The basis problem revisited

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 p^3 .