The basis problem revisited

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.

37 pages; minor revisions; to appear in Trans. Amer. Math. Soc

Cited by in corpus (4)

• Zeroes of quaternionic modular forms and central $L$-values
• Root number bias for newforms
• Exact double averages of twisted L-values
• Mass formulas and Eisenstein congruences in higher rank