Kenneth Ribet

Abstract:   Twenty years ago, Kazhdan and Mazur suggested the possibility of a geometric interpretation of a relationship among Hecke operators that Eichler and Shimizu had proved using the Selberg trace formula. In this question, there are two distinguished prime numbers and two algebraic curves. A surprise is that the reduction of one curve modulo the first prime is related to the reduction of the other curve modulo the second prime. This unexpected twist, which surfaced in a positive way as a tool in level-lowering for Galois representations, seems to convey the unwelcome news that there is no simple relation between the Jacobians of the two curves. Nonetheless, a recent construction of David Helm of Berkeley shows how one might construct an abelian variety isomorphic to one Jacobian in terms of the arithmetic of the other.