Atkin-Lehner Decompositions for Quaternionic modular forms

TL;DR

This paper develops Atkin-Lehner decompositions for quaternionic modular forms using representation theory and establishes isomorphisms with Hilbert modular forms via Jacquet--Langlands correspondence.

Contribution

It introduces a representation-theoretic approach to Atkin-Lehner decompositions for quaternionic modular forms and connects these spaces to Hilbert modular forms.

Findings

Atkin-Lehner decompositions are established for quaternionic modular forms.

Isomorphisms between quaternionic and Hilbert modular form spaces are demonstrated.

Representation theory techniques are effectively applied to modular form decompositions.

Abstract

In this paper, we obtain Atkin--Lehner decompositions for spaces of modular forms on definite quaternion algebras. Similar to Casselman's approach our methods are representation theoretic. Using Jacquet--Langlands correspondence we also obtain isomorphisms between spaces of quaternionic modular forms and corresponding spaces of Hilbert modular forms.