The Integral Coefficient Geometric Satake Equivalence in Mixed Characteristic and its Arithmetic Applications

Abstract

The first main result of this thesis is the proof of the integral coefficient geometric Satake equivalence in mixed characteristic setting. Our proof can be divided into three parts: the construction of the monoidal structure of the hypercohomology functor on the category of integral coefficient equivariant perverse sheaves on the mixed characteristic affine Grassmannian; a generalized Tannakian formalism; and, the identification of group schemes. In particular, our proof does not employ Scholze’s theory of diamonds. We derive a geometric construction of the Jacquet-Langlands transfer for weighted automorphic forms as an application of the geometric Satake equivalence in the above setting. Our strategy follows the recent work of Xiao-Zhu [XZ17]. We relate the geometry and (l-adic) cohomology of the mod $p$ fibers of the canonical smooth integral models of different Hodge type Shimura varieties, and obtain a Jacquet-Langlands transfer for weighted automorphic forms.