Daniel Loughran (Leibniz Universität Hannover)

In this course we discuss the theory of height zeta functions. Such zeta functions naturally arise when one is interested in counting rational points of bounded height on algebraic varieties, and there is a conjecture of Manin concerning their analytic behaviour. After some basic examples we shall illustrate some of the range of behaviour which occurs via the cases of flag varieties and toric varieties, where the presence of a group structure allows one to address the zeta function using tools from harmonic analysis. If time permits, we shall also consider some related counting problems where the associated height zeta functions have branch point singularities.

Immanuel Halupczok (University of Leeds)

On ℚ_p, the field of p-adic numbers, one has a natural Lebesgue measure, giving a meaning to the integrals of functions from ℚ_p^n to ℂ. If we are given one such function f_p: ℚ_p^n -> ℂ for each prime p in a suitable uniform way, then motivic integration provides a way to also compute the integrals uniformly in p; I will explain how this works. As an example, I will show how this can be applied to Igusa Zeta functions, which fall into this framework. As a result, one obtains that a rationality result about Igusa Zeta functions can be made uniform in ℚ_p. (All this will also be explained.)

Christopher Voll (Universität Bielefeld)

I will give a very concise introduction to various zeta functions used in asymptotic group and ring theory, concentrating on those enumerating subobjects of various kinds. A prototype of such a function is the zeta function enumerating the normal subgroups of finite index in a finitely generated nilpotent group. The zeta functions in question typically satisfy Euler product decompositions, with ("local") factors indexed by places of number fields. In the first talk I will explain a number of intringuing features these Euler factors tend to exhibit (such as rationality and functional equations), talk about methods from $p$-adic integration to compute them, and introduce their reduced and topological counterparts. In the second talk I will concentrate on zeta functions enumerating submodules invariant under a nilpotent algebra of endomorphisms. These generalize the normal subgroup zeta functions mentioned above. I will discuss a sufficient condition for these zeta functions to satisfy local functional equations, shedding new light on a conjecture of du Sautoy and Woodward.