## Abstract

In this paper we rewrite a work of Sorensen to include nontrivial types at the infinite places. This work extends results of K. Ribet and R. Taylor on level-raising for algebraic modular forms on D^{x}, where D is a definite quaternion algebra over a totally real field F. We do this for any automorphic representations π of an arbitrary reductive group G over F which is compact at infinity. We do not assume π∞ is trivial. If λ is a finite place of Q, and ω is a place where π_{ω} is unramified and π _{ω} (mod λ), then under some mild additional assumptions (we relax requirements on the relation between ω and l which appear in previous works) we prove the existence of a π ≡ π (mod λ) such that π_{ω} has more parahoric fixed vectors than π_{ω}. In the case where G_{ω} has semisimple rank one, we sharpen results of Clozel, Bellaiche and Graftieaux according to which π _{ω} is Steinberg. To provide applications of the main theorem we consider two examples over F of rank greater than one. In the first example we take G to be a unitary group in three variables and a split place ω. In the second we take G to be an inner form of GSp(2). In both cases, we obtain precise satisfiable conditions on a split prime ω guaranteeing the existence of a π ≡ π (mod λ) such that the component π_{ω} is generic and Iwahori spherical. For symplectic G, to conclude that π _{ω} is generic, we use computations of R. Schmidt. In particular, if π is of Saito-Kurokawa type, it is congruent to a π which is not of Saito-Kurokawa type.

Original language | English |
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Pages (from-to) | 151-180 |

Number of pages | 30 |

Journal | Asian Journal of Mathematics |

Volume | 13 |

Issue number | 2 |

DOIs | |

State | Published - 2009 |

Externally published | Yes |

## Keywords

- Algebraic modular forms
- Congruences
- Level-raising