Zeta functions of twisted modular curves and twisted quaternionic Shimura varieties

We determine the zeta functions of twisted modular curves and twisted quaternionic Shimura varieties as a product of automorphic L-functions and then continue meromorphically their zeta functions under mild conditions. More precisely, we consider the compactified modular curve X( n) and quaternionic Shimura variety SG&d4; n associated to the principal congruence subgroups G (n) and G&d4; (n), respectively. We twist these varieties by a mod n representation rho of the absolute Galois group of canonical field of definition of these varieties and the new varieties that we obtain are called "twisted modular curves" and "twisted quaternionic Shimura varieties", respectively. The zeta functions for modular curves was computed by Shimura. The zeta function for Hilbert modular varieties was computed by Brylinski and Labesesse and then their result was generalized by Reimann to quaternionic Shimura varieties. Using their computation and multiplicity one for automorphic representations, we determine the zeta function of the "twisted" varieties. Recently, Blasius determined the zeta function for quaternionic Shimura varieties at all places. Using his result we are able to determine the zeta functions of the twisted varieties at all places. We prove the meromorphic continuation of the zeta functions when the dimension of the varieties is at most 2 and the fixed field K of the kernel of rho is a solvable extension of a totally real field. More exactly, we prove the existence of the base change for Hilbert modular forms of GL(2) /F, where F is the totally real field that corresponds to the Shimura variety, to a solvable extension of a totally real field that contains K. The proof uses deformation theory and level lowering for mod p Galois representations arising from the reductions of representations associated to Hilbert modular forms.