ ❝♦♥❝❧✉❞❡ ◗✵(❊✹, . . . , ❊✶✷) = ✵ ❛♥❞ ◗✶(❊✹, . . . , ❊✶✷) = ✵✳ ◮ ❊✹, . . . , ❊✶✷ ❛r❡ ❛❧❣❡❜r❛✐❝❛❧❧② ✐♥❞❡♣❡♥❞❡♥t✳ ❚❤❡r❡❢♦r❡ ◗✵ = ◗✶ = ✵✱

✇❤✐❝❤ ❝♦♠♣❧❡t❡s t❤❡ ♣r♦♦❢✳

■♥❣♦ ❑❧ö❝❦❡r ✭❘❲❚❍ ❆❛❝❤❡♥✮ ❏❛♣❛♥❡s❡✲●❡r♠❛♥ ◆✉♠❜❡r ❚❤❡♦r② ❲♦r❦s❤♦♣ ✷✽ ✴ ✷✾

▼♦❞✉❧❛r ❋♦r♠s ❢♦r ❖(✷, ✺) ❚❤❡ ●r❛❞❡❞ ❘✐♥❣ ♦❢ ▼♦❞✉❧❛r ❋♦r♠s

Pr♦♦❢ ♦❢ t❤❡ ❙❡❝♦♥❞ ▼❛✐♥ ❘❡s✉❧t ✭❝♦♥t✬❞✮

Pr♦♦❢✳

◮ ❲❡ ❤❛✈❡ ψ✷ ✺✹ ∈ [Γ❆✸, ✶✵✽, ✶]✳ ❚❤✉s ψ✷ ✺✹ = ♣(❊✹, ❊✻, ψ✽, ψ✾, ❊✶✵, ❊✶✷)✳ ◮ ❲❡ ✇❛♥t t♦ s❤♦✇ t❤❛t A(Γ′ ❙) ∼

= C[❳✶, . . . , ❳✼]/(❳ ✷

✼ − ♣(❳✶, . . . , ❳✻))✳

❙♦ ❧❡t ◗ ∈ C[❳✶, . . . , ❳✼] s✉❝❤ t❤❛t ◗(❊✹, . . . , ❊✶✷, ψ✺✹) = ✵✳

◮ ❚❤❡r❡ ❡①✐st ◗✵, ◗✶ ∈ C[❳✶, . . . , ❳✻] s✉❝❤ t❤❛t

◗ ∈ ◗✵ + ❳✼◗✶ + (❳ ✷

✼ − ♣(❳✶, . . . , ❳✻))✳ ◮ ❚❤✉s ◗✵(❊✹, . . . , ❊✶✷) + ψ✺✹ · ◗✶(❊✹, . . . , ❊✶✷) = ✵✳ ◮ ❚❤❡r❡ ❡①✐sts ❛ ♠♦❞✉❧❛r s✉❜st✐t✉t✐♦♥ ♠❛♣♣✐♥❣ ψ✺✹ t♦ −ψ✺✹ ❛♥❞ ❧❡❛✈✐♥❣

❊✹, . . . , ❊✶✷ ✐♥✈❛r✐❛♥t❀ ❤❡♥❝❡ ◗✵(❊✹, . . . , ❊✶✷) − ψ✺✹ · ◗✶(❊✹, . . . , ❊✶✷) = ✵✳

◮ ❲❡ ❝♦♥❝❧✉❞❡ ◗✵(❊✹, . . . , ❊✶✷) = ✵ ❛♥❞ ◗✶(❊✹, . . . , ❊✶✷) = ✵✳ ◮ ❊✹, . . . , ❊✶✷ ❛r❡ ❛❧❣❡❜r❛✐❝❛❧❧② ✐♥❞❡♣❡♥❞❡♥t✳ ❚❤❡r❡❢♦r❡ ◗✵ = ◗✶ = ✵✱

✇❤✐❝❤ ❝♦♠♣❧❡t❡s t❤❡ ♣r♦♦❢✳

■♥❣♦ ❑❧ö❝❦❡r ✭❘❲❚❍ ❆❛❝❤❡♥✮ ❏❛♣❛♥❡s❡✲●❡r♠❛♥ ◆✉♠❜❡r ❚❤❡♦r② ❲♦r❦s❤♦♣ ✷✽ ✴ ✷✾

▼♦❞✉❧❛r ❋♦r♠s ❢♦r ❖(✷, ✺) ❚❤❡ ●r❛❞❡❞ ❘✐♥❣ ♦❢ ▼♦❞✉❧❛r ❋♦r♠s

Pr♦♦❢ ♦❢ t❤❡ ❙❡❝♦♥❞ ▼❛✐♥ ❘❡s✉❧t ✭❝♦♥t✬❞✮

Pr♦♦❢✳

◮ ❲❡ ❤❛✈❡ ψ✷ ✺✹ ∈ [Γ❆✸, ✶✵✽, ✶]✳ ❚❤✉s ψ✷ ✺✹ = ♣(❊✹, ❊✻, ψ✽, ψ✾, ❊✶✵, ❊✶✷)✳ ◮ ❲❡ ✇❛♥t t♦ s❤♦✇ t❤❛t A(Γ′ ❙) ∼

= C[❳✶, . . . , ❳✼]/(❳ ✷

✼ − ♣(❳✶, . . . , ❳✻))✳

❙♦ ❧❡t ◗ ∈ C[❳✶, . . . , ❳✼] s✉❝❤ t❤❛t ◗(❊✹, . . . , ❊✶✷, ψ✺✹) = ✵✳

◮ ❚❤❡r❡ ❡①✐st ◗✵, ◗✶ ∈ C[❳✶, . . . , ❳✻] s✉❝❤ t❤❛t

◗ ∈ ◗✵ + ❳✼◗✶ + (❳ ✷

✼ − ♣(❳✶, . . . , ❳✻))✳ ◮ ❚❤✉s ◗✵(❊✹, . . . , ❊✶✷) + ψ✺✹ · ◗✶(❊✹, . . . , ❊✶✷) = ✵✳ ◮ ❚❤❡r❡ ❡①✐sts ❛ ♠♦❞✉❧❛r s✉❜st✐t✉t✐♦♥ ♠❛♣♣✐♥❣ ψ✺✹ t♦ −ψ✺✹ ❛♥❞ ❧❡❛✈✐♥❣

❊✹, . . . , ❊✶✷ ✐♥✈❛r✐❛♥t❀ ❤❡♥❝❡ ◗✵(❊✹, . . . , ❊✶✷) − ψ✺✹ · ◗✶(❊✹, . . . , ❊✶✷) = ✵✳

◮ ❲❡ ❝♦♥❝❧✉❞❡ ◗✵(❊✹, . . . , ❊✶✷) = ✵ ❛♥❞ ◗✶(❊✹, . . . , ❊✶✷) = ✵✳ ◮ ❊✹, . . . , ❊✶✷ ❛r❡ ❛❧❣❡❜r❛✐❝❛❧❧② ✐♥❞❡♣❡♥❞❡♥t✳ ❚❤❡r❡❢♦r❡ ◗✵ = ◗✶ = ✵✱

✇❤✐❝❤ ❝♦♠♣❧❡t❡s t❤❡ ♣r♦♦❢✳

■♥❣♦ ❑❧ö❝❦❡r ✭❘❲❚❍ ❆❛❝❤❡♥✮ ❏❛♣❛♥❡s❡✲●❡r♠❛♥ ◆✉♠❜❡r ❚❤❡♦r② ❲♦r❦s❤♦♣ ✷✽ ✴ ✷✾

▼♦❞✉❧❛r ❋♦r♠s ❢♦r ❖(✷, ✺) ❚❤❡ ●r❛❞❡❞ ❘✐♥❣ ♦❢ ▼♦❞✉❧❛r ❋♦r♠s

❋✐❡❧❞s ♦❢ ❖rt❤♦❣♦♥❛❧ ▼♦❞✉❧❛r ❋✉♥❝t✐♦♥s

❈♦r♦❧❧❛r②

✶✳ ❚❤❡ ✜❡❧❞ K(Γ❆✸) ♦❢ ♦rt❤♦❣♦♥❛❧ ♠♦❞✉❧❛r ❢✉♥❝t✐♦♥s ✇✐t❤ r❡s♣❡❝t t♦ Γ❆✸ ❛♥❞ t❤❡ tr✐✈✐❛❧ ❝❤❛r❛❝t❡r ✐s ❛ r❛t✐♦♥❛❧ ❢✉♥❝t✐♦♥ ✜❡❧❞ ✐♥ t❤❡ ❣❡♥❡r❛t♦rs ❊ ✷

✻

❊ ✸

✹

, ψ✽ ❊ ✷

✹

, ❊✶✵ ❊✹ ❊✻ , ❊✶✷ ❊ ✸

✹

❛♥❞ ψ✷

✾

❊ ✸

✻

. ✷✳ ❚❤❡ ✜❡❧❞ K(Γ′

❆✸) ♦❢ ❛❧❧ ♦rt❤♦❣♦♥❛❧ ♠♦❞✉❧❛r ❢✉♥❝t✐♦♥s ✇✐t❤ r❡s♣❡❝t t♦

Γ′

❆✸ ✐s ❛♥ ❡①t❡♥s✐♦♥ ♦❢ ❞❡❣r❡❡ ✷ ♦✈❡r K(Γ❆✸) ❣❡♥❡r❛t❡❞ ❜② ψ✺✹/ψ✻ ✾✳

■♥❣♦ ❑❧ö❝❦❡r ✭❘❲❚❍ ❆❛❝❤❡♥✮ ❏❛♣❛♥❡s❡✲●❡r♠❛♥ ◆✉♠❜❡r ❚❤❡♦r② ❲♦r❦s❤♦♣ ✷✾ ✴ ✷✾

▼♦❞✉❧❛r ❋♦r♠s ❢♦r ❖(✷, ✺) ❚❤❡ ●r❛❞❡❞ ❘✐♥❣ ♦❢ ▼♦❞✉❧❛r ❋♦r♠s

❋✐❡❧❞s ♦❢ ❖rt❤♦❣♦♥❛❧ ▼♦❞✉❧❛r ❋✉♥❝t✐♦♥s

❈♦r♦❧❧❛r②

✶✳ ❚❤❡ ✜❡❧❞ K(Γ❆✸) ♦❢ ♦rt❤♦❣♦♥❛❧ ♠♦❞✉❧❛r ❢✉♥❝t✐♦♥s ✇✐t❤ r❡s♣❡❝t t♦ Γ❆✸ ❛♥❞ t❤❡ tr✐✈✐❛❧ ❝❤❛r❛❝t❡r ✐s ❛ r❛t✐♦♥❛❧ ❢✉♥❝t✐♦♥ ✜❡❧❞ ✐♥ t❤❡ ❣❡♥❡r❛t♦rs ❊ ✷

✻

❊ ✸

✹

, ψ✽ ❊ ✷

✹

, ❊✶✵ ❊✹ ❊✻ , ❊✶✷ ❊ ✸

✹

❛♥❞ ψ✷

✾

❊ ✸

✻

. ✷✳ ❚❤❡ ✜❡❧❞ K(Γ′

❆✸) ♦❢ ❛❧❧ ♦rt❤♦❣♦♥❛❧ ♠♦❞✉❧❛r ❢✉♥❝t✐♦♥s ✇✐t❤ r❡s♣❡❝t t♦

Γ′

❆✸ ✐s ❛♥ ❡①t❡♥s✐♦♥ ♦❢ ❞❡❣r❡❡ ✷ ♦✈❡r K(Γ❆✸) ❣❡♥❡r❛t❡❞ ❜② ψ✺✹/ψ✻ ✾✳

■♥❣♦ ❑❧ö❝❦❡r ✭❘❲❚❍ ❆❛❝❤❡♥✮ ❏❛♣❛♥❡s❡✲●❡r♠❛♥ ◆✉♠❜❡r ❚❤❡♦r② ❲♦r❦s❤♦♣ ✷✾ ✴ ✷✾