# Arithmetic and Groups

### Research Area A

Bays, Cuntz, Deninger, Hartl, Hellmann, Hille, Hils, Jahnke, Kwiatkoswka, Nikolaus, Scherotzke (until 2020), R. Schindler, Schlutzenberg, Schneider, Scholbach, Schürmann, Tent.

Research Area A will focus on problems of algebraic nature with methods ranging from representation and cohomology theory to model theory and mathematical logic. The p-adic and characteristic p Langlands correspondences that relate arithmetic invariants with automorphic representations are major research topics. We also aim at developing new powerful cohomology theories in algebra and arithmetic geometry. One of the long-term goals is to understand Hasse-Weil zeta functions cohomologically.

In group theory, we plan to develop new techniques for small cancellation theory in order to construct sharply 2-transitive groups in sufficiently large finite characteristic. On the model theoretic side, the focus will be on understanding and classifying NIP groups and fields as well as on classifying and constructing ample strongly minimal structures. In set theory and descriptive set theory, we will be studying Varsovian models as well as Polish group actions.

• ## A1. Arithmetic, geometry and representations.

The Langlands programme relates representations of (the adele valued points of) reductive groups G over Q - so-called automorphic representations - with certain representations of the absolute Galois group of Q. This programme includes the study of these objects over general global fields (finite extension of Q or Fp (t)) and local fields as well. In its local form the classical programme only considered l-adic Galois representations of p-adic fields for unequal primes l neq p. In order to allow for a p-adic variation of the objects, it is absolutely crucial to extend it to the case l = p. In the global situation, the automorphic representations in question can often be realised in (or studied via) the cohomology of a tower of Shimura varieties (or related moduli spaces) attached to the group G.

We will focus on the following directions within this programme:
The p-adic and mod p Langlands programme asks for an extension of such a correspondence involving certain continuous representations with p-adic respectively mod p coefficients. Broadening the perspective to p-adic automorphic forms should, for example, enable us to capture all Galois representations, not just those having a particular Hodge theoretic behaviour at primes dividing p. This extended programme requires the introduction of derived categories. We will study differential graded Hecke algebras and their derived categories on the reductive group side. On the Galois side, we hope to use derived versions of the moduli spaces of p-adic Galois representations introduced by Emerton and Gee.

The geometric Langlands programme is a categorification of the Langlands programme. We plan to unify the different approaches using motivic methods.
In another direction, we study the geometry and arithmetic of moduli stacks of global G-shtukas over function fields. Their cohomology has been the crucial tool to establish large parts of the local and global Langlands programme over function fields. Variants of G-shtukas are also used to construct and investigate families of p-adic Galois representations.

Cohomology theories are a universal tool pervading large parts of algebraic and arithmetic geometry. We will develop and study cohomology theories, especially in mixed characteristic, that generalise and unify étale cohomology, crystalline cohomology and de Rham cohomology as well as Hochschild cohomology in the non-commutative setting. Developing (topological) cyclic homology in new contexts is an important aim. A main goal is to construct a cohomology theory that can serve the same purposes for arithmetic schemes as the l-adic or crystalline cohomology with their Frobenius actions for varieties over finite fields. Ideas from algebraic geometry, algebraic topology, operator algebras and analysis blend in these investigations.

• ## A2. Groups, model theory and sets.

Model theory and, more generally, mathematical logic as a whole has seen striking applications to arithmetic geometry, topological dynamics and group theory. Such applications as well as fundamental questions in geometric group theory, on the foundations of model theory and in set theory are at the focus of our work.
The research in our group reaches from group theoretic questions to the model theory of groups and valued fields as well as set theory. The study of automorphism groups of first order structures as topological groups, for examples, uses tools from descriptive set theory leading to pure set theoretic questions.

## Further research projects of Research Area A members

$\bullet$ Christopher Deninger, Thomas Nikolaus: CRC 1442: Geometry: Deformation and Rigidity - A04: New cohomology theories for arithmetic schemes (2020-2024)
We develop, study and compute certain global cohomology theories for schemes. These cohomology theories may be viewed as deformations of ‘classical’ cohomology theories over mixed characteristic or over the sphere spectrum. For example, de Rham–Witt cohomology is a deformation of de Rham cohomology over mixed characteristic. Topological Hochschild homology is a deformation of ordinary Hochschild homology over the sphere spectrum. The motivation for the project is to use these cohomology theories to attack deep problems in algebra, topology and arithmetic geometry. Our most ambitious application concerns zeta functions.

$\bullet$ Eugen Hellmann, Peter Schneider: CRC 1442: Geometry: Deformation and Rigidity - A01: Automorphic forms and the p-adic Langlands programme (2020-2024)
The p-adic Langlands programme aims to establish a relation between p-adic representations of p-adic reductive groups and p-adic representations of Galois groups of p-adic local fields. We plan an in depth study of the smooth mod p representation theory of reductive groups on the level of derived categories. In first relevant test cases we want to construct functors from representations of reductive groups to sheaves on deformation spaces of Galois representations.

$\bullet$ Eugen Hellmann, Urs Hartl, Peter Schneider: CRC 1442: Geometry: Deformation and Rigidity - A02: Moduli spaces of p-adic Galois representations (2020-2024)
p-adic Galois representations in finite Zp-modules are equivalent to (phi,Gamma)-modules for Qp. In this project, we develop the theory of (phi,Gamma)-modules further in the direction of finite extensions of Qp and their function field analogues. We will also use (phi,Gamma)-modules to construct moduli spaces of p-adic Galois representations. We aim to decompose special fibres on these moduli spaces into cycles in a way that mirrors multiplicity formulas in representation theory.

$\bullet$ Urs Hartl: CRC 1442: Geometry: Deformation and Rigidity - A03: Special cycles on moduli spaces of G-shtukas (2020-2024)
Moduli stacks of global G-shtukas are the function field analogue of Shimura varieties. Their geometry was investigated by the PI and his collaborators. The long term goal of the project is to investigate arithmetic properties of these moduli stacks, special cycles on them and the intersection numbers of the latter. In particular, we want to develop a function field analogue of the Kudla programme for Shimura varieties and prove arithmetic fundamental lemmas for function fields.

$\bullet$ Katrin Tent: CRC 1442: Geometry: Deformation and Rigidity - C04: Hyperbolic groups acting sharply 2- or 3-transitively and the Burnside problem (2020-2024)
Until recent 'free' constructions of the PI and her collaborators, the only known sharply 2- and 3-transitive permutation groups were those arising from linear transformations of the affine or projective line. We will investigate the limitations of this class of groups. The quest for sharply 2- and 3-transitive groups in positive characteristic not arising from linear transformations leads us to the Burnside problem, for which we propose a new approach yielding much lower bounds on the exponent for infinite such groups.

$\bullet$ Aleksandra Kwiatkowska: CRC 1442: Geometry: Deformation and Rigidity - C05: Rigidity of group topologies and universal minimal flows (2020-2024)
We will study automatic continuity and universal minimal flows for topological groups acting on geometric objects. First, we focus on rigidity of group topologies, motivated by a question due to Rosendal on the existence of a locally compact infinite group with the automatic continuity property. We want to study this question for Burger–Mozes groups. Furthermore, motivated by a question of Evans–Hubi?ka–Nešet?il, we want to understand when universal minimal flows of kaleidoscopic groups are metrisable. Our methods will involve interactions between group theory, topological dynamics and Ramsey theory.

$\bullet$ Thomas Nikolaus: CRC 1442: Geometry: Deformation and Rigidity - C02: Homological algebra for stable ?-categories (2020-2024)
The goal of this project is to study the emerging area of homological algebra for stable infinity-categories. Concretely the major objective of this project is to study non-commutative motives as introduced by Blumberg–Gepner–Tabuada and the homotopy theory of chain complexes of stable infinity-categories that will be developed as part of the project, following up on pioneering work of Dyckerhoff. Moreover, we will explore the notion of a stable (infinity,infinity)-category and the corresponding higher version of spectra, which will besides its general importance also be relevant in setting up a rigid higher version of Quinn’s Ad-theories.

$\bullet$ Martin Bays, Katrin Tent, Franziska Jahnke, Martin Hils: Geometric and Combinatorial Configurations in Model Theory (2020-2022)
Model theory studies structures from the point of view of first-order logic. It isolates combinatorial properties of definable sets and uses these to obtain algebraic consequences. A key example is the group configuration theorem, a powerful tool in geometric stability used, e.g., to prove the trichotomy for Zariski geometries and in recent applications to combinatorics. Valued fields are an example of the confluence of stability theory and algebraic model theory. While Robinson studied algebraically closed valued fields already in 1959, the tools from geometric stability were only made available in this context in work of Haskell-Hrushovski-Macpherson, brought to bear in Hrushovski-Loeser's approach to non-archimedean geometry. In the project, we aim to strengthen the recent relations between model theory and combinatorics, develop the model theory of valued fields using tools from geometric stability and carry out an abstract study of the configurations which are a fundamental tool in these two areas.