Let E be an elliptic curve over Q, with L-function LE(s). For any primitive Dirichlet character χ, let LE(s, χ) be the L-function of E twisted by χ. In this paper, we use random matrix theory to study vanishing of the twisted L-functions LE(s, χ) at the central value s = 1. In particular, random matrix theory predicts that there are infinitely many characters of order 3 and 5 such that LE(1, χ)...

The initial version of the Birch and Swinnerton-Dyer conjecture concerned asymptotics for partial Euler products for an elliptic curve L-function at s = 1. Goldfeld later proved that these asymptotics imply the Riemann hypothesis for the L-function and that the constant in the asymptotics has an unexpected factor of √ 2. We extend Goldfeld’s theorem to an analysis of partial Euler products for ...

is the Hecke L-series attached to the eigenform f . Hecke’s theory shows that L(f, s) has an Euler product expansion identical to (2), and also that it admits an integral representation as a Mellin transform of f . This extends L(f, s) analytically to the whole complex plane and shows that it satisfies a functional equation relating its values at s and 2− s. The modularity of E thus implies tha...

For the integer D = pq of the product of two distinct odd primes, we construct an elliptic curve E2rD : y 2 = x3 − 2rDx over Q, where r is a parameter dependent on the classes of p and q modulo 8, and show, under Birch and Swinnerton-Dyer conjecture, that the elliptic curve has rank one and vp(x([k]Q)) 6= vq(x([k]Q)) for odd k and a generator Q of the free part of E2rD(Q). Thus we can get p or ...

Fix an elliptic curve E/Q, and assume the Riemann Hypothesis for the Lfunction L(ED, s) for every quadratic twist ED of E by D ∈ Z. We combine Weil’s explicit formula with techniques of Heath-Brown to derive an asymptotic upper bound for the weighted moments of the analytic rank of ED. We derive from this an upper bound for the density of low-lying zeros of L(ED, s) which is compatible with the...

Following Katz-Sarnak [KS1], [KS2], Iwaniec-Luo-Sarnak [ILS], and Rubinstein [Ru], we use the 1and 2-level densities to study the distribution of low lying zeros for one-parameter rational families of elliptic curves over Q(t). Modulo standard conjectures, for small support the densities agree with Katz and Sarnak’s predictions. Further, the densities confirm that the curves’ L-functions behave...

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 ...

In this paper we explore the arithmetic correspondence between, on the one hand, (isogeny classes of) elliptic curves E defined over an imaginary quadratic field K of class number one, and on the other hand, rational newforms F of weight two for the congruence subgroups r0(n), where n is an ideal in the ring of integers R of K . This continues work of the first author and forms part of the Ph.D...

The equation y2 = x3 +k, for k ∈ Z, is called Mordell’s equation on account of Mordell’s long interest in it throughout his life. A natural number-theoretic task is the description of all rational and integral solutions to such an equation, either qualitatively (decide if there are finitely or infinitely many solutions in Z or Q) or quantitatively (list or otherwise conveniently describe all su...

Recently, all Birch and Swinnerton-Dyer invariants, except for the order of , have been computed curves genus 2 contained in L-functions Modular Forms Database [LMFDB]. This report explains improvements made to implementation algorithm described [vBom19] that were needed do computation Tamagawa numbers real period reasonable time. We also explain some more technical details algorithm, give a br...