PhD Advisor: Florian Herzig
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Let $K$ be a $p$-adic field. Given a continuous Galois representation $\bar{\rho}: \mathrm{Gal}(\overline{K}/K)\rightarrow \mathrm{GL}_n(\overline{\mathbb{F}}_p)$, the mod $p$ Langlands program hopes to associate with it a smooth admissible $\overline{\mathbb{F}}_p$-representations $\Pi_p(\bar{\rho})$ of $\mathrm{GL}_n(K)$ in a natural way. When $\bar{\rho}=\bar{r}|_{G_{F_w}}$ is the local $w$-part of a global automorphic Galois representation $\bar{r}:\mathrm{Gal}(\overline{F}/F)\rightarrow \mathrm{GL}_n(\overline{\mathbb{F}}_p)$, for some CM field $F/F^+$ and place $w|p$, it is possible to construct a candidate $H^0(\bar{r})$ for $\Pi_p(\bar{r}|_{G_{F_w}})$ using spaces of mod $p$ automorphic forms on definite unitary groups.
Assume that $F_w$ is unramified. When $\bar{r}|_{G_{F_w}}$ is semisimple, it is possible to recover the data of $\bar{r}|_{G_{F_w}}$ from the $\mathrm{GL}_n(\mathcal{O}_{F_w})$-socle of $H^0(\bar{r})$ (also known as the set of Serre weights of $\bar{r}$). But when $\bar{r}|_{G_{F_w}}$ is wildly ramified this socle does not contain enough information. In this thesis we give an explicit recipe to find the missing data of $\bar{r}|_{G_{F_w}}$ inside the $\mathrm{GL}_3(F_w)$-action on $H^0(\bar{r})$ when $n=3$ and $\bar{r}|_{G_{F_w}}$ is maximally nonsplit, Fontaine-Laffaille, and generic. This generalizes work of Herzig, Le and Morra who found analogous results when $F_w=\mathbb{Q}_p$ as well as work of Breuil and Diamond in the case of unramified $\mathrm{GL}_2$.