Date & Time: Wednesday, August 16, 2006, 16:00-17:00.
Venue: Ramanujan Hall
Title: The Integral Basis Problem of Eichler
Speaker: Haruzo Hida, UCLA
Abstract: The basis problem of Eichler is to find an explicit basis (over C) of an appropriate space of elliptic modular forms among the theta series of the norm forms of definite quaternion algebras. Eichler solved this problem in the 1950s by comparing trace of Hecke operators acting on automorphic forms on such quaternion algebras and on elliptic modular forms. The basis problem has its origin in Jacobi's celebrated formula of the number S_4(n) of ways of expressing a given integer n as a sum of four squares. For this Jacobi's example, the basis problem has solution over the rational integer ring (i.e., theta series = integral Eisenstein series). We try to extend this integral solution to cusp forms and to more general cases of definite quaternions whose discriminant is a single prime.