The taxonomic and systematic history of the South American silversides has changed significantly since Campos (1984) review. As a result of phylogenetic studies of the subfamilies Menidiinae (Chernoff, 1986b), Atherinopsinae (White, 1985; Crabtree, 1987; Dyer 1997, 1998) and of the order Atheriniformes (Saeed et al., 1994; Dyer and Chernoff, 1996) the taxonomy and classification of silversides ...

1. A brief history of nonvanishing theorems. The nonvanishing of a Dirichlet series 2 a(n)n~\ or the existence of a pole, at a particular value of s often has applications to arithmetic. Euler gave the first example of this, showing that the infinitude of the primes follows from the pole of Ç(s) at s = 1. A deep refinement was given by Dirichlet, whose theorem on primes in an arithmetic progres...

For N and k positive integers, let M(N, k)C denote the C-vector space of cuspidal modular forms of level N and weight k. This vector space is equipped with the usual Hecke operators Tn, n ≥ 1. If we need to consider several levels or weights at the same time, we will denote this Tn by T N n , or T N,k n . If p is a prime number dividing N , our Tp is also known under the name Up. One of our mai...

The group of rational points on an elliptic curve is one of the more fascinating number theoretic objects studied in recent times. The description of this group in terms of the special value of the L-function, or a derivative of some order, at the center of the critical strip, as enunciated by Birch and Swinnerton-Dyer is surely one of the most beautiful relationships in all of mathematics; als...

Chen's biharmonic conjecture is well-known and stays open: The only biharmonic submanifolds of Euclidean spaces are the minimal ones. In this paper, we consider an advanced version of the conjecture, replacing $Delta$ by its extension, $L_1$-operator ($L_1$-conjecture). The $L_1$-conjecture states that any $L_1$-biharmonic Euclidean hypersurface is 1-minimal. We prove that the $L_1$-conje...

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...

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-...

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...

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...