Let $l$ and $p$ be primes, let $F/{\mathbb{Q}}_p$ be a finite extension with absolute Galois group $G_F$, let $\mathbb{F}$ be a finite field of characteristic $l$, and let

$$\overline{\rho }:G_F\to {GL}_n\left(\mathbb{F}\right)$$ be a continuous representation. Let $R^{\square }\left(\overline{\rho }\right)$ be the universal framed deformation ring for $\overline{\rho }$. If $l=p$, then the Breuil–Mézard conjecture (as recently formulated by Emerton and Gee) relates the mod $l$ reduction of certain cycles in $R^{\square }\left(\overline{\rho }\right)$ to the mod $l$ reduction of certain representations of ${GL}_n\left({\mathcal{O}}_F\right)$. We state an analogue of the Breuil–Mézard conjecture when $l\ne p$, and we prove it whenever $l>2$ using automorphy lifting theorems. We give a local proof when $l$ is “quasibanal” for $F$ and $\overline{\rho }$ is tamely ramified. We also analyze the reduction modulo $l$ of the types $\sigma \left(\tau \right)$ defined by Schneider and Zink.

The object of this article is to study ${GL}_2\mathbb{R}$ orbit closures in hyperelliptic components of strata of Abelian differentials. The main result is that all higher-rank affine invariant submanifolds in hyperelliptic components are branched covering constructions; that is, every translation surface in the affine invariant submanifold covers a translation surface in a lower genus hyperelliptic component of a stratum of Abelian differentials. This result implies the finiteness of algebraically primitive Teichmüller curves in all hyperelliptic components for genus greater than two. A classification of all ${GL}_2\mathbb{R}$ orbit closures in hyperelliptic components of strata (up to computing connected components and up to finitely many nonarithmetic rank one orbit closures) is provided. Our main theorem resolves a pair of conjectures of Mirzakhani in the case of hyperelliptic components of moduli space.

In this article, we study $p$-adic torus periods for certain $p$-adic-valued functions on Shimura curves of classical origin. We prove a $p$-adic Waldspurger formula for these periods as a generalization of recent work of Bertolini, Darmon, and Prasanna. In pursuing such a formula, we construct a new anti-cyclotomic $p$-adic $L$-function of Rankin–Selberg type. At a character of positive weight, the $p$-adic $L$-function interpolates the central critical value of the complex Rankin–Selberg $L$-function. Its value at a finite-order character, which is outside the range of interpolation, essentially computes the corresponding $p$-adic torus period.