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By V. Dokchitser, Sebastian Pancratz

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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 .

K→C ❲❡ ♥♦t❡ t❤❛t ❝♦♠♣❧❡① ❝♦♥❥✉❣❛t❡ ❡♠❜❡❞❞✐♥❣s ❣✐✈❡ r✐s❡ t♦ t❤❡ s❛♠❡ ❛❜s♦❧✉t❡ ✈❛❧✉❡s✳ ■♥ ❢❛❝t✱ t❤✐s ✐s t❤❡ ♦♥❧② ❝❛s❡ ✇❤❡♥ t✇♦ ♦❢ t❤❡s❡ ❡♠❜❡❞❞✐♥❣s ❣✐✈❡ ❡q✉✐✈❛❧❡♥t ❛❜s♦❧✉t❡ ✈❛❧✉❡s✱ ❛♥❞ t❤❡s❡ ❛r❡ ❛❧❧ t❤❡ ❛r❝❤✐♠❡❞❡❛♥ ❛❜s♦❧✉t❡ ✈❛❧✉❡s ♦♥ K ✉♣ t♦ ❡q✉✐✈❛❧❡♥❝❡✳ ❚❤❡ ♥✉♠❜❡r ♦❢ ✐♥✜♥✐t❡ ♣❧❛❝❡s ✐s r1 + r2 ✳ ❋✐♥✐t❡ ♣❧❛❝❡s✱ ✐✳❡✳✱ ♥♦♥✲❛r❝❤✐♠❡❞❡❛♥ ❛❜s♦❧✉t❡ ✈❛❧✉❡s✱ ❝♦rr❡s♣♦♥❞ t♦ ♣r✐♠❡s ✐♥ K ❛s ❢♦❧❧♦✇s✳ ■❢ P ✐s ❛ ♣r✐♠❡✱ s❡t |x|P = N (P )− ordP (x) ✇❤❡r❡ ordP (x) ✐s✱ ❢♦r x ∈ OK ✱ t❤❡ ♣♦✇❡r ♦❢ P ✐♥ t❤❡ ❢❛❝t♦r✐s❛t✐♦♥ ♦❢ (x) ❛♥❞ ❡①t❡♥❞ t❤✐s ♠✉❧t✐♣❧✐❝❛t✐✈❡❧② t♦ K ∗ ✳ ■t ✐s ❛ ❢❛❝t t❤❛t t❤❡s❡ ❛r❡ ✐♥❡q✉✐✈❛❧❡♥t ❢♦r ❞✐✛❡r❡♥t P ❛♥❞ ❣✐✈❡ ❛❧❧ t❤❡ ♥♦♥✲❛r❝❤✐♠❡❞❡❛♥ ❛❜s♦❧✉t❡ ✈❛❧✉❡s ✉♣ t♦ ❡q✉✐✈❛❧❡♥❝❡✳ ◆♦t❡ t❤❛t |·|v ♠❛❦❡s K ❛ ♠❡tr✐❝ s♣❛❝❡ ❛♥❞ ✐ts ❝♦♠♣❧❡t✐♦♥ Kv ✐s ❛ ❝♦♠♣❧❡t❡ ❧♦❝❛❧ ✜❡❧❞✳ ❍❡♥❝❡❢♦rt❤ ❛ss✉♠❡ t❤❛t v ✐s ✜♥✐t❡✳ ❊①❛♠♣❧❡✳ • ■❢ K = Q ❛♥❞ v ❝♦rr❡s♣♦♥❞s t♦ P = (p) t❤❡♥ |·|v = |·|p ❛♥❞ Kv = Qp ✳ • ■❢ K ✐s ❛ ♥✉♠❜❡r ✜❡❧❞ ❛♥❞ v ❝♦rr❡s♣♦♥❞s t♦ Q ❛❜♦✈❡ p ∈ Q t❤❡♥ |·|v r❡str✐❝t❡❞ t♦ Q ✐s ❡q✉✐✈❛❧❡♥t t♦ |·|p ✳ ❚❤❡r❡❢♦r❡✱ Kv ✐s ❛ ✜♥✐t❡ ❡①t❡♥s✐♦♥ ♦❢ Q✳ ✸✳✽✳✶ ❘❡s✐❞✉❡ ✜❡❧❞s ❛♥❞ r❛♠✐✜❝❛t✐♦♥ ❲❡ ❝♦♥s✐❞❡r t❤❡ ❢♦❧❧♦✇✐♥❣ s❡tt✐♥❣✳ ▲❡t K ❜❡ ❛ ♥✉♠❜❡r ✜❡❧❞✱ v ❛ ✜♥✐t❡ ♣❧❛❝❡ ❝♦rr❡s♣♦♥❞✐♥❣ t♦ Q ❛♥❞ Kv ✐ts ❝♦♠♣❧❡t✐♦♥✳ ▼♦r❡♦✈❡r✱ ❧❡t OKv ❜❡ ✐ts ✈❛❧✉❛t✐♦♥ r✐♥❣ ❛♥❞ Mv ✐ts ✉♥✐q✉❡ × ♠❛①✐♠❛❧ ✐❞❡❛❧✳ ❋✐♥❛❧❧②✱ OK ✐s t❤❡ s❡t ♦❢ ✉♥✐ts ✐♥ OKv ❛♥❞ kv = OKv /Mv t❤❡ r❡s✐❞✉❡ v ✜❡❧❞✳ ❲❡ ♦❜s❡r✈❡ t❤❛t ✐❢ Q ⊂ Mv ❛♥❞ OK ⊂ OKv t❤❡♥ t❤❡ ♠❛♣ OK /Q → OKv /Mv = kv ✐s ✐♥❥❡❝t✐✈❡✱ ❛s ✐t ✐s ❜❡t✇❡❡♥ ✜❡❧❞s✱ ❛♥❞ s✉r❥❡❝t✐✈❡✱ ❛s ❡✈❡r② ❡❧❡♠❡♥t ♦❢ Kv ❝❛♥ ❜❡ ❛♣♣r♦①✲ ✐♠❛t❡❞ ❜② ❛♥ ❡❧❡♠❡♥t ♦❢ K ✳ ❚❤✉s OK /Q ∼ = kv ✳ ✸✻ L✲❙❡r✐❡s ▲❡t L/K ❜❡ ❛ ✜♥✐t❡ ❡①t❡♥s✐♦♥ ♦❢ ♥✉♠❜❡r ✜❡❧❞s ❛♥❞ s✉♣♣♦s❡ t❤❛t R ❧✐❡s ❛❜♦✈❡ Q ✇✐t❤ ♣❧❛❝❡ w ❝♦rr❡s♣♦♥❞✐♥❣ t♦ R✳ ❖♥❡ ❝❛♥ ❝❤❡❝❦ t❤❛t |·|w ❡①t❡♥❞s |·|v ✳ ❚❤❡♥ Lw /Kv ✐s ❛ ✜♥✐t❡ ❡①t❡♥s✐♦♥ ❛♥❞✱ ❜② ❝♦♠♣❛r✐♥❣ ✈❛❧✉❛t✐♦♥s✱ eR/Q = ew/v , fR/Q = fw/v .

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 .

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