Spherochromatism in representation theory and arithmetic geometry

Abstract

The goal of this thesis is to explain some applications of the perspective of chromatic homotopy theory to geometric representation theory and to arithmetic geometry. In the first half of this thesis, we study how the derived geometric Satake equivalence (due to Bezrukavnikov-Finkelberg, building on work of Ginzburg and Mirkovic-Vilonen) changes when one considers the category of constructible equivariant sheaves of $k$-modules on the affine Grassmannian of a complex (simply-laced) reductive group $G$, where $k$ is a commutative ring \textit{spectrum}. We state a conjecture describing the ``spectral side'' in terms of the Langlands dual group $\ld{G}$ and the $1$-dimensional formal group associated to $k$ via chromatic homotopy theory, and we make progress towards proving this conjecture. We also explore consequences of our conjecture in relation to the relative Langlands program recently elucidated by Ben-Zvi--Sakellaridis--Venkatesh. In the second half of this thesis, we describe some joint work with Arpon Raksit, in which we refine work of B"okstedt-Madsen to provide a complete description of the topological Hochschild homology of the ring $\Z_p$ of $p$-adic integers in terms of the classical image of J spectrum. This result has several applications, both to homotopy theory and to arithmetic geometry, which we outline. We also describe some joint work with Jeremy Hahn, Arpon Raksit, and Allen Yuan, which aims to extend the theory of prismatization recently developed by Bhatt-Lurie-Drinfeld to the setting of ring spectra. This is tightly related to the theory of equivariant formal groups, and we provide some explicit calculations of these objects by generalizing rudiments of $q$-deformed calculus. The constructions we describe also have applications to classical arithmetic geometry; for example, we explain how our work can be used to provide a higher-dimensional refinement of Drinfeld's recent reinterpretation of Deligne-Illusie's work on Hodge theory in characteristic $p>0$.