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

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Families of L-functions and symmetry types• Unitary G(N) = U(N) :{ζ(1/2 + it), t ≥ 0}• Symplectic G(N) = Sp(2N) :{L(1/2, χ d ), χ d quadratic character}• Orthogonal G(N) = SO(2N) :{L(1/2, f), f ∈ S k (Γ 0 (N)) with k fixed}For the family of cubic twists, we have unitarysymmetries as- the signs of the functional equation areuniformly distributed on the unit circle (Heath-Brown and Patterson);- behavior of the moments of |L E (1, χ)|.7
MomentsLet k be a positive integer.2kth momentsM E (2k, X) = 1N(X)∑f χ ≤XWe consider the|L E (1, χ)| 2kwhere ∑ f χ ≤X runs over all cubic characters ofconductor f χ ≤ X.Conjecture (Keating and Snaith)M E (2k, X) ∼ a E (k) g k (2 log X) k2as X → ∞whereg k =k−1 ∏j=0j!(j + k)!and a E (k) is an arithmetical factor related tothe curve E.8
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: Random matricesPhilosophy for a fam
Page 12 and 13: Random matricesKeating and Snaith:
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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