Let `$F$`

be a totally real field and `$p$`

be an odd prime unramified in `$F$`

. We will give an overview of the problem of determining the explicit mod `$p$`

structure of a modular `$p$`

-adic Galois representation and determining the associated local Serre weights. The Galois representations are attached to Hilbert modular forms over `$F$`

, more precisely to eigenforms on a Shimura curve over `$F$`

. The weight part of the Serre’s modularity conjecture for Hilbert modular forms relates the local Serre weights at a place `$v|p$`

to the structure of the mod `$p$`

Galois representation at the inertia group over `$v$`

. Thus, local Serre weights give good information on the structure of the modular mod `$p$`

Galois representation. The eigenforms considered are of small slope at a fixed place `$\mathbf{p}|p$`

, and with certain constraints on the weight over `$\mathbf{p}$`

. This is based on a joint work with Shalini Bhattacharya.

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Last updated: 20 Mar 2023