Barcelona Fall Workshop on Number Theory

Serre weights and wild ramification in two-dimensional Galois representations


Lassina Dembélé


Let $F$ be a totally real number fi eld, $p\geq 2$ a rational prime, and $\overline{\rho} : Gal(\overline{F}/F)\rightarrow GL_2(\overline{\mathbb{F}}_p)$ a Galois representation arising from a Hilbert modular form. The weight part of Serre's Conjecture (proved by Gee et al) determines the set of weights $\mathcal{D}(\overline{\rho})$ of forms from which $\overline{\rho}$ arises. However the dependence of the set $\mathcal{D}(\overline{\rho})$ on wild ramifi cation is not explicit. I'll discuss joint work with Diamond and Roberts towards making it so. This amounts to describing wild rami cation in
reductions of two-dimensional crystalline Galois representations.


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