On Level-Raising Congruences

Abstract

A work of Sorensen is rewritten here to include nontrivial types at the infinite places. This extends results of K. Ribet and R. Taylor on level-raising for algebraic modular forms on D^{\times}, where D is a definite quaternion algebra over a totally real field F. This is done for any automorphic representations \pi of an arbitrary reductive group G over F which is compact at infinity. It is not assumed that \pi_\infty is trivial. If \lambda is a finite place of \bar{\Q}, and w is a place where \pi_w is unramified and \pi_w is congruent to the trivial representation mod \lambda, then under some mild additional assumptions (relaxing requirements on the relation between w and \ell which appear in previous works) the existence of a \tilde{\pi} congruent to \pi mod \lambda such that \tilde{\pi}_w has more parahoric fixed vectors than \pi_w, is proven. In the case where G_w has semisimple rank one, results of Clozel, Bellaiche and Graftieaux according to which \tilde{\pi}_w is Steinberg, are sharpened. To provide applications of the main theorem two examples over F of rank greater than one are considered. In the first example G is taken to be a unitary group in three variables and a split place w. In the second G is taken to be an inner form of GSp(2). In both cases, precise satisfiable conditions on a split prime w guaranteeing the existence of a \tilde{\pi} congruent to \pi mod \lambda such that the component \tilde{\pi}_w is generic and Iwahori spherical, are obtained. For symplectic G, to conclude that \tilde{\pi}_w is generic, computations of R. Schmidt are used. In particular, if \pi is of Saito-Kurokawa type, it is congruent to a \tilde{\pi} which is not of Saito-Kurokawa type.