On Level-Raising Congruences

TL;DR

This paper extends level-raising results for automorphic forms on reductive groups over totally real fields, allowing nontrivial types at infinite places and providing explicit conditions for congruences with more fixed vectors.

Contribution

It generalizes previous level-raising theorems to include nontrivial infinite types and applies these results to specific examples involving unitary and symplectic groups.

Findings

Established existence of congruent automorphic representations with increased parahoric fixed vectors.

Sharpened results for cases where G_w has semisimple rank one, showing eformations can be Steinberg.

Provided explicit conditions for level-raising in examples involving unitary and symplectic groups.

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.