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.