Arithmetic geometry and representation theory

Our research group is concerned with two lines of investigation: the construction and study of (new) cohomology theories for algebraic varieties and the study of various aspects of the Langlands programme.

Algebraic varieties over $\mathbb{Q}$ are among the fundamental objects in arithmetic geometry. We aim to study them by establishing and studying new cohomology theories. Here, one of our main guidelines is the relation of the newly-established prismatic cohomology with topology - in particular K-theory.

In the framework of the Langlands programme we study the envisioned correspondence between automorphic forms and Galois representations through a geometric and a categorical lens.
The geometric approach aims at understanding the geometry of the spaces whose cohomology should realize Langlands correspondences: local and global Shimura varieties and related spaces (affine Deligne-Lusztig varieties and moduli spaces of shtukas).

The categorical approach (in which we focus on $p$-adic aspects) wants to upgrade the envisioned bijection between isomorphism classes of representations to an equivalence of categories, by replacing Galois representations with coherent sheaves on their moduli spaces.

What is ... ?

• a Galois representation

  One of the ultimate goals of algebraic number theory is to understand the structure of the absolute Galois group $\text{Gal}(\overline{\mathbb{Q}}/\mathbb{Q})$ of the rational numbers (as well as the absolute Galois groups of finite field extensions of $\mathbb{Q}$). Often a group can be understood by looking at its representations, that is: actions on (finite dimensional) vector spaces. Such Galois representations naturally arise in the study of elliptic curves, more generally in the cohomology of algebraic varieties, or as $p$-adic limits of the latter.

  The Langlands program aims to unravel some of the mysteries of these Galois representations by linking them to modular forms and automorphic forms.

• a modular/automorphic form

  Modular forms are holomorphic functions on the complex upper half plane that have a prescribed transformation behavior under the action of $SL_2(\mathbb{Z})$ on the upper half plane.  They are strongly intertwined with algebraic number theory: the Taniyma-Shimura-Weil conjecture (also known as the modularity theorem of Wiles) asserts that every elliptic curve over the rationals is associated to a modular (eigen-)form.

  Modular forms provide an explicit model to study special representations of the group $GL_2$; so called automorphic representations.

  In the framework of the Langlands program (that among others tries to vastly generalize Wiles’ modularity theorem) automorphic representations of more general linear algebraic groups (like $GL_n$ or orthogonal or symplectic groups) should be related to objects in algebraic number theory like motives and Galois representations.

•  a Shimura variety

  Shimura varieties and their cohomology carry natural actions of linear algebraic groups and of a suitable Galois group. In this way, they are the natural candidates for the geometric realization of Langlands correspondences,a first example being Harris-Taylor's proof of the local Langlands correspondence for $GL_n$. Often, they can be viewed as moduli spaces of abelian varieties with additional structure or, more generally, of suitable motives. In this way, properties and invariants of the parametrized objects yield insight into the geometric properties of the Shimura variety, and into the representations arising in its cohomology.

• $p$-adic Hodge theory

  Is the study of structures that appear in the cohomology of algebraic varieties defined over the $p$-adic numbers. The cohomology is a vector space associated to the algebraic variety that can be thought of as a linearization of the variety; the additional structures on this vector space (e.g. a filtration, a monodromy operator or an action of the Galois group) can be used to get a better understanding of the variety itself. $p$-adic Hodge theory aims to understand the relation between various cohomology theories and their interplay. It has recently been lifted to a new level by the developments of perfectoid spaces and prismatic cohomology.

• a perfectoid space

  Perfectoid spaces are a recent development that have developed into a key technique in arithmetic geometry.

  One of their key features is that they can be used to study arithmetic and geometric in characteristic zero, while having features that are familiar from objects in characteristic p where the Frobenius automorphisms ($ x \mapsto x^p $) is a tool of invaluable importance. Typical examples are rings that have many $p$-power roots such as $\mathbb{Q}_p(\mu_{p^\infty})$, the extension of $\mathbb{Q}_p$ obtained by adjoining a compatible system of $p$-power roots of unity; or $\mathbb{Q}_p(\mu_{p^\infty})\langle T^{1/p^\infty} \rangle$ the $p$-adic completion of the ring obtained from the polynomial ring by extracting $p$-power roots of the variable.

  Prefectoid spaces have proven to be a key technique in all aspects of $p$-adic arithmetic in particular in the study of Shimura varieties and in $p$-adic Hodge theory and prismatic cohomology.