L3

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.