Locally Split Galois Representations and Hilbert Modular Forms of Partial Weight One

Abstract: The $p$-adic Galois representation attached to a $p$-ordinary eigenform is upper triangular when restricted to a decomposition group at $p$. A natural question to ask is under what conditions this upper triangular decomposition splits as a direct sum. Ghate and Vatsal have shown that for the Galois representation attached to a Hida family of $p$-ordinary eigenforms, the restriction to a decomposition group at $p$ is split if and only if the family has complex multiplication; in their proof, the weight one members of the family play a key role.

I'll talk about work in progress which aims to answer similar questions in the case of Galois representations for a totally real field which are split at only some of the decomposition groups at primes above $p$. In this work Hilbert modular forms of partial weight one play a central role; I'll discuss what is known about them and to what extent the techniques of Ghate and Vatsal can be adapted to this situation.

number theory

Audience: researchers in the topic

( slides )

UCLA Number Theory Seminar