Fractal zeta functions for $$p$$-adic affine variaties

Abstract

Abstract We investigate the fractal zeta function of a $$p$$ p -adic affine variety. We identify it with the distance zeta function. We also prove that for smooth varieties with good reduction it is a rational function of $t=p^{-s}$ t = p - s and that the abscissa of convergence is the dimension of the variety. We conjecture that this remains so even in the presence of singularities and show that this conjecture holds for a generic family of singular varieties.