# Project A: Algebra and logic

In research area A two fundamental principles are used. On the one
hand, group representation theory is employed for the investigation of
objects coming from number theory and algebraic geometry. On the
other hand, the logic based projects A4 and A5 use non-linear group
actions for further investigations of algebraic groups. Using methods
from model theory and set theory, these projects also reach into topological
and geometric questions.

In Project A1 the primary goal is to understand the category of semistable
vector bundles on algebraic varieties, in particular curves, over
p-adic fields. Vector bundles contain a lot of interesting geometric information
about the variety. A natural group which is associated with
a variety is its fundamental group. In the analogous classical situation
over the complex numbers the Narasimhan-Seshadri correspondence
describes stable vector bundles in terms of unitary representations of
the topological fundamental group. Over p-adic fields this principle
of setting vector bundles into correspondence with representations of
fundamental groups is only partially realized so far. One is still far
from understanding the precise domain of this correspondence as well
as from giving a characterization of its range. These are two of the
central questions to be addressed in A1.

Project A2 aims at expanding a program devised by Langlands to
understand nonabelian extensions of number fields by means of the
representation theory of reductive algebraic groups. The underlying
principle again is a correspondence between representations of Galois
groups of local or global number fields and representations of reductive
groups. The classical Langlands program only deals, in its local
form, with l-adic Galois representations of a p-adic field when the
prime numbers l and p are different. The category of p-adic Galois
representations of a p-adic field is considerably more complicated. The
main challenge in A2 is to find an extension of the local Langlands correspondence
to p-adic Galois representations.

One of the most important algebraic methods to investigate group
representations is to transform them into modules over a group algebra.
In the p-adic context the arising p-adic group algebras are still
rather mysterious objects which need to be further investigated in their
own right. This all the more so as they also turn up in other contexts,
for example in global number theory where they act on Selmer groups
and are used to construct p-adic L-functions. For the latter purpose it
is crucial to compute algebraic K-groups of p-adic group rings. This is
the second goal in A2. It builds very much on methods from algebraic
topology.

One of the important advantages which one expects from a p-adic
extension of the Langlands program is that objects on both sides of
the correspondence may vary in families. This leads naturally to the
consideration of various kinds of moduli spaces and period domains.
This phenomenon also plays a role in A1. Semistable vector bundles on
an algebraic curve, for example, form a moduli space which itself is an
algebraic variety. One may even pursue the idea that correspondences
like the ones described above derive from actual maps between moduli
spaces.

Project A3 will be devoted to a systematic study of certain moduli
spaces over local and global function fields in positive characteristics
which are analogs of Shimura varieties over characteristic zero number
fields. The construction of such a moduli space involves a specific
reductive group. A main goal of the project is to describe the bad reduction
of these moduli spaces in group theoretical terms. Tools are
the affine buildings also studied in project B4. It is expected that the
picture is somewhat similar to the bad reduction of Shimura varieties.
Over function fields however, group theoretic methods are much more
powerful. A thorough group theoretic understanding of the situation
will most likely also refertilize the theory of Shimura varieties.
For certain groups the local moduli spaces also have analogs in characteristic zero,
which are moduli spaces of Barsotti-Tate groups.
In cooperation between A2 and A3 it will be investigated
whether these latter spaces make it possible to understand the p-adic
local Langlands correspondence as a map between spaces.

In contrast to projects A1, A2 and A3, project A4 investigates nonlinear
actions of algebraic and other, abstractly given groups. The
questions addressed here concern characterizations of groups acting
on certain natural geometries, like Tits buildings, associated to algebraic
groups. Model theory has a natural interest in these questions
as from the model theoretic point of view one wants to find characterizations
which imply that an abstractly given group is (isomorphic
to) an algebraic group. This yields information about the first-order
structures involved which can then be used for questions concerning
algebraic groups and geometry, like the questions about the definable
subsets in valued fields.

Project A5, while still investigating group actions will also use methods
from descriptive set theory and set theory. We will investigate actions
of Polish groups which resemble algebraic groups in the way they
can act and plan to construct new examples. This leads to questions
about Borel actions and orbit equivalence relations also considered in
Project B7. Our emphasis will be more on the model and set theoretic
aspects of these questions than Project A4.

## Subprojects

- A1: Algebraic vector bundles
- A2: p-adic group algebras
- A3: Moduli spaces of G-shtukas and the Langlands Program
- A4: Model theory, stable groups and corresponding complexes and geometries
- A5: Descriptive set theory and model theory
- A6: Definable reducibility
- A7: Generalized cohomology theories
- A8: Quasi-hereditary algebras, toric geometry and derived categories
- A9: Geometric Model Theory of Valued Fields

## Principal investigators

- Prof. Dr. Christopher Deninger
- Prof. Dr. Urs Hartl
- apl. Prof. Dr. Lutz Hille
- Prof. Dr. Martin Hils
- Prof.Dr. Ben Miller
- Prof. Dr. Ralf Schindler
- Prof. Dr. Peter Schneider
- apl. Prof. Dr. Joerg Schuermann
- Prof. Dr. Dr. Katrin Tent

## Participating scientists

- Dr. Esmail Arasteh Rad
- Dr. Martin Bays
- Dr. Mark Blume
- Marten Bornmann
- Prof. Dr. Siegfried Bosch
- Martin Brandenburg
- Fabiana Castiblanco
- Yong Cheng
- Dr. John Clemens
- Dr. Sean D. Cox
- Javier de la Nuez Gonzalez
- Dr. Antongiulio Fornasiero
- Dr. Shelly Garion
- Dr. Zaniar Ghadernezhad
- Dr. Immanuel Halupczok
- Simon Huesken
- Manuel Inselmann
- Dr. Itay Kaplan
- Marius Kley
- Dr. Jan Kohlhaase
- Prof. Dr. Linus Kramer
- Dominik Leukers
- Philipp Luecke
- Marius Moeller
- Isabel Mueller
- Dr. Gonenc Onay
- Dr. Daniel Palacin
- Felix Roetting
- Matthias Rother
- Tim Schauch
- Dr. Lars Scheele
- Dr. Tobias Schmidt
- Torsten Schoeneberg
- Dr. Jakob Scholbach
- Rajneesh Kumar Singh
- Jonas Stelzig
- Carsten Szardenings
- Pierre Touchard
- Dimitri Wegner
- Anna Weiss
- Matthias Wulkau

## Publications and preprints of the SFB

- Katsurada, : Complete asymptotic expansions for certain multiple q-integrals and q-differentials of Thomae-Jackson type
- Katsurada, : Complete asymptotic expansions for the product averges of higher derivetives of Lerch zeta-functions
- Deninger, : Regulators, entropy and infinite determinants
- Deninger, : Determinants on von Neumann algebras, Mahler measures and Ljapunov exponents
- Scholbach, : Mixed Artin-Tate motives over number rings
- Scholbach, Andreas Holmstrom: Arakelov motivic cohomology
- Scholbach, : f-cohomology and motives over number rings
- Scholbach, : Special L-values of geometric motives
- Deninger, : The Hilbert-Polya strategy and height pairings
- Deninger, : Invariant measures on the circle and functional equations
- Deninger, Wegner, : Horizontal factorizations of certain Hasse-Weil zeta functions - a remark on a paper by Taniyama
- Scholbach, : Algebraic K-theory of the infinite place
- Scholbach, : Arakelov motivic cohomology II
- Pavlov, Scholbach, : Symmetric operads in abstract symmetric spectra
- Pavlov, Scholbach, : Admissibility and rectification of colored symmetric operads
- Schneider, : Smooth representations and Hecke modules in characteristic p
- Schneider, Venjakob, Otmar: Coates-Wiles homomorphisms and Iwasawa cohomology for Lubin-Tate extensions
- Schneider, : Galois representations and (phi,Gamma)-modules
- Schneider, Berger, Laurent, Xie, Bingyong: Rigid character groups, Lubin-Tate theory, and (phi,Gamma)-modules
- Schneider, Ollivier, Rachel: A canonical torsion theory for pro-p Iwahori-Hecke modules
- Hartl, Eva Viehmann (Univ. Bonn): The Newton stratification on deformations of local G-shtukas
- Hartl, : Period Spaces for Hodge Structures in Equal Characteristic
- Hartl, Matthias Bornhofen: Pure Anderson Motives and Abelian tau-Sheaves
- Hartl, : On a Conjecture of Rapoport and Zink
- Hartl, Eva Viehmann (Univ. Bonn): Foliations in deformation spaces of local G-shtukas
- Halupczok, : A language for quantifier elimination in ordered abelian groups
- Halupczok, Raf Cluckers: Approximations and Lipschitz continuity in p-adic semi-algebraic and subanalytic geometry
- Halupczok, : Non-Archimedean Whitney-stratifications
- Halupczok, Cluckers, Raf: Quantifier elimination in ordered abelian groups
- Halupczok, Fornasiero, : Dimension in topological structures: topological closure and local property
- Garion, Tatiana Bandman: Surjectivity and equidistribution of the word x^ay^b on PSL(2,q) and SL(2,q)
- Garion, Yair Glasner: Highly Transitive Actions of Out(Fn)
- Garion, Matteo Penegini: New Beauville surfaces and finite simple groups
- Garion, Matteo Penegini: Beauville surfaces, moduli spaces and finite groups
- Halupczok, Raf Cluckers, Julia Gordon: Transfer principles for integrability and boundedness conditions for motivic exponential functions
- Raf Cluckers, Julia Gordon: Definability results for invariant distributions on a reductive unramified p-adic group
- Garion, : On Beauville Structures for PSL(2,q)
- Garion, Tatiana Bandman, Boris Kunyavskii: Equations in simple matrix groups: algebra, geometry, arithmetic, dynamics
- Garion, : Expansion of conjugacy classes in PSL(2,q)
- Garion, : Beauville surfaces and probabilistic group theory
- Tent, : On polygons, twin trees and CAT(1)-spaces
- Tent, Martin Ziegler: On computable functions on the reals
- Tent, Chris Parker: Completely reducible subcomplexes of spherical buildings
- Tent, Dugald Macpherson: Simplicity of some automorphism groups
- Tent, Dugald Macpherson: Pseudofinite groups with NIP theory
- Tent, Martin Ziegler: On the isometry group of the Urysohn space
- Tent, : The free pseudospace is n-ample, but not n+1-ample
- Schuermann, J.-P. Brasselet, S. Yokura: Motivic and derived motivic Hirzebruch classes
- Schuermann, L. Maxim: Cohomology representations of external and symmetric products of varieties
- Schuermann, Laurentiu Maxim, Morihiko Saito: Spectral Hirzebruch-Milnor classes of singular hypersurfaces