The meromorphic continuation of the zeta function of a product of Hilbert and Picard modular surfaces over CM-fields

Abstract

In this paper we prove in particular the meromorphic continuation to the entire complex plane of the zeta function of a product of a Hilbert modular surface and a Picard modular surface regarded over arbitrary CM-fields (see the end of the Introduction for more results regarding the meromorphic continuation of the zeta functions associated to products of Shimura curves, quaternionic Shimura surfaces and Picard modular surfaces). In order to obtain the meromorphic continuation we show the simultaneous potential modularity for an arbitrary finite number of l -adic representations associated to Hilbert modular forms and Picard modular surfaces.