Families of padic 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 padic automorphic forms. Conversely, the rich geometry of these varieties gives insights about padic (and thereby also about classical) automorphic forms. Recent techniques from perfectoid geometry, locally analytic representation theory and the point of view of the padic Langlands program give new insights and impulses.
The spring school will give an introduction to both padic automorphic forms and eigenvarieties as well as the necessary background in padic 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 padic analytic geometry, the analogue of complex analytic geometry over padic 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 padic automorphic forms and eigenvarieties. We will introduce and compare several approaches to padic automorphic forms.
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 ringtheoretic 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 presheafs of rings $\mathcal{O}_X$ and $\mathcal{O}^+_X$. We discuss the relevant cases when these presheaves 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/quasicoherent 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 DrinfeldCerednik uniformization of Shimura curves.
Ben Heuer
In this course, we first define perfectoid algebras, which are certain kinds of topological rings over the padic 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 proétale and the vtopology. We then use these to discuss applications to padic Hodge theory.
Judith Ludwig
In this course we explain the eigenvariety machine. This is an abstract gadget that eats spaces of padic 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 padic functional analysis, which we will develop first. We will learn about compact operators on padic 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 minicourse will explain the construction of eigenvarieties in various different contexts, including the prototypical example of the ColemanMazur 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) padic 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 GaussManin 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 padically 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 HodgeTate period map to the perfectoid modular curve of infinite plevel 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 padic weight $k$, let us denote by $(W_k, \mathrm{Fil}_k,\nabla_k)$ the result of the construction at 1) above. The KodairaSpencer isomorphism allows us to see the connection, i.e. the family \( (\nabla_{k+2(n1)}\circ \dots \circ \nabla_{k+2}\circ \nabla_k:W_k \to W_{k+2n})_{n\in \mathbb{N}}. \) This are interesting applications of this construction to a) triple product padic Lfunctions, for finite slope (versus ordinary) families of modular forms and b) Katztype padic Lfunctions for cases when p is nonsplit 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 Liealgebra methods à la BGG and find a simpler complex, quasiisomorphic 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 EichlerShimura 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 representationtheoretic approach to eigenvarieties using completed cohomology of locally symmetric spaces(e.g. the tower of modular curves) and
Emeton's locally analytic Jacquetmodule.
We will start by introducing the representation theoretic background, the theory of socalled locally analytic representations developed by SchneiderTeitelbaum. After that we will
describe locally analytic parabolic induction (as well as a close relative of this construction) and Emerton's Jacquetmodule. 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 Jacquetmodule
of the completed cohomology and have an interpretation in terms of overconvergent padic automorphic forms.
We will finish the lecture course by relating the above construction to the padic 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 (padic) local Langlands correspondence.
Mingjia Zhang
$p$Adic Shimura varieties with infinite level at $p$ are important examples of perfectoid spaces. We discuss the example of the modular curve, the HodgeTate period map on it and its geometry revealed by the HodgeTate map. If time permits, we will review briefly some applications.
Lucas Mann
Condensed mathematics is a recent theory by ClausenScholze 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.
Vincent Pilloni
We describe the ordinary part of the coherent cohomology of Hilbert modular varieties. Along the way we establish a geometric JacquetLanglands correspondence. Joint with G. Boxer.
Chris Birkbeck
Following a construction of ChojeckiHansenJohansson, we show how to use Scholze's infinite level modular varieties and the HodgeTate 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 EichlerShimura maps in these settings. This is joint work with Ben Heuer and Chris Williams.
Christian Johansson
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 nonconstructive 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.
Rebecca Bellovin
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.
Joaquín Rodrigues Jacinto
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 padic 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.
All lectures and talks take place in the “Hörsaal” of the Mathematikon (Address: Im Neuenheimer Feld 205). Coffee breaks take place in the seminar room SR A+B close to the Hörsaal. The seminar room SR C will be open throughout the spring school as working space.
Monday  Tuesday  Wednesday  Thursday  Friday  

8:00 
Registration


9:30 
Bergdall
1/4

Bergdall
3/4

Johansson
2/4

Heuer
2/4

Johansson
4/4

Coffee break  
11:00 
Bergdall
2/4

Hübner
2/4

Bergdall
4/4

Hübner
4/4

Zhang

Lunch break 
Problem session
Coherent sheaves

Lunch break  
14:00 
Hübner
1/4

Hübner
3/4

Conference hike

Johansson
3/4

Birkbeck

Coffee break  Coffee break  
15:30 
Heuer
1/4

Johansson
1/4

Heuer
3/4

Mann


17:00 
Problem session
Huber pairs

Problem session
Adic spaces

Problem session
Perfectoid spaces

Monday  Tuesday  Wednesday  Thursday  Friday  

8:00 
Registration


9:30 
Ludwig
1/4

Ludwig
3/4

Ludwig
4/4

Hellmann
3/4

Hellmann
4/4

Coffee break  
11:00 
Ludwig
2/4

Newton
2/4

Iovita
3/4

Newton
4/4

Johansson

Lunch break  
14:00 
Newton
1/4

Iovita
2/4

Hellmann
2/4

Iovita
4/4

Bellovin

Coffee break  
15:30 
Iovita
1/4

Hellmann
1/4

Newton
3/4

Pilloni

Rodrigues Jacinto

17:00  Heuer
4/4


19:00 
Conference dinner

All talks are 60 minutes long, with coffee breaks at 10:30 and 15:00. Colored slots again refer to the mini courses, all other talks are research talks.
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 KlausTschiraPlatz.
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 SchmoettenJonas
Tel.: +4962215414241
Fax: +4962215414243
Email: springschool2023@mathi.uniheidelberg.de
RuprechtsKarlsUniversität Heidelberg
Mathematisches Institut, Raum 03417
Im Neuenheimer Feld 205
D69120 Heidelberg
Germany