My research interests are in number theory where I use mostly analytic tools to study objects from algebraic number theory and arithmetic geometry. I am interested in topics such as modular forms, class numbers, quadratic forms, and finite fields. However, most of my current work is focused on elliptic curves, and in particular on their reductions. The theory of elliptic curves figured strongly...

This paper consists of two parts. In the first we present a general theory of Euler systems. The main results (see §§3 and 4) show that an Euler system for a p-adic representation T gives a bound on the Selmer group associated to the dual module Hom(T, μp∞). These theorems, which generalize work of Kolyvagin [Ko], have been obtained independently by Kato [Ka1], Perrin-Riou [PR2], and the author...

Here we calculate the value distribution of the first derivative of characteristic polynomials of matrices from SO(2N + 1) at the point 1, the symmetry point on the unit circle of the eigenvalues of these matrices. The connection between the values of random matrix characteristic polynomials and values of the L-functions of families of elliptic curves implies that this calculation in random mat...

This article is the first in a series devoted to studying generalised Gross-KudlaSchoen diagonal cycles in the product of three Kuga-Sato varieties and the Euler system properties of the associated Selmer classes, with special emphasis on their application to the Birch–Swinnerton-Dyer conjecture and the theory of Stark-Heegner points. The basis for the entire study is a p-adic formula of Gross-...

Given an elliptic curve E defined over Q, we are motivated by the 2-part of the Birch and Swinnerton-Dyer formula to study the relation between the 2-Selmer rank of E and the 2-Selmer rank of an abelian variety A obtained by Ribet’s level raising theorem. For certain imaginary quadratic fields K satisfying the Heegner hypothesis, we prove that the 2-Selmer ranks of E and A over K have different...

In this paper we examine diierences between the two standard methods for computing the 2-Selmer group of an elliptic curve. In particular we focus on practical diierences in the timings of the two methods. In addition we discuss how to proceed if one fails to determine the rank of the curve from the 2-Selmer group. Finally we mention brieey ongoing research i n to generalizing such methods to t...

Let A be a quotient of J0(N) associated to a newform f such that the special L-value of A (at s = 1) is non-zero. We give a formula for the ratio of the special L-value to the real period of A that expresses this ratio as a rational number. We extract an integer factor from the numerator of this formula; this factor is non-trivial in general and is related to certain congruences of f with eigen...

Let E be an elliptic curve over Q attached to a newform f of weight 2 on 00(N ), and let K be a real quadratic field in which all the primes dividing N are split. This paper relates the canonical R/Z-valued “circle pairing” on E(K ) defined by Mazur and Tate [MT1] to a period integral I ( f, K ) defined in terms of f and K . The resulting conjecture can be viewed as an analogue of the classical...

We study the elliptic curve E given by y = x(x+1)(x+ t) over the rational function field k(t) and its extensions Kd = k(μd, t). When k is finite of characteristic p and d = p + 1, we write down explicit points onE and show by elementary arguments that they generate a subgroup Vd of rank d − 2 and of finite index in E(Kd). Using more sophisticated methods, we then show that the Birch and Swinner...