On the vanishing of twists of L-functions of elliptic curves Chantal ...

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$MATH 208 - Department of Mathematics and Statistics - Concordia ...$

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Random matricesKeating and Snaith: statistics on the values ofL f (s) in F should fit those of the characteristicpolynomials of random matrices in G(N).For each A ∈ G(N), letλ 1 = e iθ 1, . . . , λ N = e iθ Nbe the eigenvalues of A ordered such that0 ≤ θ 1 ≤ . . . ≤ θ N < 2πand letP A (λ) = det (A − λI)be the characteristic polynomial of A.Notice thatP A (1) = 0 ⇐⇒ λ 1 = e iθ 1 = 1 ⇐⇒ θ 1 = 011
The unitary groupJ. P. Keating, N. C. Snaith, Random matrixtheory and ζ(1/2 + it).Moments for |P A (1)| in U(N):M U (s, N) ==∫U(N) |P A(1)| s dHaarN∏j=1Γ(j)Γ(j + s)Γ 2 (j + s/2)is analytic for Re(s) > −1 with meromorphiccontinuation to the whole complex plane.Distribution function:P U (x, N) = 12πifor some c > −1.∫(c) M U(s, N)x −s−1 ds.For x smallP U (x, N) ∼ 1Γ(N)N∏j=1Γ(j) 2Γ 2 (j − 1/2) = R(N).12
Page 1 and 2: On the vanishing of twists of L-fun
Page 3 and 4: Quadratic charactersLet χ d be a q
Page 5 and 6: Higher order twistsLet k be an inte
Page 7 and 8: Random matricesPhilosophy for a fam
Page 9: MomentsLet k be a positive integer.
Page 14 and 15: When N → ∞R(N) → N 1/4 G 2 (1
Page 16 and 17: Distribution of the integers n χCu
Page 18: Empirical results for E11A and E14A

On the vanishing of twists of L-functions of elliptic curves Chantal ...

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