Past Number Theory Seminar

We consider certain q-series depending on parameters (A,B,C), where A is

a positive definite r times r matrix, B is a r-vector and C is a scalar,

and ask when these q-series are modular forms. Werner Nahm (DIAS) has

formulated a partial answer to this question: he conjectured a criterion

for which A's can occur, in terms of torsion in the Bloch group. For the

case r=1, the conjecture has been show to hold by Don Zagier (MPIM and

CdF). For r=2, Masha Vlasenko (MPIM) has recently found a

counterexample. In this talk we'll discuss various aspects of Nahm's conjecture.

While studying growth of lattices in semisimple Lie groups we

encounter many interesting number theoretic problems. In some cases we

can show an equivalence between the two classes of problems, while in

the other the true relation between them is unclear. On the talk I

will give a brief overview of the subject and will then try to focus

on some particularly interesting examples.

Given a polynomial function f defined on a variety X,

we consider two questions, which are non-commutative analogues

of the Prime Number Theorem and the Linnik Theorem:

- how often the values of f(x) at integral points in X are almost prime?

- can one effectively solve the congruence equation f(x)=b (mod q)

with f(x) being almost prime?

We discuss a solution to these questions when X is a homogeneous

variety (e.g, a quadratic surface).

Let C be a smooth plane cubic curve over the rationals. The Mordell--Weil Theorem can be restated as follows: there is a finite subset B of rational points such that all rational points can be obtained from this subset by successive tangent and secant constructions. It is conjectured that a minimal such B can be arbitrarily large; this is indeed the well-known conjecture that there are elliptic curves with arbitrarily large ranks. This talk is concerned with the corresponding problem for cubic surfaces.


Artin formalism gives an equality of certain L-functions of elliptic curves or of zeta-functions of number fields. When combined with the Birch and Swinnerton-Dyer conjecture, this can give interesting results about the Galois module structure of the Selmer group of an elliptic curve. When combined with the analytic class number formula, this can help determine the Galois module structure of the group of units of a number field. In this talk, I will introduce the main technique, which is completely representation theoretic, for extracting such information

In the first part of the talk we briefly describe an algorithm which computes a relative algebraic K-group as an abstract abelian group. We also show how this representation can be used to do computations in these groups. This is joint work with Steve Wilson.

Our motivation for this project originates from the study of the Equivariant Tamagawa Number Conjecture which is formulated as an equality of an analytic and an algebraic element in a relative algebraic K-group. As a first application we give some numerical evidence for ETNC in the case of the base change of an elliptic curve defined over the rational numbers. In this special case ETNC is an equivariant version of the Birch and Swinnerton-Dyer conjecture

The "large sieve" was invented by Linnik in order to attack problems involving the distribution of integers subject to certain constraints modulo primes, for which earlier methods of sieve theory were not suitable. Recently, the arithmetic large sieve inequality has been found to be capable of much wider application, and has been used to obtain results involving objects not usually considered as related to sieve theory. A form of the general sieve setting will be presented, together with sample applications; those may involve arithmetic properties of random walks on discrete groups, zeta functions over finite fields, modular forms, or even random groups.

We report on work joint with Scott Parsell in which estimates are obtained for the set of real numbers not closely approximated by a given form with real coefficients. "Slim"

technology plays a role in obtaining the sharpest estimates.

In a paper from 1994, 'The density of rational points on non-singular hypersurfaces', Heath-Brown developed a `multi-dimensional q-analogue'

of van der Corput's method of exponential sums, giving good bounds for the density of solutions to Diophantine equations in many variables. I will discuss this method and present some generalizations.

"Stickelberger's famous theorem (from 1890) gives an explicit ideal which annihilates the imaginary part of the class group of an abelian field as a module for the group-ring of the Galois group. In the 1980s Tate and Brumer proposed a generalisation of Stickelberger's Theorem (and his ideal) to other abelian extensions of number fields, the so-called `Brumer-Stark conjecture'.

I shall discuss some of the many unresolved issues connected with the annihilation of class groups of number fields. For instance, should the (generalised) Stickelberger ideal be the full annihilator, the Fitting ideal or what? And what can we say in the plus part (where Stickelberger's Theorem is trivial)?"

We show how the circle method with a suitably chosen Gaussian weight can be used to count unweighted zeros of polynomials. Tschinkel's problem asks for the density of solutions to Diophantine equations with S-unit and integral variables.

Let E be an elliptic curve over the rationals. To get an asymptotic to the number of primes p

Recently there has been increasing interest in discrete analogues of classical operators in harmonic analysis. Often the difficulties one encounters in the discrete setting require completely new approaches; the most successful current approaches are motivated by ideas from classical analytic number theory. This talk will describe a menagerie of new results for discrete analogues of operators ranging from twisted singular Radon transforms to fractional integral operators both on R^n and on the Heisenberg group H^n. Although these are genuinely analytic results, key aspects of the methods come from number theory, and this talk will highlight the roles played by theta functions, Waring's problem, the Hypothesis K* of Hardy and Littlewood, and the circle method.

We recall that an elliptic curve is a curve of genus one with a rational point on it. Certain algorithms for determining the structure of the group of rational points on an elliptic curve produce a whole set of curves of genus one and then require that we determine which of these curves has a rational point.

Unfortunately no algorithm which has been proved to terminate is known for doing this. Such an algorithm or proof would probably have profound implications for the study of elliptic curves and may shed light on the Birch and Swinnerton-Dyer conjecture.

This talk will be about joint work with Samir Siksek (Warwick) on the development of a new algorithmic criterion for determining that a given curve of genus one has no rational points. Both the theory behind the criterion and recent attempts to make the criterion computationally practical, will be detailed.