We consider the problem of classifying elliptic curves over Q with prescribed torsion group and conductor of the form p^a q^b where p and q are arbitrary primes. This can be achieved under certain technical conditions, using results from the theory of Galois representations and modular forms. This is joint work with Nike Vatsal and Soroosh Yazdani.

(Joint work with Alexandru Zaharescu)

Let a and b be positive real numbers and s a real number satisfying 0 £ s < 1. Let ëxû denote the greatest integer £ x. Define Y_k(a,b;s) to be the k-th positive integer n such that ëna+sû ¹ ënb+sû. For i=1,2 we compute asymptotics for the probability that Y_i(a,b;0) > Q for Q large as a and b range independently over a subinterval of [0,1). We find the expected value of Y₁(b,a;0) as a and b range independently over [0,1). When a, b, and s are fixed, the algebraic structure of the set of natural numbers {Y_i(b,a;s) | i Î Z⁺} is characterized.

For decades the merit factor problem of binary sequences has been stuck at a value of 6 for the maximum asymptotic merit factor and indeed a number of authors speculated that this was best possible. We construct an infinite family of binary sequences that we conjecture have merit factors greater than 6.34. The numerical experimentation that led to this example is a significant and interesting part of the story. This problem is related to finding small L₄ norm of polynomials with +1 or -1 coefficients.

This is a joint work with Peter Borwein and Jonathan Jedwab.

Let E be an elliptic curve defined over the rationals and without complex multiplication. Let K be a fixed imaginary quadratic field. We use the square sieve to find nontrivial upper bounds for the number of primes p of ordinary reduction for E such that Q(p_p)=K, where p_p is the Frobenius endomorphism of E at p. This represents progress towards a 1976 Lang-Trotter conjecture.

(This is joint work with E. Fouvry and M. Ram Murty)

(joint work with J. Fearnley and H. Kisilevsky)

Let E be an elliptic curve over the rationals with L-function L_E(s). Let c be a Dirichlet character, and let L_E(s, c) be the L-function of E twisted by the character c. For quadratic characters c, L_E(1, c) vanishes for at least half of the characters (where the sign of the functional equation is -1), and Goldfeld conjectured that the density of vanishing is exactly 1/2 in this case. For higher order characters, the functional equation now relates L_E(1, c) and L_E(1, [`(c)]), and there is no reason to predict a positive density of vanishing. We present in this talk some evidence for the case of twists by cubic character c, based on empirical computations and random matric theory.

We consider systems of cubic Diophantine inequalities. In particular, we have that if s is any integer with s ³ (10R)^g, where g = (10R)⁵, then given any R real cubic forms C₁,¼,C_R in s variables, there is a nonzero integral solution x of the simultaneous Diophantine inequalities |C₁(x)| < 1,|C₂(x)| < 1,¼,|C_R(x)| < 1.

Let m be a positive integer, a and g integers relatively prime to m. We give estimates for the exponential sum

We shall discuss the local structure of PEL moduli spaces-the moduli spaces parameterizing abelian varieties with polarization, endomorphism and level structure. Two main techniques will be exposed: local models and displays. Both very geometric in nature, though eventually very algebraic.

We shall explain a general theorem, due to Andreatta and the speaker, that allows calculation of universal displays and show how it recaptures also previously known cases.

For a quadratic character c over Q and an integer n > 0 the values of the L-function of c at 1-n are non-zero rational numbers if c has parity (-1)ⁿ. Most of the time the values are 2-integral, and in these cases one can prove general divisibility properties by powers of 2. This has been done by Fox, Urbanowicz and K. S. Willia ms using sophisticated identities for generalized Bernouilli numbers. We will discuss a purely algebraic approach a la Gauss, which also allows to generalize the results to quadratic characters over arbitrary abelian fields.

Let F(s) be a Dirichlet series, F(s)=å_n=1 a_n n^-s, Âs > 1. Define the the summatory function S(x) to be å_{n £ x} a_n. We assume that F(s) satisfies the following conditions. First, for all e > 0, |a_n|=O(n^e). In addition, it admits analytic continuations and functional equations. More precisely, there is a function D(s) = Q^sÕG(a_is+g_i), Q > 0, a_i > 0, Âg_i > 0, such that F(s)D(s)=w[`(F)](1-s)[`(D)](1-s), |w|=1. Furthermore, assume that F(s) is entire. Twice of the summation of a_i is called the degree d_F of F. In this talk, I will derive an estimation of S(x) without extra conditions. The trivial estimation is S(x)=O(x^1+e), "e > 0.

I will provide two estimations of S(x). One is a joint work with Ram Murty; we prove that for d_F ³ 1, S(x) = O(Q^1-q+ex^q+e), where q = d/(d+2). For the larger d_F ³ 2, I can get a better result: S(x) = O(Q^1-q¢+ex^q¢+e), where q = (d-1)/(d+1). In both cases, the implied constants are independent of Q.

Let E be an elliptic curve defined over Q. We construct cohomology classes from quadratic twists of E and apply the local-global duality theorem (a reformulation of the reciprocity law) to these cohomology classes. As a result, we get a bound for rank of E. The technique of using the reciprocity law was used by Kolyvagin to bound the size of Selmer group and study Tate-Shafarevich group. This work was discussed with V. Kolyvagin, R. Murty and J. Shalika.

A set S of integers is called a B^*[g] set if for any given m there are at most g ordered pairs (s₁,s₂) Î S×S with s₁+s₂ = m; in the case g = 2, these are better known as Sidon sets. It is trivial to show that any B^*[g] set contained in {1,2,...,n} has at most Ö[(2gn)] elements, but proving a lower bound of the same order of magnitude is more difficult. This problem, surprisingly, is intimately related to the following problem concerning measurable subsets of the real numbers: given 0 < e < 1, estimate the supremum of those real numbers d such that every subset of [0,1] with measure e contains a symmetric subset with measure d. Using harmonic analysis and relationships among L^p norms as well as methods from combinatorial and probabilistic number theory, we establish fairly tight upper and lower bounds for these two interconnected problems.

In this talk, I will describe a general result computing the number of rational points of bounded height on an algebraic variety V which is covered by lines. The main technical result used to achieve this is an upper bound on the number of rational points of bounded height on a line. This bound varies in an pleasantly controllable manner as the line varies, and hence can be used to sum the counting functions of the lines which cover V.

It is well known that an irreducible polynomial over the integers may become reducible mod p for every prime p. In this talk, we shall discuss the analogue for Abelian varieties. Given an absolutely simple Abelian variety over a number field, does it stay absolutely simple modulo infinitely many primes?

If f is a polynomial with integer coefficients which is a prime number for infinitely many specializations, then it is clear that f must be irreducible over the rational number field. An analogous result over number fields is not true due to the possible presence of infinitely many units. However, using Siegel's theorem on integral points of curves of genus ³ 1, we show that an analogous result is ``almost true'' and the obstruction is the presence of ``Mersenne-like'' primes in a number field. We also discuss the case of a function field over a finite field. (This is joint work with Jasbir Chahal.)

Eisenstein series twisted by modular symbols were introduced by Goldfeld to study the distribution of modular symbols in connection to a weak form of the ABC conjecture. I will present distribution results that follow from the study of the pole at s=1 of such series. The proof uses families of Eisenstein series twisted by characters and perturbation methods of the Laplace operator.

Erdos and Moser investigated the problem of finding sets of positive integers A with the property that a+b is a square whenever a and b are distinct elements of A. With Rivat and Sarkozy we showed that if A is a subset of the first N positive integers then A has cardinality at most 37logN provided that N is large enough. We shall discuss recent joint work with Gyarmati and Sarkozy where we replace the requirement that a+b be a square with the requirement that a+b be a pure power.

We estimate the number of integer solutions to inequalities of the form |F(x)| £ m, where F(X) is a homogeneous polynomial with integer coefficients which factors completely over the complex field as a product of linear forms. We give asymptotic estimates as the parameter m®¥ which have a good dependency on F.

Given an analytic function of one complex variable f, we investigate the arithmetic nature of the values of f at algebraic points. A typical question is whether f(a) is a transcendental number for each algebraic number a. Since there exist transcendental entire functions f such that f^(k)(a) Î Q[a] for any k ³ 0 and any algebraic number a, one needs to restrict the situation by adding hypotheses, either on the functions, or on the points, or else on the set of values.

Among the topics we discuss are recent results due to Andrea Surroca on algebraic values of analytic functions and Diophantine properties of special values of polylogarithms.

http://www.institut.math.jussieu.fr/ ~ miw/