On level-raising congruences

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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 Dx, 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 πω 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 πω has more parahoric fixed vectors than πω. In the case where Gω 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 πω 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 languageEnglish
Pages (from-to)151-180
Number of pages30
JournalAsian Journal of Mathematics
Issue number2
StatePublished - 2009
Externally publishedYes


  • Algebraic modular forms
  • Congruences
  • Level-raising


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