Mahler measures and Fuglede--Kadison determinants

The Mahler measure of a function on the real d-torus is its geometric mean over the torus. It appears in number theory, ergodic theory and other fields. The Fuglede-Kadison determinant is defined in the context of von Neumann algebra theory and can be seen as a noncommutative generalization of the Mahler measure. In the paper we discuss and compare theorems in both fields, especially approximation theorems by finite dimensional determinants. We also explain how to view Fuglede-Kadison determinants as continuous functions on the space of marked groups

Horizontal factorizations of certain Hasse--Weil zeta functions - a remark on a paper by Taniyama

In one of his papers, using arguments about l-adic representations, Taniyama expresses the zeta function of an abelian variety over a number field as an infinite product of modified Artin L-functions. The latter can be further decomposed as products of modified Dedekind zeta functions. After recalling Taniyama's work, we give a simple geometric proof of the resulting product formula for abelian and more general group schemes