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Let and be primes, let be a finite extension with absolute Galois group , let be a finite field of characteristic , and let
be a continuous representation. Let be the universal framed deformation ring for . If , then the Breuil–Mézard conjecture (as recently formulated by Emerton and Gee) relates the mod reduction of certain cycles in to the mod reduction of certain representations of . We state an analogue of the Breuil–Mézard conjecture when , and we prove it whenever using automorphy lifting theorems. We give a local proof when is “quasibanal” for and is tamely ramified. We also analyze the reduction modulo of the types defined by Schneider and Zink.
The object of this article is to study 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 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 -adic torus periods for certain -adic-valued functions on Shimura curves of classical origin. We prove a -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 -adic -function of Rankin–Selberg type. At a character of positive weight, the -adic -function interpolates the central critical value of the complex Rankin–Selberg -function. Its value at a finite-order character, which is outside the range of interpolation, essentially computes the corresponding -adic torus period.