Geometric level raising for p-adic automorphic forms

TL;DR

This paper proves a level raising theorem for p-adic automorphic forms on quaternion algebras, extending classical results and introducing new dual space techniques due to the infinite-dimensional setting.

Contribution

It establishes a level raising result for p-adic automorphic forms using dual coefficient systems and an analogue of Ihara's lemma, addressing challenges in defining pairings.

Findings

Families of old forms at prime l intersect with l-new forms at non-classical points.

Introduces a dual space framework to define pairings in infinite-dimensional settings.

Provides an analogue of Ihara's lemma showing asymmetry between spaces.

Abstract

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.