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.
展开▼
机译：我们确定扭曲的模块曲线和扭曲的四元离子Shimura品种的zeta函数作为自同构L函数的乘积，然后在温和条件下继续亚纯地维持其zeta函数。更准确地说，我们考虑了紧实的模块化曲线X（n）和四元离子Shimura品种SG＆d4; n与主同余子组G（n）和G＆d4相关； （n）。我们用这些变量定义的标准规范域的绝对Galois组的mod n表示rho扭曲这些变量，我们获得的新变量分别称为“扭曲模块曲线”和“扭曲四元离子Shimura变量”。模块化曲线的zeta函数由Shimura计算。 Hilbert模块化品种的zeta函数由Brylinski和Labesesse计算，然后他们的结果由Reimann推广到四元离子Shimura品种。使用他们的计算和多重性之一来表示同构，我们确定了“扭曲”变种的zeta函数。最近，Blasius在所有地方确定了四元Shimura品种的zeta函数。使用他的结果，我们可以确定所有地方的扭曲品种的zeta函数。我们证明了当变种的维数最大为2且rho核的固定场K是整个实场的可解扩展时，zeta函数的亚纯连续性。更确切地说，我们证明了GL（2）/ F的希尔伯特模块化形式的基本变化的存在，其中F是与Shimura品种相对应的完全实地，是包含K的完全实地的可解扩展。该证明使用变形理论和水平降低，以降低与希尔伯特模块化形式相关联的表示形式而引起的mod p Galois表示形式。
展开▼
