By V. Dokchitser, Sebastian Pancratz

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1 − T )dim ρ ❍❡♥❝❡ |aP j ||N (P )−js | ≤ P ♣r✐♠❡ ❛❜♦✈❡ p j≥0 ✇❤❡r❡ σ = [K:Q] 1 (1 − p−σ )dim ρ (s) ❛♥❞ ✇❡ ♥♦t❡ a(1) = 1✱ ✇❤❡♥❝❡ |aP j ||N (P )−js | ≤ p P j≥0 1 1 − p−σ (dim ρ)[K:Q] = ζ(s)(dim ρ)[K:Q] ❛s (s) > 1✳ ❊①❛♠♣❧❡✳ ✭✐✮ ▲❡t K = Q✱ F ❛r❜✐tr❛r②✱ ρ = I✳ ❚❤❡♥ ❢♦r ❛ ♣r✐♠❡ P = (p) ♦❢ K ✱ ρIP = ρ ❛♥❞ FrobP ❛❝ts ❛s t❤❡ ✐❞❡♥t✐t② ♦♥ ρIP ✳ ❙♦ PP (ρ, T ) = det(1 − T |I) = 1 − T ✳ ❚❤✉s L(I, s) = p 1−p1 −s = ζ(s)✳ ✭✐✐✮ ▲❡t K, F ❜❡ ❛r❜✐tr❛r②✱ ρ = I✳ ❚❤❡♥ L(I, s) = P 1 = ρK (s) 1 − N (P )−s t❤❡ ❉❡❞❡❦✐♥❞ ρ✲❢✉♥❝t✐♦♥ ♦❢ K ✳ ✭✐✐✐✮ ▲❡t K = Q✱ F = Q(ζN )✱ ✇❤❡r❡ N ✐s ♣r✐♠❡✱ ❛♥❞ ρ 1✲❞✐♠❡♥s✐♦♥❛❧ ♥♦♥tr✐✈✐❛❧✳ ❚❤❡♥ L(ρ, s) = LN (ψ, s) ✇❤❡r❡ ψ ✐s t❤❡ ❉✐r✐❝❤❧❡t ❝❤❛r❛❝t❡r ♠♦❞✉❧♦ N ❞❡✜♥❡❞ ❜② ψ(n) = ρ(σn ) ✇❤❡r❡ n✳ σn ∈ Gal Q(ζN )/Q ✇✐❝❤ σn ζN = ζN ◆♦t❛t✐♦♥✳ ■❢ ρ : G → GLn (C) ✐s ❛ r❡♣r❡s❡♥t❛t✐♦♥ t❤❡♥ ✇r✐t❡ Tr λi gi ρ = Tr det λi ρ(gi ) = λi gi ρ = det λi χρ (gi ), λi ρ(gi ) .

Dn ) ✐♥ t❤❡ ❛❝t✐♦♥ ♦♥ r♦♦ts}| . |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 }| = = .

Dn ) ✐♥ t❤❡ ❛❝t✐♦♥ ♦♥ r♦♦ts}| . |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 }| = = .