On the Frequency of Vanishing of Quadratic Twists of Modular L-functions
نویسندگان
چکیده
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 by the real, primitive, Dirichlet character associated with the discriminant d, vanish at the center of the critical strip to order at least 2? This question is of particular interest in the case that the L-function is associated with an elliptic curve, in light of the conjecture of Birch and Swinnerton-Dyer. This case corresponds to weight k = 2. We will focus on this case for most of the paper, though we do make some remarks about higher weights (see (26) and below). Suppose that E/Q is an elliptic curve with associated L-function
منابع مشابه
Yoshida lifts and simultaneous non-vanishing of dihedral twists of modular L-functions
Given elliptic modular forms f and g satisfying certain conditions on their weights and levels, we prove (a quantitative version of the statement) that there exist infinitely many imaginary quadratic fields K and characters χ of the ideal class group ClK such that L( 1 2 , fK × χ) 6= 0 and L( 1 2 , gK × χ) 6= 0. The proof is based on a non-vanishing result for Fourier coefficients of Siegel mod...
متن کاملComputing Central Values of Twisted L-series the Case of Composite Levels Ariel Pacetti and Gonzalo Tornaría
We will assume that the twisted L-series are primitive (i.e. the corresponding twisted modular forms are newforms). There is no loss of generality in making this assumption: if this is not the case, then f would be a quadratic twist of a newform of smaller level, which we can choose instead. The question of efficiently computing the family of central values L(f,D, 1), for fundamental discrimina...
متن کاملA Variant of the Bombieri-vinogradov Theorem in Short Intervals with Applications
We generalize the classical Bombieri-Vinogradov theorem to a short interval, non-abelian setting. This leads to variants of the prime number theorem for short intervals where the primes lie in arithmetic progressions that are “twisted” by a splitting condition in a Galois extension L/K of number fields. Using this result in conjunction with recent work of Maynard, we prove that rational primes ...
متن کاملThe 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...
متن کامل