By V. Dokchitser, Sebastian Pancratz

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D ❛r❡ t❤❡ ❡✐❣❡♥✈❛❧✉❡s ✭✇✐t❤ ♠✉❧t✐♣❧✐❝✐t②✮ ♦❢ Frobp ♦♥ ρ✱ t❤❡♥ log 1 = log Pp (ρ, p−s ) = i = 1 −s i (1 − λi p ) 1 log 1 − λi p−s λi p−s + 1 2 λ2i p−2s + · · · 1 = χρ (Frobp )p−s + χρ (Frob2p )p−2s + · · · . 2 ✸✺ ❚❤❡ ❉✐r✐❝❤❧❡t s❡r✐❡s p ✉♥r❛♠✐✜❡❞ n≥1 χρ (Frobnp ) −ns p n ❤❛s ❜♦✉♥❞❡❞ ❝♦❡✣❝✐❡♥ts✱ s♦ ❜② Pr♦♣♦s✐t✐♦♥ ✸✳✽ ❛♥❞ ✐ts ♣r♦♦❢ ❞❡✜♥❡s ❛♥ ❛♥❛❧②t✐❝ ❜r❛♥❝❤ ♦❢ log L∗ (ρ, s) ♦♥ (s) > 1❀ ❜② t❤❡ ✜rst ♣❛rt✱ ✐t ♠✉st ❜❡ ❜♦✉♥❞❡❞ ❛s s → 1 ♦♥ (s) > 1✳ ✸✳✽ ❉❡✜♥✐t✐♦♥✳ ✈❛❧✉❡s✳ ❆♣♣❡♥❞✐① ✭▲♦❝❛❧ ❋✐❡❧❞s✮ ❆ ♣❧❛❝❡ v ✐♥ ❛ ♥✉♠❜❡r ✜❡❧❞ K ✐s ❛♥ ❡q✉✐✈❛❧❡♥❝❡ ❝❧❛ss ♦❢ ♥♦♥✲tr✐✈✐❛❧ ❛❜s♦❧✉t❡ ❚❤❡r❡ ❛r❡ t✇♦ t②♣❡s✿ ■♥✜♥✐t❡ ♣❧❛❝❡s✱ ✐✳❡✳✱ ❛r❝❤✐♠❡❞❡❛♥ ❛❜s♦❧✉t❡ ✈❛❧✉❡s✱ ❝♦♠❡ ❢r♦♠ ❡♠✲ ❜❡❞❞✐♥❣s K → R ♦r K → C ❛♥❞ t❛❦❡ |x|v = |x| |x|2 K→R .

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 . 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 .

N ) ❙♦ p∈Pa 1 = ps χ χ(a) φ(N ) χ(p)p−s . p ♣r✐♠❡ ✷✽ L✲❙❡r✐❡s ❊❛❝❤ t❡r♠ ♦♥ t❤❡ r✐❣❤t✲❤❛♥❞ s✐❞❡ ✐s ❜♦✉♥❞❡❞ ❛s s → 1 ❡①❝❡♣t ❢♦r t❤❡ ❝♦♥tr✐❜✉t✐♦♥ ❢r♦♠ χ = I✱ s♦ 1 1 1 1 ∼ p−s ∼ log ps φ(N ) φ(N ) s−1 p∈Pa pN ❛s s → 1 ❜② ❈♦r♦❧❧❛r② ✸✳✾✳ ✸✳✹ ❉✐r✐❝❤❧❡t ❈❤❛r❛❝t❡rs ❘❡❝❛❧❧ t❤❛t ∼ (Z/N Z)× − → Gal Q(ζN )/Q a → σa a σa (ζN ) = ζN p → σp p σp (ζN ) = ζN ■❢ Q ✐s ❛ ♣r✐♠❡ ♦❢ Q(ζN ) ❛❜♦✈❡ p N t❤❡♥ σP = FrobQ/P ✳ ◆♦t❛t✐♦♥✳ ■❢ F/K ✐s ❛ ●❛❧♦✐s ❡①t❡♥s✐♦♥ ♦❢ ♥✉♠❜❡r ✜❡❧❞s ✇✐t❤ Gal(F/K) ❛❜❡❧✐❛♥✱ ❛♥❞ P ✐s ❛ ♣r✐♠❡ ♦❢ K ✉♥r❛♠✐✜❡❞ ✐♥ F/K ✱ ✇r✐t❡ FrobP ∈ Gal(F/K) ❢♦r t❤❡ ❋r♦❜❡♥✐✉s ❡❧❡♠❡♥t ♦❢ ❛♥② ♣r✐♠❡ ❛❜♦✈❡ P ✱ ✐♥❞❡♣❡♥❞❡♥t ♦❢ Q ❛❜♦✈❡ P ❛s t❤❡ ❞❡❝♦♠♣♦s✐t✐♦♥ ❣r♦✉♣s ❛r❡ ❝♦♥❥✉❣❛t❡✱ ❛♥❞ I = 1 ❛s P ✐s ✉♥r❛♠✐✜❡❞✳ ❚❤❡♦r❡♠ ✸✳✶✶ ✭❍❡❝❦❡✱ ✶✾✷✵✱ ❈❧❛ss ❋✐❡❧❞ ❚❤❡♦r② ✮✳ ▲❡t F/K ❜❡ ❛ ●❛❧♦✐s ❡①t❡♥s✐♦♥ ♦❢ ♥✉♠❜❡r ✜❡❧❞s ✇✐t❤ Gal(F/K) ❛❜❡❧✐❛♥✱ ❛♥❞ ψ : Gal(F/K) → C× ❛ ❤♦♠♦♠♦r♣❤✐s♠✳ ❚❤❡♥ 1 L∗ (ψ, s) = 1 − ψ(FrobP )N (P )−s p ♣r✐♠❡s ♦❢ K ✉♥r❛♠✐✜❡❞ ✐♥ F/K ❤❛s ❛♥ ❛♥❛❧②t✐❝ ❝♦♥t✐♥✉❛t✐♦♥ t♦ C✱ ❡①❝❡♣t ❢♦r ❛ s✐♠♣❧❡ ♣♦❧❡ ❛t s = 1 ✇❤❡♥ ψ = I✳ Pr♦♦❢✳ ❖♠✐tt❡❞✳ ❘❡♠❛r❦✳ ❲❤❡♥ K = Q✱ F = Q(ζN ) t❤✐s r❡❝♦✈❡rs ❚❤❡♦r❡♠ ✸✳✻✱ ❛♥❞ ♠♦r❡✳ ✸✳✺ ◆♦t❛t✐♦♥✳ ❆rt✐♥ L✲❋✉♥❝t✐♦♥s ■❢ I ≤ D ❛r❡ ✜♥✐t❡ ❣r♦✉♣s✱ ρ ❛ D✲r❡♣r❡s❡♥t❛t✐♦♥✱ ✇r✐t❡ ρI = {v ∈ ρ : ∀g ∈ I gv = v} ❢♦r t❤❡ s✉❜s♣❛❝❡ ♦❢ I ✲✐♥✈❛r✐❛♥t ✈❡❝t♦rs✳ ❘❡♠❛r❦✳ ■❢ I D t❤❡♥ ρI ✐s ❛ D✲s✉❜r❡♣r❡s❡♥t❛t✐♦♥✳ ■❢ v ∈ ρI ✱ g ∈ D✱ i ∈ I t❤❡♥ i(gv) = g(i v) = gv ❢♦r s♦♠❡ i ∈ I ✱ s♦ gv ∈ ρI ✳ ✷✾ ❉❡✜♥✐t✐♦♥✳ ▲❡t F/K ❜❡ ❛ ●❛❧♦✐s ❡①t❡♥s✐♦♥ ♦❢ ♥✉♠❜❡r ✜❡❧❞s✱ ❧❡t ρ ❜❡ ❛ Gal(F/K)✲ r❡♣r❡s❡♥t❛t✐♦♥✳ ▲❡t P ❜❡ ❛ ♣r✐♠❡ ♦❢ K ✱ ❛♥❞ ❝❤♦♦s❡ Q ❛ ♣r✐♠❡ ♦❢ F ❛❜♦✈❡ K ✱ ❝❤♦♦s❡ FrobP t♦ ❜❡ ❛♥ ❡❧❡♠❡♥t ♦❢ DQ/P ✇❤✐❝❤ ✐♥ DQ/P /IQ/P ✐s FrobQ/P ✱ ✐✳❡✳✱ FrobP ❛❝ts ♦♥ t❤❡ r❡s✐❞✉❡ ✜❡❧❞ ❛s ❋r♦❜❡♥✐✉s✳ ❚❤❡♥ t❤❡ ❧♦❝❛❧ ♣♦❧②♥♦♠✐❛❧ ♦❢ ρ ❛t P ✐s PP (ρ, T ) = PP (F/K, ρ, T ) ρIP = det 1 − T FrobP Gal F/K ✇❤❡r❡ IP = IQ/P ❛♥❞ t❤❡ r✐❣❤t✲❤❛♥❞ s✐❞❡ ✐s det(1 − T FrobP ) ❛❝t✐♥❣ ❛t ResDQ/P ▲❡♠♠❛ ✸✳✶✷✳ ρ IP ✳ PP (ρ, T ) ✐s ✐♥❞❡♣❡♥❞❡♥t ♦❢ t❤❡ ❝❤♦✐❝❡ ♦❢ Q ❛♥❞ FrobP ✳ Pr♦♦❢✳ ❋♦r ✜①❡❞ Q✱ ✐♥❞❡♣❡♥❞❡♥❝❡ ♦❢ ❝❤♦✐❝❡ ♦❢ FrobP ✐s ❝❧❡❛r✿ ❛♥♦t❤❡r ❝❤♦✐❝❡ ❞✐✛❡rs ❜② ❛♥ ❡❧❡♠❡♥t ♦❢ IQ/P ✇❤✐❝❤ ❛❝ts ❛s t❤❡ ✐❞❡♥t✐t② ❛t ρIQ/P ✳ ■❢ Q = gQ ✐s ❛♥♦t❤❡r ♣r✐♠❡✱ g ∈ Gal(F/K)✱ t❤❡♥ ✇❡ ❝❛♥ t❛❦❡ FrobP ❢♦r Q t♦ ❜❡ −1 g FrobP g −1 ❛♥❞ ♦❜s❡r✈❡ t❤❛t ❡✐❣❡♥✈❛❧✉❡s ✭✇✐t❤ ♠✉❧t✐♣❧✐❝✐t✐❡s✮ ♦❢ g FrobP g −1 ♦♥ ρgIP g ❛❣r❡❡ ✇✐t❤ ❡✐❣❡♥✈❛❧✉❡s ♦❢ FrobP ♦♥ ρIP ✱ s♦ ❤❛✈❡ t❤❡ s❛♠❡ ♠✐♥✐♠❛❧ ♣♦❧②♥♦♠✐❛❧ ❛♥❞ ❤❡♥❝❡ ❣✐✈❡ t❤❡ s❛♠❡ ❧♦❝❛❧ ❢❛❝t♦rs✳ ❉❡✜♥✐t✐♦♥✳ ▲❡t F/K ❜❡ ❛ ●❛❧♦✐s ❡①t❡♥s✐♦♥ ♦❢ ♥✉♠❜❡r ✜❡❧❞s✱ ❛♥❞ ρ ❜❡ ❛ Gal(F/K)✲ r❡♣r❡s❡♥t❛t✐♦♥✳ ❚❤❡ ❆rt✐♥ L✲❢✉♥❝t✐♦♥ ♦❢ ρ ✐s ❞❡✜♥❡❞ ❜② t❤❡ ❊✉❧❡r ♣r♦❞✉❝t L(F/K, ρ, s) = L(ρ, s) = K ♣r✐♠❡ ♦❢ K 1 .

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Algebraic Number Theory by V. Dokchitser, Sebastian Pancratz


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