Geometric level raising for p-adic automorphic forms

Details Geometric level raising for p-adic automorphic forms Geometric level raising for p-adic automorphic forms

2009-03-20

arxiv.org/abs/0903.3541v4

Compositio Mathematica 147 (2011) no. 2, 335-354

Mathematics

Number Theory

19 pages. Minor changes and references updated

Scientific paper

10.1112/S0010437X10004999

We present a level raising result for families of p-adic automorphic forms for a definite quaternion algebra D over the rational numbers. The main theorem is an analogue of a theorem for classical automorphic forms due to Diamond and Taylor. We show that certain families of forms old at a prime l intersect with families of l-new forms (at a non-classical point). One of the ingredients in the proof of Diamond and Taylor's theorem (which also played a role in earlier work of Taylor) is the definition of a suitable pairing on the space of automorphic forms. In our situation one cannot define such a pairing on the infinite dimensional space of p-adic automorphic forms, so instead we introduce a space defined with respect to a dual coefficient system and work with a pairing between the usual forms and the dual space. A key ingredient is an analogue of Ihara's lemma which shows an interesting asymmetry between the usual and the dual spaces.

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