Let K be a number field, and let A/K be an abelian variety. Let c denote the product of the Tamagawa numbers of A/K, and let A(K)tors denote the finite torsion subgroup of A(K). The quotient c/|A(K)tors| is a factor appearing in the leading term of the L-function of A/K in the conjecture of Birch and Swinnerton-Dyer. We investigate in this article possible cancellations in this ratio. Precise r...

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 value distribution of derivatives of characteristic polynomials of matrices from SO(N) is calculated at the point 1, the symmetry point on the unit circle of the eigenvalues of these matrices. We consider subsets of matrices from SO(N) that are constrained to have n eigenvalues equal to 1, and investigate the first non-zero derivative of the characteristic polynomial at that point. The conn...

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 ...

We study a subgroup of the Shafarevich-Tate group of an abelian variety known as the visible subgroup. We explain the geometric intuition behind this subgroup, prove its finiteness and describe several techniques for exhibiting visible elements. Two important results are proved one what we call the visualization theorem, which asserts that every element of the Shafarevich-Tate group of an abeli...

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...

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...

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...