Let p be a prime number and K a finite unramified extension of $\mathbb{Q}_p$. Let $\pi$ be an admissible smooth mod p representation of $\operatorname{GL}_2(K)$ occurring in some Hecke eigenspaces of the mod p cohomology and $\overline{r}$ be its underlying global two-dimensional Galois representation. When $\overline{r}$ satisfies some Taylor–Wiles hypotheses and is sufficiently generic at p, we compute explicitly certain constants appearing in the diagram associated to $\pi$, generalizing the results of Dotto and Le. As a result, we prove that the associated étale $(\varphi,\mathcal{O}_K^{\times})$-module $D_A(\pi)$ defined by Breuil, Herzig, Hu, Morra and Schraen is explicitly determined by the restriction of $\overline{r}$ to the decomposition group at p, generalizing the results of Breuil et al. and the author.

In an infinitesimal variant of Guo–Jacquet trace formulae, the regular semi-simple terms are expressed as noninvariant weighted orbital integrals on two global infinitesimal symmetric spaces. We prove some relations between the Fourier transforms of invariant weighted orbital integrals on the corresponding local infinitesimal symmetric spaces. These relations should be useful in the noninvariant comparison of the infinitesimal variant of Guo–Jacquet trace formulae.

We prove the Jacquet–Rallis fundamental lemma for spherical Hecke algebras over local function fields using multiplicative Hitchin fibrations. Our work is inspired by the proof by Zhiwei Yun in the Lie algebra case and builds upon the general framework of multiplicative Hitchin fibrations by the first-named author.

We study the classification of $\mathbb{Z}$-DGAs (differential graded algebras) with polynomial homology $\mathbb{F}_p[x]$ with $\lvert x \rvert \gt0$, motivated by computations in algebraic K-theory. This classification problem was left open in work of Dwyer, Greenlees, and Iyengar. We prove that there are infinitely many such DGAs for even $\lvert x\rvert$ and that for $\lvert x \rvert \geq 2p-2$ any such DGA is formal as a ring spectrum. Through this, we obtain examples of triangulated categories with infinitely many DG enhancements and a classification of prime DG division rings. Combining our results with earlier work of the second author and Tamme, we obtain new (relative) algebraic K-theory computations for rings such as the mixed characteristic coordinate axes $\mathbb{Z}[x]/px$ and the group ring $\mathbb{Z}[C_{p^n}]$.

We study the basic locus of Shimura varieties associated to the group of spinor similitudes of a quadratic space over ${{\mathbb{Q}}}$ with level structure given by the stabilizer of a vertex lattice. We give a description of the underlying reduced scheme of the associated Rapoport–Zink space, generalizing results of Howard–Pappas and Oki, in the case of self-dual, and almost self-dual, level structure.