Number Theory Seminar

Eichler described classical modular forms of squarefree level in terms of theta series coming from quaternion algebras. This may be viewed as an explicit special case of the Jacquet-Langlands correspondence, which relates automorphic representations on GL(2) with those on its inner forms. First I will explain how to explicitly compare modular forms on quaternion algebras and GL(2) beyond squarefree level. Then I will discuss ongoing joint work with Gefei Dang on extending this to the unitary group U(2), which is an outer form of GL(2). The main local ingredient needed is a suitable newform theory for inner and outer forms of GL(2).