Department of

March 2020 April 2020 May 2020 Su Mo Tu We Th Fr Sa Su Mo Tu We Th Fr Sa Su Mo Tu We Th Fr Sa 1 2 3 4 5 6 7 1 2 3 4 1 2 8 9 10 11 12 13 14 5 6 7 8 9 10 11 3 4 5 6 7 8 9 15 16 17 18 19 20 21 12 13 14 15 16 17 18 10 11 12 13 14 15 16 22 23 24 25 26 27 28 19 20 21 22 23 24 25 17 18 19 20 21 22 23 29 30 31 26 27 28 29 30 24 25 26 27 28 29 30 31

Thursday, April 30, 2020

**Abstract:** The 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 Galois representations coming from families 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 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.