The basis problem

Kimball Martin – University of Oklahoma

Modular forms arose as a natural object of study in number theory
due to connections with quadratic forms and theta series.  The basis problem, studied in depth by Eichler and others, conversely asks what spaces of modular forms can be generated by theta series, especially theta series attached to
quaternion algebras.  We will discuss a representation-theoretic approach to the basis problem using the Jacquet-Langlands correspondence.  This approach both refines the solution to the basis problem by Hijikata-Pizer-Shemanske for elliptic modular forms and solves the basis problem for Hilbert modular forms.  This has practical applications to computing spaces of modular forms and arithmetic applications to Eisenstein congruences.