Speaker: Anthony Guzman (Arizona)

Title: Slopes of Modular Forms and Galois Representations

Abstract: Modular forms of weight k are complex analytic functions that have miraculous connections to algebra and number theory. An example of this occurs when one considers the action of Hecke operators on the space of modular forms giving rise to Hecke eigenvalues a_p. The p-adic valuation of these eigenvalues are known as slopes, and conjectures exist regarding special behavior occurring when these slopes are all integers. Surprisingly, one of the more fruitful avenues to understanding this behavior lies in the field of Galois representations. Indeed, one can attach to a modular form (under suitable restrictions) a p-adic Galois representation whose structure is determined by the weight k and Hecke eigenvalue a_p. It turns out that the integrality of the slopes also seem to affect the reducibility of the mod p reduction of these representations. In this talk, we will introduce these objects, conjectures and connections before considering a generalization of these ideas to Hilbert modular forms where life is much more complicated.