By V. Dokchitser, Sebastian Pancratz

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N ✭✐✐✮ ❋♦❧❧♦✇s ❢r♦♠ ❚❤❡♦r❡♠ ✸✳✼✳ ■❢ ψ ✐s ♥♦♥✲tr✐✈✐❛❧ t❤❡♥ L(ψ, s) ❝♦♥✈❡r❣❡s t♦ ❛ ♥♦♥✲③❡r♦ ✈❛❧✉❡ ❛s s → 1✱ ❤❡♥❝❡ ✐ts ❧♦❣❛r✐t❤♠ ✐s ❜♦✉♥❞❡❞ ♥❡❛r 1✳ λ 1 L(I, s) ❤❛s ❛ s✐♠♣❧❡ ♣♦❧❡ ❛t s = 1✱ ❤❡♥❝❡ L(I, s) ∼ 1−s ✱ s♦ log L(I, s) ∼ s−1 ❛s s → 1✳ ❈♦r♦❧❧❛r② ✸✳✾✳ tr✐✈✐❛❧ t❤❡♥ ∞✳ ■❢ ψ ✐s ♥♦♥✲tr✐✈✐❛❧ t❤❡♥ ψ(p)p−s = p N p−s ∼ log −s p ♣r✐♠❡ ψ(p)p 1 s−1 ❛s s → 1✳ ✐s ❜♦✉♥❞❡❞ ❛s s → 1✳ ■❢ ψ ✐s ■♥ ♣❛rt✐❝✉❧❛r✱ ✐t ❝♦♥✈❡r❣❡s t♦ ✷✼ Pr♦♦❢✳ ψ(p)p−s = log LN (ψ, s) − p n≥2 p ♣r✐♠❡ ψ(p)n −ns p n s♦ ✐t s✉✣❝❡s t♦ ♣r♦✈❡ t❤❛t t❤❡ ❧❛st t❡r♠ ✐s ❜♦✉♥❞❡❞ ♦♥ n≥2 p ♣r✐♠❡ ψ(p)n −ns p ≤ n pN n≥2 ≤ p ≤ p (s) > 1✱ ψ(p)n −ns p n n≥2 p ♣r✐♠❡ ≤ (s) > 1✳ ❇✉t ♦♥ 1 pns |ps | 1 |ps | − 1 1 p(p − 1) ≤2 k∈N 1 <∞ k2 ❚❤❡♦r❡♠ ✸✳✶✵ ✭❉✐r✐❝❤❧❡t✬s ❚❤❡♦r❡♠ ♦♥ Pr✐♠❡s ✐♥ ❆r✐t❤♠❡t✐❝ Pr♦❣r❡ss✐♦♥s✮✳ ▲❡t a, N ≥ 1 ❜❡ ❝♦♣r✐♠❡✳ ❚❤❡♥ t❤❡r❡ ❛r❡ ✐♥✜♥✐t❡❧② ♠❛♥② ♣r✐♠❡s p ✇✐t❤ p ≡ a (mod N )✳ ▼♦r❡♦✈❡r✱ ✐❢ Pa ✐s t❤❡ s❡t ♦❢ t❤❡s❡ ♣r✐♠❡s t❤❡♥ 1 1 1 ∼ log ps φ(N ) s−1 p∈Pa ❛s s → 1✳ Pr♦♦❢✳ ◆♦t✐❝❡ t❤❛t t❤❡ ✜rst st❛t❡♠❡♥t ❢♦❧❧♦✇s ❢r♦♠ t❤❡ s❡❝♦♥❞✳ ▲❡t Ca ❜❡ t❤❡ ❝❧❛ss ❢✉♥❝t✐♦♥ Ca : (Z/N Z)× → C× ✱ 1 n=a .

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 .

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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