We study the non-vanishing on the line Â(s)=1 of the convolution series associated to two Dirichlet series in a certain class of Dirichlet series. The non-vanishing of various L-functions on the line Â(s)=1 will be simple corollaries of our general theorems.

This is joint work with Shahab Shahabi.

For any number field K which is not totally imaginary, Elkies showed that any elliptic curve E over K has infinitely many supersingular primes. The result is also believed to be true for number fields which are totally imaginary, but the proof does not seem to generalise to this case. One reason may be that over totally imaginary fields, we can build examples where the supersingular primes must be inert primes, and then the set of supersingular primes is much thinner than what is expected for elliptic curves over Q. We then consider the problem of enumerating the inert primes of the quadratic imaginary field Q(i) for which a given elliptic curve E has supersingualr reduction. We prove that on average, the number of such primes up to x is in accordance with the standard conjectures. This gives further evidence that there should be infinitely many supersingular primes in this case as well. (joint work with F. Pappalardi)

Resultants and discriminants have always been important tools in number theory. Departing from a theorem of Emma Lehmer on resultants of cyclotomic polynomials, we make the following observation: When one takes the resultant of a certain pair of polynomials closely related to the Chebyshev polynomials (of the second kind) and involving a parameter, one surprisingly obtains again a Chebyshev polynomial in this parameter. We prove an identity that contains this observation and others as special cases. The proofs use a "chain rule" for resultants and an identity related to the Euclidean algorithm for polynomials; these last results, though not new, do not appear to be widely known in the literature. (Joint work with K. B. Stolarsky).

We discuss recent joint work with S. Adhikari, S. Konyagin, and F. Pappalardi concerning a zero-sum problem in the residue class ring of integers modulo n.

We continue our examination of the vanishing of the central values of L-functions of elliptic curves twisted by Dirichlet characters.

We explicitly exhibit the unit group of certain infinite families of fields L of degree 12 over Q which are composita of real quadratic fields and pure sextic fields. This is achieved with the help of the unit groups of the three subfields of L which are of degree 6.

When computing upper bounds for the number of rational points of bounded height on a (compact) projective algebraic variety defined over Q, it is an easy observation that there either must be few points, or those points must be packed closely together. In this talk, I will describe a few results describing how close rational points on an algebraic variety can get to one another, and the implications these results have for the density of rational points on these varieties.

We will use the theory of modular forms and Dirichlet series, more specifically, those attached to Hecke grossencharacters of imaginary quadratic fields to settle a recent conjecture of Borwein and Choi.

This is joint work with R. Osburn.

In this talk we consider the square of the Riemann zeta function times a Dirichlet polynomial averaged over the zeros of the zeta function. We give an asymptotic evaluation of this moment, thus generalizing earlier work of Conrey, Ghosh, and Gonek. As a consequence, we will deduce that there are infinitely many consecutive zeros of the zeta function on the critical line whose gaps are larger than the average.

We present results of simultaneous approximation of several numbers from a field of transcendence degree one by conjugate algebraic numbers. In some cases, we also derive, by duality, new Gel'fond type criterions for polynomials taking small values at distinct points.

In this talk we plan to discuss some joint work with A. Sarkozy concerning pseudorandom binary sequences. We shall construct a large family of such sequences with a remarkable uniformity within the family.

We describe an algorithm to compute all occurrences of a given integer in a family of ternary recurrence sequences which arise from units in pure cubic fields. The algorithm involves computing all rational integer points on a parametric family of elliptic curves, which just happen to contain a parametric point of order three. Rational torsion points of higher order on these curves correspond to integer solutions of certain diophantine equations F(x,y)=0, where the height and degrees of each F are quite large. Fortunately, these polynomials satisfy some unexpected properties, allowing for the complete description of the rational torsion subgroup of the original family of curves. This is joint work with Emanuel Herrmann.

One of the most important invariants of an order of an algebraic number field is the regulator of that order. This object is often very difficult to evaluate, even in a field as seemingly simple as a real quadratic field. Indeed this is one of the oldest problems of number theory, and it can be traced as far back as Archimedes. A very useful technique for evaluating the regulator of a real quadratic order is the method of continued fractions. It has been known for over two hundred years that the fundamental unit of such an order can be determined from the periodic continued fraction expansion of the generator of the order. In this case the regulator is simply the logarithm of the fundamental unit, and this is roughly the same as the length of the period of the corresponding continued fraction.

For most real quadratic orders the regulator tends to be rather large, but in certain, unusual and infrequent cases it is small, ensuring that the ideal class number will be large. Such orders are of very great interest to number theorists because their ideal class groups often have exotic structures. The search for such orders is a very old problem in number theory; indeed, Dickson's History of the Theory of Numbers devotes several pages to the work of many authors concerning this problem. In this talk I will describe some families of radicands D(X), given by D(X)=AX2+BX+C such that the value of the period length l(D(X)) of the regular continued fraction expansion of vD(X) tends to be small.