The basis problem revisited

TL;DR

This paper provides a new, representation-theoretic approach to the basis problem for modular forms, explicitly describing the Jacquet-Langlands correspondence and solving the basis problem for Hilbert modular forms.

Contribution

It offers a simpler, more flexible solution to Eichler's basis problem for general levels and extends the basis problem solution to Hilbert modular forms using quaternionic newforms.

Findings

Explicit description of Jacquet-Langlands correspondence at the modular forms level

Solution to Eichler's basis problem for general level

Basis problem solved for Hilbert modular forms

Abstract

We explicitly describe the Jacquet-Langlands correspondence at the level of modular forms. This gives a simpler and more flexible solution to Eichler's basis problem for general level than earlier work of Hijikata-Pizer-Shemanske for elliptic modular forms, and solves the basis problem for Hilbert modular forms. The approach is representation theoretic rather than the classical approach of Eichler and Hijikata-Pizer-Shemanske, and involves both a local and global theory of quaternionic newforms.