Frobenius eigenvalues in a space of mod-p automorphic forms

Abstract

For a given mod-p local Galois representation, one can construct a space of mod-p automorphic forms. One believes that this space is a candidate of the automorphic representation corresponding to the given mod-p local Galois representation for mod-p Langlands program. It is believed that Frobenius eigenvalues of certain potentially crystalline lifts of the given mod-p local Galois representation capture the extension classes of the given mod-p local Galois representation. In this talk, we introduce the actions of U_p-operators and discuss how one can use those U_p-operators to capture the Frobenius eigenvalues in the space of mod-p automorphic forms.