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By V. Dokchitser, Sebastian Pancratz

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J j ■❢ ρ, I = 0 t❤❡♥ τi ❝❛♥ ❜❡ ❝❤♦s❡♥ t♦ ❜❡ ♥♦♥✲tr✐✈✐❛❧✳ Pr♦♦❢✳ ❙❧✐❣❤t❧② ♥♦♥✲tr✐✈✐❛❧ ❡①❡r❝✐s❡✳ ❚❤❡♦r❡♠ ✸✳✶✻ ✭❆rt✐♥✮✳ ▲❡t F/K ❜❡ ❛ ●❛❧♦✐s ❡①t❡♥s✐♦♥ ♦❢ ♥✉♠❜❡r ✜❡❧❞s ❛♥❞ ρ ❛ r❡♣r❡✲ s❡♥t❛t✐♦♥ ♦❢ Gal(F/K)✳ ❚❤❡♥ t❤❡r❡ ❡①✐sts n ≥ 1 s✉❝❤ t❤❛t L(ρ, s)n ❛❞♠✐ts ❛ ♠❡r♦♠♦r✲ ♣❤✐❝ ❝♦♥t✐♥✉❛t✐♦♥ t♦ C✱ ❛♥❛❧②t✐❝ ❛♥❞ ♥♦♥✲③❡r♦ ❛t s = 1 ✐❢ ρ, I = 0✳ Pr♦♦❢✳ Pr♦♣♦s✐t✐♦♥ ✸✳✶✺ ❛♥❞ ❆rt✐♥ ❋♦r♠❛❧✐s♠ r❡❞✉❝❡ t❤❡ ♣r♦❜❧❡♠ t♦ s❤♦✇✐♥❣ t❤❛t L(τ, s) ❤❛s ❛♥❛❧②t✐❝ ❝♦♥t✐♥✉❛t✐♦♥ t♦ C ✇❤❡♥ τ ✐s 1✲❞✐♠❡♥s✐♦♥❛❧✱ ❡①❝❡♣t ♣♦ss✐❜❧② ❛ ♣♦❧❡ ❛t s = 1 ✇❤❡♥ τ = I✳ ❚❤✐s ✐s tr✉❡ ❜② ❍❡❝❦❡✬s ❚❤❡♦r❡♠ ❛♥❞ t❤❡ ❢❛❝t t❤❛t ♦♥❧② ✜♥✐t❡❧② ♠❛♥② ♣r✐♠❡s r❛♠✐❢② ✐♥ ❛♥② ❡①t❡♥s✐♦♥ ♦❢ ♥✉♠❜❡r ✜❡❧❞s✱ ❛♥❞ L(τ, s) ✐s ♥♦♥✲③❡r♦ ❛t s = 1✳ ✸✸ ❈♦r♦❧❧❛r② ✸✳✶✼✳ ♥❡❛r s = 1✳ ■❢ ρ ✐s ✐rr❡❞✉❝✐❜❧❡ ❛♥❞ ♥♦♥✲tr✐✈✐❛❧ t❤❡♥ L(ρ, s) ✐s ❜♦✉♥❞❡❞ ❛♥❞ ♥♦♥✲③❡r♦ Pr♦♦❢✳ ❖❜s❡r✈❡ ✐❢ F/K ✐s ❝②❝❧✐❝ t❤❡♥ t❤✐s ✐s tr✉❡ ❜② ❚❤❡♦r❡♠ ✸✳✶✶✳ ❈♦♥❥❡❝t✉r❡✳ t♦ C✳ ■❢ ρ ✐s ✐rr❡❞✉❝✐❜❧❡ ❛♥❞ ♥♦♥✲tr✐✈✐❛❧ t❤❡♥ L(ρ, s) ❤❛s ❛♥❛❧②t✐❝ ❝♦♥t✐♥✉❛t✐♦♥ ❘❡♠❛r❦✳ ❚❤❡♦r❡♠ ✸✳✶✻ ✐♠♣❧✐❡s t❤❛t L(ρ, s)n ✐s ♠❡r♦♠♦r♣❤✐❝✳ ❆ t❤❡♦r❡♠ ♦❢ ❇r❛✉❡r st❛t❡s t❤❛t L(ρ, s) ✐s ♠❡r♦♠♦r♣❤✐❝✳ ✸✳✼ ❉❡✜♥✐t✐♦♥✳ ❉❡♥s✐t② ❚❤❡♦r❡♠s ▲❡t S ❜❡ ❛ s❡t ♦❢ ♣r✐♠❡ ♥✉♠❜❡rs✳ ❚❤❡♥ S ❤❛s ❉✐r✐❝❤❧❡t ❞❡♥s✐t② α ✐❢ 1 p∈S ps 1 log s−1 →α ❛s s → 1 ❢r♦♠ ❛❜♦✈❡ ✐♥ R✳ ❊①❛♠♣❧❡✳ ❛♥❞ Sa,N ❇② ❉✐r✐❝❤❧❡t✬s ❚❤❡♦r❡♠ ✭❚❤❡♦r❡♠ ✸✳✶✵✮✱ t❤❡ s❡t ♦❢ ❛❧❧ ♣r✐♠❡s ❤❛s ❞❡♥s✐t② 1 = {p : p ≡ a (mod N )} ❤❛s ❞❡♥s✐t② 1/φ(N ) ❢♦r a ❛♥❞ N ❝♦♣r✐♠❡✳ ◆♦t❛t✐♦♥✳ ❋♦r F/Q ●❛❧♦✐s ❛♥❞ P ✉♥r❛♠✐✜❡❞ ✐♥ F ✱ ✇r✐t❡ FrobP ∈ Gal(F/Q) ❢♦r t❤❡ ❋r♦❜❡♥✐✉s ❡❧❡♠❡♥t FrobQ/P ♦❢ s♦♠❡ Q ❛❜♦✈❡ P ✳ ◆♦t❡ t❤❛t FrobP ❧✐❡s ✐♥ ❛ ✇❡❧❧✲❞❡✜♥❡❞ ❝♦♥❥✉❣❛❝② ❝❧❛ss ♦❢ Gal(F/Q)✱ ❜❡❝❛✉s❡ FrobQ /P = x FrobQ/P x−1 ✇❤❡r❡ Q = xQ ❢♦r s♦♠❡ x ∈ Gal(F/Q)✳ ❊①❛♠♣❧❡✳ a t❤❡♥✱ ❢♦r p N ✱ Frob = F = Q(ζN ) ❛♥❞ σa ∈ Gal(F/Q) ✇✐t❤ σa (ζN ) = ζN P p σa ✐❢ ❛♥❞ ♦♥❧② ✐❢ p ≡ a (mod N )✱ ❜❡❝❛✉s❡ FrobP (ζN ) = ζN ✳ ❙♦ ❜② ❉✐r✐❝❤❧❡t✬s ❚❤❡♦r❡♠✱ t❤❡ s❡t SN,σ = {p ✉♥r❛♠✐✜❡❞ ✐♥ Q(ζN )/Q : Frobp = σ} ❤❛s ❉✐r✐❝❤❧❡t ❞❡♥s✐t② 1/φ(N ) = 1/|Gal Q(ζN )/Q | ❢♦r ❡✈❡r② σ ∈ Gal Q(ζN )/Q ✳ ❚❤❡♦r❡♠ ✸✳✶✽ ✭❈❤❡❜♦t❛r❡✈✬s ❉❡♥s✐t② ❚❤❡♦r❡♠✮✳ ▲❡t F/Q ❜❡ ❛ ✜♥✐t❡ ●❛❧♦✐s ❡①t❡♥s✐♦♥ ❛♥❞ C ❛ ❝♦♥❥✉❣❛② ❝❧❛ss ✐♥ G = Gal(F/Q)✳ ❚❤❡♥ t❤❡ s❡t SC = {p ✉♥r❛♠✐✜❡❞ ✐♥ F/Q : Frobp ∈ C} ❤❛s ❉✐r✐❝❤❧❡t ❞❡♥s✐t② |C|/|G|✳ ❈♦r♦❧❧❛r② ✸✳✶✾✳ ▲❡t f (X) ∈ Z[X] ❜❡ ♠♦♥✐❝ ❛♥❞ ✐rr❡❞✉❝✐❜❧❡✳ ❚❤❡♥ t❤❡ s❡t ♦❢ ♣r✐♠❡s p s✉❝❤ t❤❛t f (X) (mod p) ❢❛❝t♦r✐s❡s ✐♥t♦ ✐rr❡❞✉❝✐❜❧❡ ♣♦❧②♥♦♠✐❛❧s ♦❢ ❞❡❣r❡❡s d1 , .

Gal(f )| Pr♦♦❢✳ f (X) (mod p) ❤❛s ❛ r❡♣❡❛t❡❞ r♦♦t ✐♥ F¯ p ❢♦r ♦♥❧② ✜♥✐t❡❧② ♠❛♥② p✳ ❋♦r t❤❡ r❡st✱ Frobp ❛❝ts ❛s ❛♥ ❡❧❡♠❡♥t ♦❢ ❝②❝❧❡ t②♣❡ (d1 , . . , dn ) ✇❤❡r❡ t❤❡s❡ ❛r❡ t❤❡ ❞❡❣r❡❡s ♦❢ t❤❡ ✐rr❡❞✉❝✐❜❧❡ ❢❛❝t♦rs ♦❢ f (X) (mod p)✱ ❜② ❈♦r♦❧❧❛r② ✷✳✺ ❛♥❞ ✐ts ♣r♦♦❢✳ ❊①❛♠♣❧❡✳ ❙✉♣♣♦s❡ f (X) ✐s ❛♥ ✐rr❡❞✉❝✐❜❧❡ q✉✐♥t✐❝ ✇✐t❤ Gal(f ) = S5 ✳ ✸✹ L✲❙❡r✐❡s • ❚❤❡ s❡t ♦❢ ♣r✐♠❡s s✉❝❤ t❤❛t f (X) (mod p) ✐s ❛ ♣r♦❞✉❝t ♦❢ ❧✐♥❡❛r ❢❛❝t♦rs ❤❛s ❞❡♥s✐t② 1/120✳ • ❚❤❡ s❡t ♦❢ ♣r✐♠❡s s✉❝❤ t❤❛t f (X) (mod p) ❢❛❝t♦r✐s❡s ✐♥t♦ ❛ ❝✉❜✐❝ ❛♥❞ ❛ q✉❛❞r❛t✐❝ ❤❛s ❞❡♥s✐t② 1 20 1 |{❡❧❡♠❡♥ts ♦❢ t❤❡ ❢♦r♠ (··)(· · ·) ✐♥ S5 }| = = .

R♠❛❧❧②✱ s✉❜st✐t✉t✐♥❣ t❤✐s ✐♥t♦ t❤❡ ❛❜♦✈❡ ♣r♦❞✉❝t ❣✐✈❡s t❤❡ s❡r✐❡s ❡①♣r❡ss✐♦♥ ✭❆rt✐♥ L✲s❡r✐❡s✮ (1 + aP N (P )−s + aP 2 N (P )−2s + · · · ) L(ρ, s) = P aN N (N )−s = (0)=N ⊂OK N ✐❞❡❛❧ ❢♦r s♦♠❡ aN ∈ C✳ ◆♦t❡ t❤❛t ❣r♦✉♣✐♥❣ ✐❞❡❛❧s ✇✐t❤ ❡q✉❛❧ ♥♦r♠ ②✐❡❧❞s ❛♥ ❡①♣r❡ss✐♦♥ ❢♦r L(ρ, s) ❛s ❛♥ ♦r❞✐♥❛r② ❉✐r✐❝❤❧❡t s❡r✐❡s✳ ▲❡♠♠❛ ✸✳✶✸✳ ❚❤❡ L✲s❡r✐❡s ❡①♣r❡ss✐♦♥ ❢♦r L(ρ, s) ❛❣r❡❡s ✇✐t❤ t❤❡ ❊✉❧❡r ♣r♦❞✉❝t ♦♥ (s) > 1✱ ✇❤❡r❡ ❜♦t❤ ❝♦♥✈❡r❣❡ ❛❜s♦❧✉t❡❧② t♦ ❛♥ ❛♥❛❧②t✐❝ ❢✉♥❝t✐♦♥✳ ✸✵ L✲❙❡r✐❡s Pr♦♦❢✳ ■t s✉✣❝❡s t♦ ❝❤❡❝❦ t❤❛t t❤❡ ❞♦✉❜❧❡ s❡r✐❡s (1 + aP N (P )−s + aP 2 N (P )−2s + · · · ) P ❝♦♥✈❡r❣❡s ❛❜s♦❧✉t❡❧② ♦♥ (s) > 1 ✖ t❤✐s ❥✉st✐✜❡s ❜♦t❤ t❤❡ ❊✉❧❡r ♣r♦❞✉❝t ❛♥❞ t❤❡ s❡r✐❡s ❡①♣r❡ss✐♦♥s ♦♥ (s) > 1✱ t❤❡♥ t❤❡ ❛♥❛❧②t✐❝✐t② ❢♦❧❧♦✇s ❢r♦♠ t❤❡ ❡①♣r❡ss✐♦♥ ♦❢ L(ρ, s) ❛s ❛♥ ♦r❞✐♥❛r② ❉✐r✐❝❤❧❡t s❡r✐❡s ❜② Pr♦♣♦s✐t✐♦♥ ✸✳✷✳ ❚❤❡ ♣♦❧②♥♦♠✐❛❧ PP (ρ, T ) ❢❛❝t♦r✐s❡s ♦✈❡r C ❛s PP (ρ, T ) = (1 − λ1 T ) · · · (1 − λk T ) ✇✐t❤ |λi | = 1 ❛♥❞ k ≤ dim ρ✳ ❙♦ t❤❡ ❝♦❡✣❝✐❡♥ts ♦❢ 1 = PP (ρ, T ) 1 = 1 + aP T + aP 2 T 2 + · · · (1 − λ T ) i i ❛r❡ ❜♦✉♥❞❡❞ ✐♥ ❛❜s♦❧✉t❡ ✈❛❧✉❡ ❜② t❤♦s❡ ♦❢ 1 = (1 + T + T 2 + · · · )dim ρ .

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