## On Level-Raising Congruences

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Flicker, Yuval Z.

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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.

Comment: 36 pages

Comment: 36 pages

##### Keywords

Mathematics - Number Theory, Mathematics - Representation Theory, 11F33 (Primary) 22E55, 11F70, 11F85, 11F46, 20G25, 22E35 (Secondary)