We prove a version of the weight part of Serre’s conjecture for mod $p$ Galois representations attached to automorphic forms on rank 2 unitary groups which are nonsplit at $p$. More precisely, let $F∕F^{+}$ denote a CM extension of a totally real field such that every place of $F^{+}$ above $p$ is unramified and inert in $F$, and let $\overline{r}:\text{Gal}(\overline{F^{+}}/F^{+}){\to }^C{U}_2({\overline{\mathbb{𝔽}}}_p)$ be a Galois parameter valued in the $C$-group of a rank $2$ unitary group attached to $F∕F^{+}$. We assume that $\overline{r}$ is semisimple and sufficiently generic at all places above $p$. Using base change techniques and (a strengthened version of) the Taylor–Wiles–Kisin conditions, we prove that the set of Serre weights in which $\overline{r}$ is modular agrees with the set of Serre weights predicted by Gee, Herzig and Savitt.

We give several criteria to decide whether a given tensor category is the abelian envelope of a fixed symmetric monoidal category. As a main result we prove that the category of finite-dimensional representations of a semisimple simply connected algebraic group is the abelian envelope of the category of tilting modules. Benson and Etingof conjectured that a certain limit of finite symmetric tensor categories is tensor equivalent to the finite-dimensional representations of ${\text{SL}}_2$ in characteristic $2$. We use our results on the abelian envelopes to prove this conjecture and its variants for any prime $p$.

Let $F$ be a local field of characteristic $p>0$. By adapting methods of Scholze (2013), we give a new proof of the local Langlands correspondence for ${\text{GL}}_n$ over $F$. More specifically, we construct $\ell$-adic Galois representations associated with many discrete automorphic representations over global function fields, which we use to construct a map $\pi \mapsto \text{rec}(\pi )$ from isomorphism classes of irreducible smooth representations of ${\text{GL}}_n(F)$ to isomorphism classes of $n$-dimensional semisimple continuous representations of $W_F$. Our map $\text{rec}$ is characterized in terms of a local compatibility condition on traces of a certain test function $f_{\tau ,h}$, and we prove that $\text{rec}$ equals the usual local Langlands correspondence (after forgetting the monodromy operator).

We fix motivic data $(K∕F,E)$ consisting of a Galois extension $K∕F$ of characteristic $p$ global fields with arbitrary abelian Galois group $G$ and a Drinfeld module $E$ defined over a certain Dedekind subring of $F$. For this data, we define a $G$-equivariant motivic $L$-function ${\mathrm{\Theta }}_{K∕F}^E$ and prove an equivariant Tamagawa number formula for appropriate Euler product completions of its special value ${\mathrm{\Theta }}_{K∕F}^E(0)$. This generalizes to an equivariant setting the celebrated class number formula proved by Taelman in 2012 for the value ${\zeta }_F^E(0)$ of the Goss zeta function ${\zeta }_F^E$ associated to the pair $(F,E)$. (See also Mornev’s 2018 work for a generalization in a very different, nonequivariant direction.) We refine and adapt Taelman’s techniques to the general equivariant setting and recover his precise formula in the particular case $K=F$. As a notable consequence, we prove a perfect Drinfeld module analogue of the classical (number field) refined Brumer–Stark conjecture, relating a certain $G$-Fitting ideal of Taelman’s class group $H(E∕K)$ to the special value ${\mathrm{\Theta }}_{K∕F}^E(0)$ in question. In upcoming work, these results will be extended to the category of $t$-modules and used in developing an Iwasawa theory for Taelman’s class groups in Carlitz cyclotomic towers.