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By V. Dokchitser, Sebastian Pancratz

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Dd dd d K P Qk SK ~~ ~~ ~ ~ ■t r❡♠❛✐♥s t♦ s❤♦✇ t❤❛t det 1 − T FrobQ/P det 1 − T fQi /P FrobQi /Si τ IQi /Si . IQ/P (IndG = H τ) Si ❙t❡♣ ✶✳ ❆ss✉♠❡ t❤❡r❡ ✐s ❛ ✉♥✐q✉❡ ♣r✐♠❡ ✐♥ F ❛❜♦✈❡ P ✳ ◆♦t❡ t❤❛t ✐t s✉✣❝❡s t♦ s❤♦✇ t❤❡ ❡q✉❛❧✐t② ✇❤❡♥ τ ✐s ✐rr❡❞✉❝✐❜❧❡✳ ❲r✐t❡ IndG Hτ = i σi ✱ ✇❤❡r❡ σi ❛r❡ ✐rr❡❞✉❝✐❜❧❡ r❡♣r❡s❡♥t❛t✐♦♥s ♦❢ G✳ • ■❢ τ IQ/S = 0 t❤❡♥ IQ/S ❛❝ts ♥♦♥✲tr✐✈✐❛❧❧② ♦♥ τ ✱ s♦ ❜② ❋r♦❜❡♥✐✉s r❡❝✐♣r♦❝✐t② IQ/P ❛❝ts I ♥♦♥✲tr✐✈✐❛❧❧② ♦♥σi ❛♥❞ σi , Ind τ = Res σi , τ ✳ ❚❤❡♥ σi Q/P = 0 s♦ (Ind τ )IQ/P = 0✱ ❛♥❞ ♥♦✇ t❤❡ r❡s✉❧t ✐s tr✐✈✐❛❧✳ • ■❢ τ IQ/S = 0 t❤❡♥ IQ/S ❛❝ts tr✐✈✐❛❧❧② ♦♥ τ ✱ s♦ τ ✐s 1✲❞✐♠❡♥s✐♦♥❛❧✱ τ (IQ/S ) = 1✱ τ (FrobQ/S ) = ζn ✱ s❛②✳ ❙♦ det 1 − T FrobQ/S τ IQ/S = 1 − ζn T f .

Pr♦♦❢✳ ■t ✐s s✉✣❝✐❡♥t t♦ ❝❤❡❝❦ ❡❛❝❤ st❛t❡♠❡♥t ♣r✐♠❡✲❜②✲♣r✐♠❡ ❢♦r t❤❡ ❧♦❝❛❧ ♣♦❧②♥♦♠✐❛❧s✳ ✭✐✮ ❖❜s❡r✈❡ (ρ ⊕ τ )IP = ρIP ⊕ τ IP ✳ ✭✐✐✮ ❙tr❛✐❣❤t ❢r♦♠ t❤❡ ❞❡✜♥✐t✐♦♥✳ ❖❜s❡r✈❡ t❤❛t ❋r♦❜❡♥✐✉s ❢♦r F/K ♣r♦❥❡❝ts t♦ ❋r♦❜❡♥✐✉s ❢♦r F N /K ❛♥❞ s✐♠✐❧❛r❧② ❢♦r ✐♥❡rt✐❛✳ ✭✐✐✐✮ ▲❡t S1 , . . , Sk ❜❡ t❤❡ ♣r✐♠❡s ♦❢ F N ❛❜♦✈❡ P ❛♥❞ t❛❦❡ Qi t♦ ❜❡ ❛ ♣r✐♠❡ ♦❢ F ❛❜♦✈❡ Si ✱ s❛② Q = Q1 ✱ Qi = xi Q ❢♦r s♦♠❡ xi ∈ Gal(F/K)✳     F Nd dd dd d ... Q1 F S1 d G=Gal(F/K) ... dd dd d K P Qk SK ~~ ~~ ~ ~ ■t r❡♠❛✐♥s t♦ s❤♦✇ t❤❛t det 1 − T FrobQ/P det 1 − T fQi /P FrobQi /Si τ IQi /Si .

Pr♦♣❡rt✐❡s ♦❢ ❆rt✐♥ L✲❋✉♥❝t✐♦♥s Pr♦♣♦s✐t✐♦♥ ✸✳✶✹✳ ▲❡t F/K ❜❡ ❛ ●❛❧♦✐s ❡①t❡♥s✐♦♥ ♦❢ ♥✉♠❜❡r ✜❡❧❞s ❛♥❞ ρ ❜❡ ❛ r❡♣r❡✲ s❡♥t❛t✐♦♥ ♦❢ Gal(F/K)✳ ✭✐✮ ■❢ τ ✐s ❛♥♦t❤❡r Gal(F/K)✲r❡♣r❡s❡♥t❛t✐♦♥ t❤❡♥ L(ρ ⊕ τ, s) = L(ρ, s)L(τ, s). ✭✐✐✮ ■❢ N Gal(F/K) ❛♥❞ ρ ✐s tr✐✈✐❛❧ ♦♥ N s♦ ρ ❝♦♠❡s ❢r♦♠ ❛ r❡♣r❡s❡♥t❛t✐♦♥ ρ ♦❢ Gal(F/K)/N ∼ = Gal(F N /K) t❤❡♥ L(F/K, ρ, s) = L(F N /K, ρ , s). Gal(F/K) ✭✐✐✐✮ ✭❆rt✐♥ ❋♦r♠❛❧✐s♠✮ ■❢ ρ = IndH τ ❢♦r s♦♠❡ H ≤ G✱ τ ❛♥ H ✲r❡♣r❡s❡♥t❛t✐♦♥ t❤❡♥ L(F/K, ρ, s) = L(F/F H , τ, s). Pr♦♦❢✳ ■t ✐s s✉✣❝✐❡♥t t♦ ❝❤❡❝❦ ❡❛❝❤ st❛t❡♠❡♥t ♣r✐♠❡✲❜②✲♣r✐♠❡ ❢♦r t❤❡ ❧♦❝❛❧ ♣♦❧②♥♦♠✐❛❧s✳ ✭✐✮ ❖❜s❡r✈❡ (ρ ⊕ τ )IP = ρIP ⊕ τ IP ✳ ✭✐✐✮ ❙tr❛✐❣❤t ❢r♦♠ t❤❡ ❞❡✜♥✐t✐♦♥✳ ❖❜s❡r✈❡ t❤❛t ❋r♦❜❡♥✐✉s ❢♦r F/K ♣r♦❥❡❝ts t♦ ❋r♦❜❡♥✐✉s ❢♦r F N /K ❛♥❞ s✐♠✐❧❛r❧② ❢♦r ✐♥❡rt✐❛✳ ✭✐✐✐✮ ▲❡t S1 , .

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