We show that families of coverings of an algebraic curve where the associated Cayley-Schreier graphs form an expander family exhibit strong forms of geometric growth. Combining this general result with finiteness statements for rational points under such conditions, we derive results concerning the variation of Galois representations in oneparameter families of abelian varieties.

In continuation of our paper [7] we complete the description of the derived equivalence normal forms of all one-parametric selfinjective algebras over algebraically closed fields which admit simply connected Galois coverings. As a consequence, a description of the stable equivalence normal forms of these algebras is obtained. 2000 MSC: Primary: 16G10, 18E30; Secondary: 16D50, 16G60, 16G70.

A graph theoretical analogue of Brauer-Siegel theory for zeta functions of number …elds is developed using the theory of Artin L-functions for Galois coverings of graphs from parts I and II. In the process, we discuss possible versions of the Riemann hypothesis for the Ihara zeta function of an irregular graph.

Let A be a basic and connected finite dimensional algebra over a field k of characteristic zero. We show that if the quiver of A has no double bypass then the fundamental group (as defined in [17]) of any presentation of A by quiver and relations is the quotient of the fundamental group of a privileged presentation of A. Then we show that the Galois covering of A associated with this privileged...