On the Vanishing of TwistedL-Functions of Elliptic Curves (original) (raw)
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The aim of this paper is to analyze the distribution of analytic (and signed) square roots of X values on imaginary quadratic twists of elliptic curves. Given an elliptic curve E of rank zero and prime conductor N , there is a weight-3 2 modular form g associated with it such that the d-coefficient of g is related to the value at s = 1 of the L-series of the (−d)-quadratic twist of the elliptic curve E. Assuming the Birch and Swinnerton-Dyer conjecture, we can then calculate for a large number of integers d the order of X of the (−d)-quadratic twist of E and analyze their distribution.
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The aim of this paper is to analyze the distribution of analytic (and signed) square roots of X values on imaginary quadratic twists of elliptic curves. Given an elliptic curve E of rank zero and prime conductor N , there is a weight-3 2 modular form g associated with it such that the d-coefficient of g is related to the value at s = 1 of the L-series of the (−d)-quadratic twist of the elliptic curve E. Assuming the Birch and Swinnerton-Dyer conjecture, we can then calculate for a large number of integers d the order of X of the (−d)-quadratic twist of E and analyze their distribution.
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A key ingredient in our analysis is a generalization of Jutila's bound for sums of quadratic characters with the additional restriction that the fundamental discriminant be congruent to a non-zero square modulo a square-free integer M. This bound is needed for two purposes. The first is to analyze the terms in the explicit formula corresponding to characters raised to an odd power. The second is to determine the main term in the 1-level density of quadratic twists of a fixed form on GL_n. Such an analysis was performed by Rubinstein [Rub], who implicitly assumed that Jutila's bound held with the additional restriction on the fundamental discriminants; in this paper we show that assumption is justified.
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The aim of this paper is to analyze the distribution of analytic (and signed) square roots of X values on imaginary quadratic twists of elliptic curves. Given an elliptic curve E of rank zero and prime conductor N, there is a weight- 3 modular form g associated with it such that the d-coefficient of g is related to the value at s =1 of the L-series of the (�d)-quadratic twist of the elliptic curve E. Assuming the Birch and Swinnerton-Dyer conjecture, we can then calculate for a large number of integers d the order of X of the (�d)-quadratic twist of E and analyze their distribution.
Integrality of twisted L-values of elliptic curves
arXiv (Cornell University), 2020
Under suitable, fairly weak hypotheses on an elliptic curve E/Q and a primitive non-trivial Dirichlet character χ, we show that the algebraic L-value L (E, χ) at s = 1 is an algebraic integer. For instance, for semistable curves L (E, χ) is integral whenever E admits no isogenies defined over Q. Moreover we give examples illustrating that our hypotheses are necessary for integrality to hold.
Low-lying Zeros of Quadratic Dirichlet L-Functions, Hyper-elliptic Curves and Random Matrix Theory
Geometric and Functional Analysis, 2013
The statistics of low-lying zeros of quadratic Dirichlet Lfunctions were conjectured by Katz and Sarnak to be given by the scaling limit of eigenvalues from the unitary symplectic ensemble. The n-level densities were found to be in agreement with this in a certain neighborhood of the origin in the Fourier domain by Rubinstein in his Ph.D. thesis in 1998. An attempt to extend the neighborhood was made in the Ph.D. thesis of Peng Gao (2005), who under GRH gave the density as a complicated combinatorial factor, but it remained open whether it coincides with the Random Matrix Theory factor. For n ≤ 7 this was recently confirmed by Levinson and Miller. We resolve this problem for all n, not by directly doing the combinatorics, but by passing to a function field analogue, of L-functions associated to hyper-elliptic curves of given genus g over a field of q elements. We show that the answer in this case coincides with Gao's combinatorial factor up to a controlled error. We then take the limit of large finite field size q → ∞ and use the Katz-Sarnak equidistribution theorem, which identifies the monodromy of the Frobenius conjugacy classes for the hyperelliptic ensemble with the group USp(2g). Further taking the limit of large genus g → ∞ allows us to identify Gao's combinatorial factor with the RMT answer.