Tobias Rossmann

Enumerating submodules invariant under an endomorphism

We study zeta functions enumerating submodules invariant under a given endomorphism of a finitely generated module over the ring of (S-)integers of a number field. In particular, we compute explicit formulae involving Dedekind zeta functions and establish meromorphic continuation of these zeta functions to the complex plane. As an application, we show that ideal zeta functions associated with nilpotent Lie algebras of maximal class have abscissa of convergence 2.

A Framework for Computing Zeta Functions of Groups, Algebras, and Modules

We give an overview of the author’s recent work on methods for explicitly computing various types of zeta functions associated with algebraic counting problems. Among the types of zeta functions that we consider are the so-called topological ones.