The p-adic geometry of Shimura varieties and applications to the Langlands program

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This is a research project in the general area of number theory. This area of mathematics has applications to areas such as cryptography and to physics. The particular branch of number theory considered in this project is arithmetic geometry where properties of interest in number theory are studied by geometric methods. The overall theme of the research is the interplay between arithmetic geometry and the Langlands correspondence for number fields. There have been many recent breakthroughs in the field, such as new techniques for proving reciprocity (modularity lifting theorems as well as the emerging p-adic Langlands program) and the construction of Galois representations associated to torsion classes in the cohomology of locally symmetric spaces. All of these developments have depended crucially on being able to p-adically interpolate automorphic forms. The concept of p-adic automorphic forms has a natural definition in terms of the geometry and cohomology of Shimura varieties and therefore p-adic arithmetic geometry is useful for studying them.The PI investigates two intertwined areas: on one hand, applications of p-adic arithmetic geometry (specifically the theory of perfectoid spaces) to p-adic automorphic forms and, on the other hand, p-adic and mod p analogues of the classical Langlands program. Specifically, the PI will study a new approach to the p-adic local Langlands correspondence via the Taylor-Wiles method, to further study torsion occurring in the cohomology of Shimura varieties and the properties at p of the associated Galois representations and to develop a new approach to instances of the Tate conjecture in the context of Shimura varieties. Some of the new techniques that the PI intends to use are the Taylor-Wiles patching method applied to completed cohomology and matching parameters in local deformation rings with Hecke operators. Another key idea involves studying perfectoid Shimura varieties via their associated period domains. The research project lies at the intersection of algebraic number theory, representation theory and algebraic geometry, with a focus on the interplay between p-adic arithmetic geometry and the Langlands correspondence for number fields. A central motif in number theory is the classification of algebraic extensions of number fields. Class field theory addresses this for extensions with abelian Galois group. The Langlands program provides a framework for a vast generalization of class field theory to the non-abelian setting. At its heart is the conjectural correspondence between automorphic representations and Galois representations, which is often realized by geometric objects, such as Shimura varieties. Therefore, arithmetic geometry provides many important tools for studying Langlands correspondences. Many of the most spectacular recent results in number theory are instances of the Langlands correspondence, such as Fermat's last theorem, the Sato-Tate conjecture and Serre's conjecture.