Non-archimedean geometry and eigenvarieties

These lectures will focus on valuation theory for topological rings. The discussion will be organized around the class of Huber rings that underlie the theory of adic spaces. We will discuss their definition, basic properties, and ring-theoretic constructions such as the subset of topologically nilpotent elements. We will further outline the theory of continuous valuations and valuations with supports on topological rings. And, depending on goals set by other lectures, we will explain technical material on spectral spaces, topology, specialization, dimension theory, and so on. Throughout, we will gather as many examples as possible and hopefully build intuition.

Starting with a Huber pair $(A,A^+)$ we construct its adic spectrum $X=\mathrm{Spa}(A,A^+)$. This is a certain space of continuous valuations equipped with pre-sheafs of rings $\mathcal{O}_X$ and $\mathcal{O}^+_X$. We discuss the relevant cases when these pre-sheaves are actually sheaves, allowing to glue adic spectra together to obtain adic spaces. In order to understand the underlying topological space of the adic spectrum of $(A,A^+)$, we study specialization relations of points in $\mathrm{Spa}(A,A^+)$. We then take a look at analytic adic spaces. They correspond to Tate Huber pairs and are the type of adic spaces we are interested in.

These lectures will mostly focus on adic spaces with certain finiteness conditions, most importantly rigid spaces. We will discuss some aspects of sheaf theory and cohomology on adic spaces, and some obstructions to setting a theory of coherent/quasi-coherent sheaves on general adic spaces. We will then set up the theory of coherent sheaves of rigid analytic varieties and discuss the analogues of Cartan's Theorem A and B as well as finiteness of cohomology for proper rigid spaces. The latter will necessitate a discussion of various properties of morphisms of rigid spaces, such as smoothness, étaleness and properness amongst others. Finally, we will discuss some examples of particular importance in the historic development of the theory, such as Tate's uniformization of elliptic curves with multiplicative reduction and the Drinfeld-Cerednik uniformization of Shimura curves.

In this course, we first define perfectoid algebras, which are certain kinds of topological rings over the p-adic integers. We discuss some basic properties, most importantly the "tilting equivalence" which gives an equivalence between perfectoid algebras in characteristic 0 and those in characteristic p. Building on the previous lectures, we then discuss perfectoid spaces, which are adic spaces built out of adic spectra of perfectoid algebras. A basic and important result about perfectoid spaces is the almost acyclicity theorem, for which we will sketch a proof. Finally, we discuss examples of how perfectoid spaces arise in nature, and how we can use them to study rigid spaces. For this we introduce several locally perfectoid topologies on a rigid space, namely the pro-étale and the v-topology. We then use these to discuss applications to p-adic Hodge theory.

In this course we explain the eigenvariety machine. This is an abstract gadget that eats spaces of p-adic automorphic forms and returns adic spaces that parametrize systems of Hecke eigenvalues of these forms. In order to build the machine, we need some results from p-adic functional analysis, which we will develop first. We will learn about compact operators on p-adic spaces and understand how to geometrically handle their spectral theory. We will then study the eigenvariety machine and some of its basic properties.

This mini-course will explain the construction of eigenvarieties in various different contexts, including the prototypical example of the Coleman-Mazur eigencurve. We will also discuss some of the common geometric properties of eigenvarieties.

Plan of the course: We will focus on three main themes. We will explain the main ideas and constructions first for modular curves and than for Siegel modular varietes.

1) p-adic variation of de Rham classes on modular curves

Fix a prime $p>2$. If $(H,\mathrm{Fil},\nabla)$ is a triple consisting of: the relative de Rham cohomology sheaf, $H$, of the universal generalized elliptic curve over a modular curve $X$, its Hodge filtration and Gauss-Manin connection, the goal of this section is to construct sheaves of Banach modules on certain strict neighborhoods of the ordinary locus in $X$ (seen as an adic space over $\mathrm{Spa}(\mathbb{Q}_p, \mathbb{Z}_p)$, which interpolates p-adically the family of sheaves with filtrations and connections $(\mathrm{Sym}^n(H), \mathrm{Fil}_n,\nabla_n)_{n\in \mathbb{N}}.$

The neighborhoods of the ordinary locus in $X$ are constructed first as neighborhoods of $\infty$ in $\mathbb{P}^1$, are pulled back by the Hodge-Tate period map to the perfectoid modular curve of infinite p-level and descended to $X$ by Galois theory. This allows for a good understanding of the dynamic of the $U_p$-operator on the secions of our sheaves.

2) Given a p-adic weight $k$, let us denote by $(W_k, \mathrm{Fil}_k,\nabla_k)$ the result of the construction at 1) above. The Kodaira-Spencer isomorphism allows us to see the connection, i.e. the family \( (\nabla_{k+2(n-1)}\circ \dots \circ \nabla_{k+2}\circ \nabla_k:W_k \to W_{k+2n})_{n\in \mathbb{N}}. \) This are interesting applications of this construction to a) triple product p-adic L-functions, for finite slope (versus ordinary) families of modular forms and b) Katz-type p-adic L-functions for cases when p is non-split in the CM field.

3) Let $k$ be a family weight. If $(W_K, \mathrm{Fil}_k, \nabla_k)$ is the Banach sheaf with filtration and connection constructed at 1) above, we'd like to compute the finite slope part of \[ H^1_{\mathrm{dR}}(Z,W_k^*), \] where $W_k^*$ is the de Rham complex $\nabla_k:W_k\to W_k\otimes\Omega^1_Z$, and $Z$ is the neighborhood of the ordinary locus in $X$ where $W_k$ exists.

We found that the best way to do this is to use Lie-algebra methods à la BGG and find a simpler complex, quasi-isomorphic to the (uncompleted) de Rham complex, which can be described in terms of overconvergent modular forms, and which computes the finite slope part of the de Rham cohomology. This has applications to de Rham Eichler-Shimura morphisms.

4) We will explain the ideas at 1) and 3) above for Siegel threefolds (i.e. for Shimura varieteis for $\mathrm{GSp}_4$).

In this course we will discuss a representation-theoretic approach to eigenvarieties using completed cohomology of locally symmetric spaces(e.g. the tower of modular curves) and Emeton's locally analytic Jacquet-module.
We will start by introducing the representation theoretic background, the theory of so-called locally analytic representations developed by Schneider-Teitelbaum. After that we will describe locally analytic parabolic induction (as well as a close relative of this construction) and Emerton's Jacquet-module. Applying the constructions to representations obtained from the completed cohomology of modular curves we can reconstruct the eigencurve (and, if we replace the modular curves by more general Shimura varieties: more general eigenvarieties) as the support of a coherent sheaf on the (rigid analytic generic fiber of the) deformation space of Galois representaions. The gloval sections of this coherent sheaf are identified with the dual of the Jacquet-module of the completed cohomology and have an interpretation in terms of overconvergent p-adic automorphic forms.
We will finish the lecture course by relating the above construction to the p-adic local Langlands program. In particular we will sketch the relation of the coherent sheaf (whose support is identified with the eigenvariety) with coherent sheaves on stacks of Galois representations that show up in categorical approaches to a (p-adic) local Langlands correspondence.

$p$-Adic Shimura varieties with infinite level at $p$ are important examples of perfectoid spaces. We discuss the example of the modular curve, the Hodge-Tate period map on it and its geometry revealed by the Hodge-Tate map. If time permits, we will review briefly some applications.

Condensed mathematics is a recent theory by Clausen-Scholze which solves several major problems of topological algebra by redefining the notion of a topological space. We will explain the basic ideas of condensed mathematics and use them to define a category of quasicoherent sheaves on adic spaces, which seemed previously impossible. This has some immediate consequences to the theory of vector bundles (or more general coherent sheaves) on adic spaces.

We describe the ordinary part of the coherent cohomology of Hilbert modular varieties. Along the way we establish a geometric Jacquet-Langlands correspondence. Joint with G. Boxer.

Following a construction of Chojecki-Hansen-Johansson, we show how to use Scholze's infinite level modular varieties and the Hodge-Tate period map to define overconvergent elliptic and Hilbert modular forms in a way analogous to the standard construction of modular forms. As an application we show that this is one way of constructing overconvergent Eichler-Shimura maps in these settings. This is joint work with Ben Heuer and Chris Williams.

An interesting question in the theory of eigenvarieties is to recognize classical modular forms among $p$-adic overconvergent ones. For example, for \(\mathrm{GL}(2)\) it is known that an overconvergent eigenform whose Hecke eigenvalues agree with those of a classical eigenform is in fact classical itself. However, J. Ludwig showed by a non-constructive method that this need not be the case for \(\mathrm{SL}(2)\). In this talk I will explain a way to understand and quantify this phenomenon using ideas from the geometrization of Langlands correspondences. Along the way we also obtain results on the local geometry at endoscopic points. This is joint work with Judith Ludwig.

I will discuss families of Galois representations which can be constructed over the extended eigencurve. I will discuss the $p$-adic properties of such Galois representations, and give some applications to the structure of the extended eigencurve over the boundary of weight space.

I will explain a joint and ongoing project with Juan Esteban Rodríguez Camargo, where we develop new foundations for the theory of locally analytic representations of a p-adic Lie group through the use of condensed mathematics. As an application of this new formalism, I will explain some comparison results between different cohomology theories for solid representations.