Colloquium: Tuesday, March 29, 2022. Speaker: Speaker: Christopher Voll (Bielefeld). Title: “Zeta functions of groups, rings, and modules”.

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Abstract: Zeta functions are ubiquitous in mathematics. In algebra and geometry, they often have an enumerative interpretation, prominent examples being the Dedekind zeta function of a number field and the (Weil) zeta function of an algebraic variety over a finite field.
Over the last few decades, zeta functions have become powerful tools also in the study of asymptotic invariants of infinite groups and modules, notably in the study of their subgroup and submodule growth, viz. the distribution of their finite index subgroups resp. submodules.

Under favourable conditions, zeta functions of groups, rings, and modules share key features with their more classical predecessors in algebra and geometry — such as Eulerian products with rational factors, local and (sometimes) global functional equations, analytic continuation etc. — while retaining a distinct flavour.

In my talk, I will survey some recent developments in the theory of these zeta functions. I will concentrate on the phenomenon of local functional equations, combining delicate combinatorial arguments with consequences of the “Weil conjectures”, deep results on the numbers of rational points on algebraic varieties over finite fields.