One-level density of quadratic twists of $L$-functions

The Second Moment of Quadratic Twists of Modular L-functions

The family of quadratic twists of a modular form has received much attention in recent years. Motivated by the Birch-Swinnerton-Dyer conjectures, we seek an understanding of the central values of the associated L-functions, and while this question has been investigated extensively, much remains unknown. One important theme in this area concerns the moments of these central L-values. Thanks to t...

Non-vanishing of Quadratic Twists of Modular L-functions

Regulators of rank one quadratic twists

We investigate the regulators of elliptic curves with rank 1 in some families of quadratic twists of a fixed elliptic curve. In particular, we formulate some conjectures on the average size of these regulators. We also describe an efficient algorithm to compute explicitly some of the invariants of a rank one quadratic twist of an elliptic curve (regulator, order of the Tate-Shafarevich group, e...

On the Frequency of Vanishing of Quadratic Twists of Modular L-functions

In this paper we present some evidence that methods from random matrix theory can give insight into the frequency of vanishing for quadratic twists of modular L-functions. The central question is the following: given a holomorphic newform f with integral coefficients and associated L-function Lf (s), for how many fundamental discriminants d with |d| ≤ x, does Lf (s, χd), the L-function twisted ...

Nonvanishing of Quadratic Twists of Modular L-functions and Applications to Elliptic Curves

If d is square-free or is a fundamental discriminant, then let χd = χD denote the Kronecker character for the quadratic field Q( √ d) whose fundamental discriminant is D. Throughout D shall denote a fundamental discriminant. The D-quadratic twist of F , denoted F ⊗ χD, is the newform corresponding to the twist of F by the character χD. In particular, if gcd(M,D) = 1, then (F ⊗ χD)(z) = ∑∞ n=1 χ...