Remote Access Transactions of the American Mathematical Society
Green Open Access

Transactions of the American Mathematical Society

ISSN 1088-6850(online) ISSN 0002-9947(print)

 
 

 

The basis problem revisited


Author: Kimball Martin
Journal: Trans. Amer. Math. Soc. 373 (2020), 4523-4559
MSC (2010): Primary 11F27, 11F41, 11F70
DOI: https://doi.org/10.1090/tran/8077
Published electronically: April 28, 2020
Full-text PDF

Abstract | References | Similar Articles | Additional Information

Abstract: Eichler investigated when there is a basis of a space of modular forms consisting of theta series attached to quaternion algebras, and treated squarefree level. Hijikata, Pizer, and Shemanske completed the solution to Eichler's basis problem for elliptic modular forms of arbitrary level by tour-de-force trace calculations. We revisit the basis problem using the representation-theoretic perspective of the Jacquet-Langlands correspondence.

Our results include: (i) a simpler proof of the solution to the basis problem for elliptic modular forms, which also allows for more flexibility in the choice of quaternion algebra; (ii) a solution to the basis problem for Hilbert modular forms; (iii) a theory of (local and global) new and old forms for quaternion algebras; and (iv) an explicit version of the Jacquet-Langlands correspondence at the level of modular forms, which is a refinement of the Hijikata-Pizer-Shemanske solution to the basis problem. Both (i) and (ii) have practical applications to computing elliptic and Hilbert modular forms. Moreover, (iii) and (iv) are desired for arithmetic applications--to illustrate, we give a simple application to Eisenstein congruences in level $ p^3$.


References [Enhancements On Off] (What's this?)


Similar Articles

Retrieve articles in Transactions of the American Mathematical Society with MSC (2010): 11F27, 11F41, 11F70

Retrieve articles in all journals with MSC (2010): 11F27, 11F41, 11F70


Additional Information

Kimball Martin
Affiliation: Department of Mathematics, University of Oklahoma, Norman, Oklahoma 73019
Email: kimball.martin@ou.edu

DOI: https://doi.org/10.1090/tran/8077
Received by editor(s): June 11, 2018
Received by editor(s) in revised form: March 5, 2019
Published electronically: April 28, 2020
Additional Notes: This work was supported by grants from the Simons Foundation/SFARI (240605 and 512927, KM)
Article copyright: © Copyright 2020 American Mathematical Society