Spring School on
# Non-archimedean geometry and eigenvarieties

March 06 - 17, 2023 • University of Heidelberg

Families of *p*-adic automorphic forms are well studied objects of arithmetic geometry since the pioneering work of Hida and Coleman.
Their study resulted in the definition of geometric objects, called eigenvarieties, that parametrize systems of Hecke eigenvalues of
*p*-adic automorphic forms. Conversely, the rich geometry of these varieties gives insights about *p*-adic (and thereby also about classical)
automorphic forms. Recent techniques from perfectoid geometry, locally analytic representation theory and the point of view of the *p*-adic Langlands
program give new insights and impulses.

The spring school will give an introduction to both *p*-adic automorphic forms and eigenvarieties as well as the necessary background
in *p*-adic analytic geometry. The courses will be complemented by research talks that will focus on recent developments in the area.

The first week of the spring school will focus on *p*-adic analytic geometry, the analogue of complex analytic geometry
over *p*-adic base fields. We will study classical rigid analytic spaces from the point of view of adic spaces and
introduce perfectoid spaces. The second week will focus on *p*-adic automorphic forms and eigenvarieties.
We will introduce and compare several approaches to *p*-adic automorphic forms.

- John BERGDALL, University of Arkansas
- Eugen HELLMANN, University of Münster
- Ben HEUER, University Bonn
- Katharina HÜBNER, University of Heidelberg
- Adrian IOVITA, Univ. Concordia / Padua
- Christian JOHANSSON, University of Gothenburg
- Judith LUDWIG, University of Heidelberg
- James NEWTON, University of Oxford

- Rebecca BELLOVIN, University of Glasgow
- Chris BIRKBECK, University College London
- Christian JOHANSSON, University of Gothenburg
- Lucas MANN, University Bonn
- Joaquín RODRIGUES JACINTO, Univ.Paris Saclay
- Mingjia ZHANG, University Bonn
- Vincent PILLONI, Université Paris-Saclay

John Bergdall

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.

Katharina Hübner

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.

Christian Johansson

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.

Ben Heuer

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 betweeen 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.

Judith Ludwig

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.

James Newton

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.

Adrian Iovita

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 $\mathrm{GSp}_4$).

Eugen Hellmann

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.

Mathematikon

Im Neuenheimer Feld 205

69120 Heidelberg

Room TBC

Note that there are three similar buildings next to each other that are all called “Mathematikon”. The Summer School takes place in the southernmost one of them, directly across the tram station “Bunsengymnasium” and a Shell gas station. The place in front of it is called Klaus-Tschira-Platz.

Prof. Otmar Venjakob

University of Heidelberg, Germany

Dr.
Judith Ludwig

University of Heidelberg, Germany

Prof. Eugen
Hellmann

University of Münster, Germany

Prof. Sujatha Ramdorai

University of British Columbia, Vancouver, Canada

Frau Birgit Schmoetten-Jonas

Tel.: +49-6221-54-14241

Fax: +49-6221-54-14243

Email: springschool2023@mathi.uni-heidelberg.de

Ruprechts-Karls-Universität Heidelberg

Mathematisches Institut, Raum 03-417

Im Neuenheimer Feld 205

D-69120 Heidelberg

Germany