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

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Cubic twistsLet K/Q be a cyclic cubic field with charactergroup Ĝ = {1, χ, χ}. ThenL(E/K, s) = L E (s) L E (s, χ) L E (s, χ)and the vanishing of L E (s, χ) is related to therank of E over K.Theorem (Fearnley-Kisilevsky) If E has rational3-torsion, then# {K with f K ≤ X : rank E(K) > rank E(Q)}≫ X 1/2 .We denoteN(X) = {χ cubic with f χ ≤ X} ∼ c 3 XN E (X) = {χ ∈ N(X) with L E (s, χ) = 0} .Question:of N E (X)?What is the asymptotic behavior5
Random matricesPhilosophy for a family F of L-series with symmetrytype G(N):- Katz and Sarnak: statistics on the lowlyingzeroes of L f (s) in F should fit thoseof the eigenvalues of random matrices inG(N);- Keating and Snaith: statistics on the valuesof L f (s) in F should fit those of thecharacteristic polynomials of random matricesin G(N).Experimental and theoritical evidence:Montgomery, Odlyzko, Katz-Sarnak, Rudnick-Sarnak, Conrey-Ghost, Conrey-Farmer ...6
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: Higher order twistsLet k be an inte
Page 9: MomentsLet k be a positive integer.
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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