CRC 326 - GAUS

F-modules for rings with FFRT, and applications to local cohomology
2024-06-20 14:15 - 15:15 (Europe/Berlin) | Mainz, Hilbertraum (05-432)
Eamon Quinlan-Gallego (Utah)
Abstract: For a regular ring R of positive characteristic, an F-module over R is an R-module equipped with an “action of Frobenius”. The theory of these F-modules has been used successfully in establishing finiteness properties of local cohomology, and in this talk I will present an extension to the case where R is allowed to have mild singularities (known as finite F-representation type, or FFRT for short). I will then illustrate how this theory can be used to extend previous results on local cohomology to the FFRT case.

A local model for potentially Barsotti–Tate Galois representations
2024-06-21 13:30 - 15:00 | Heidelberg, Mathematikon, SR A and Livestream
Stefano Morra (Université Paris 8, St. Denis):
Potentially Barsotti–Tate deformations are an essential tool to achieve arithmetic results as the proof of the Shimura–Taniyama–Weyl conjecture, or the Breuil–M´ezard conjecture. Nevertheless, their geometry is still poorly understood, as such rings showed a rich range of complicated behavior, e.g. they may fail to be normal or regular (as shown in some examples and conjectures by Caruso–David–Mézard). In this talk we will discuss how moduli stack of Breuil–Kisin modules can be used to describe the geometry of moduli of tamely potentially Barsotti–Tate Galois representations (in rank 2 and over an unramified extension of Qp), using the theory of local models of loop groups in mixed characteristic. The main technical tool is an analysis of the p-torsion of a tangent complex to lift affine open charts for scheme–theoretic images between moduli of Breuil–Kisin modules and moduli of Galois representations. As a byproduct, we obtain an algorithmic procedure to compute explicit presentations for any tamely potentially Barsotti–Tate deformation ring for 2-dimensional Galois representations of unramified extensions of Qp.
This is joint work with B. Le Hung and A. Mézard.

Seminar on Arithmetic Geometry
2024-06-21 15:30 - 17:00 | Darmstadt, Room 401 and Zoom, Schlossgartenstraße 7, Darmstadt, 64289, Germany
Michael Rapoport (Universität Bonn): Integral models of Hecke operators
I will discuss the construction of integral models of Hecke correspondences and the extension of Hecke correspondences on automorphic vector bundles to integral models.
Zoom 635 7328 0984, Kenncode: kleinste sechsstellige Primzahl

Salem numbers and short geodesics
2024-06-25 16:00 - 17:00 | Zoom
International Seminar on Automorphic Forms
Lola Thompson (Utrecht University): Salem numbers and short geodesics
We will discuss how Mahler measure and related concepts (e.g., Salem numbers) are connected to problems about lengths of geodesics on arithmetic hyperbolic manifolds. As a result, by solving problems using tools from number theory, we are able to answer quantitative questions in spectral geometry. This talk will build towards two goals: showing that short geodesics on arithmetic hyperbolic surfaces are rare, and showing that, on average, geodesic lengths of non-compact arithmetic hyperbolic orbifolds appear with high multiplicity. This talk is based on joint work with Mikhail Belolipetsky, Matilde Lalín, and Plinio G. P. Murillo; and with Benjamin Linowitz, D. B. McReynolds, and Paul Pollack.
https://tu-darmstadt.zoom.us/j/68048280736

Symplectic singularities and vertex algebras
2024-06-25 17:00 - 18:00 | Darmstadt, Room S215 401
Tomoyuki Arakawa (RIMS): Symplectic singularities and vertex algebras
Symplectic singularities introduced by Beauville appear in various aspects of representation theory and are referred to as “21st-century Lie theory” by Okounkov. On theother hand, symplectic singularities also arise in the context of quantum field theory in physics, particularly in the Higgs and Coulomb branches of three-dimensional theories, as well as in the Higgs branches of four-dimensional theories.
Additionally, in vertex algebra theory, certain Poisson varieties called associated varieties are defined as geometric invariants, and they often turn out to be symplectic singularities. In such cases, vertex algebras can be regarded as chiral noncommutative deformations of symplectic singularities.
In particular, the 4D/2D duality proposed by Beem et al. in theoretical physics determines vertex algebras as invariants for superconformal four-dimensional theories. It is claimed that the Higgs branch of four-dimensional theories can be reconstructed as the associated variety of vertex algebras. Therefore, all vertex algebras arising from four-dimensional theories are supposed to be chiral noncommutative deformations of singular symplectic varieties.
In this lecture, we will discuss such vertex algebras and their representation theory.

Motivic cohomology of cyclic coverings
2024-06-27 14:15 - 15:15 | Mainz, Hilbertraum (05-432)
Tariq Syed (Mainz)
Abstract. The generalized Serre question asks whether algebraic vector bundles over topologically contractible smooth affine complex varieties are always trivial or not. Many examples of topologically contractible smooth affine complex varieties are given by so-called cyclic coverings. In this talk, we present new results on the motivic cohomology of such cyclic coverings which might indicate a positive answer to the generalized Serre question in low dimensions.

Counting congruences between modular forms
2024-06-28 13:30 - 15:00 | Heidelberg, Mathematikon, SR A and Livestream
Jaclyn Lang (Temple University, Philadelphia)
For a given eigenform, how many other eigenforms of the same weight and level are congruent to it modulo a prime l? We will explain how modular representation theory can be used to help answer this question. We focus especially on the case when the level is the square of a prime p such that p = -1 mod l. This is ongoing joint work with Robert Pollack and Preston Wake.

Spectral decomposition and Siegel-Veech transforms: The case of marked tori
2024-07-02 16:00 - 17:00 | Zoom
International Seminar on Automorphic Forms
Martin Möller (Goethe University Frankfurt)
https://tu-darmstadt.zoom.us/j/68048280736
Abstract: Generalizing the well-known construction of Eisenstein series on the modular curves, Siegel-Veech transforms provide a natural construction of square-integrable functions on strata of differentials on Riemannian surfaces. Even the case of marked tori, a homogeneous space but not for a reductive group provides features that we highlight in this talk with an eye on the general case.

Higher rank Teichmüller spaces, Higgs bundles and magical triples
2024-07-03 16:00 - 17:00 | Frankfurt, Robert-Mayer-Str. 10, Raum 711 groß
Oberseminar Algebra und Geometrie
André Oliveira (Universidade do Porto)
One way to realize Teichmüller space of a closed oriented hyperbolic surface Σ is as the space of discrete and faithful representations π_1(Σ) -> PSL(2,R). These constitute a connected component of the character variety of these representations. Higher rank Teichmüller space is a generalization for higher rank Lie groups. More precisely a higher rank Teichmüller space for a group G is a connected component of the G-character variety of representations π_1(Σ) -> G entirely consisting of discrete and faithful representations.

There are however two important differences comparing to the PSL(2,R)-case. Firstly, such spaces do not exist for any group G, even though their existence has long been established whenever G is a split form (the Hitchin components) or Hermitian (maximal components). Secondly, even if they exist for G, not every discrete and faithful representation in G lies in a higher rank Teichmüller space. It was conjectured by Guichard—Labourie—Wienhard that the representations comprising such spaces are positive representations — a special class of discrete and faithful representations — which only exist when G admits a certain structure called positivity.

Non-abelian Hodge theory yields corresponding connected components of the moduli space M(G) of G-Higgs bundles over an associated compact Riemann surface. We will introduce the notion of magical sl_2-triple and show that a group admits a positive structure if and only if it arises from such a magical triple. Then we we will use these magical triples and Higgs bundles to detect and parameterize components of M(G) which should be higher rank Teichmüller spaces (this was subsequently confirmed by Guichard—Labourie—Wienhard using our work). This is joint work with S. Bradlow, B. Collier, O. García-Prada and P. Gothen.

Die absolute Galoisgruppe eines p-adischen Körpers
2024-07-05 13:10 - 14:40 | Heidelberg, Mathematikon, SR A and Livestream
Moshe Jarden (Universität Tel Aviv)
Wir rechnen die Anzahl der Erzeugenden und die Anzahl der Relationen aus, die für die Darstellung der absoluten Galois Gruppen einer gegebenen endlichen Erweiterung von Q_p erforderlich ist.

Seminar on Arithmetic Geometry
2024-07-05 14:00 - 17:00 | Darmstadt, Room 401 and Zoom, Schlossgartenstraße 7, Darmstadt, 64289, Germany
Fabio Tanania (TU Darmstadt): Isotropic motivic fundamental groups
Zoom (635 7328 0984, Kenncode: kleinste sechsstellige Primzahl

Fourier-Jacobi periods on unitary groups
2024-07-09 17:00 - 18:00 | Zoom
International Seminar on Automorphic Forms
Hang Xue (Arizona University): Fourier-Jacobi periods on unitary groups
Abstract: We prove the Gan-Gross-Prasad conjecture for Fourier-Jacobi periods on unitary groups via relative trace formulae.
https://tu-darmstadt.zoom.us/j/68048280736

On DGAs with polynomial homology
2024-07-11 14:00 - 16:00 | Mainz, Hilbertraum (05-432)
Julius Frank (Bielefeld)
Abstract: It is surprisingly hard to find non-trivial examples of DGAs whose homology is polynomial in one generator. I will discuss one such example, which is a derived quotient of a discrete ring by an odd prime. I will also give one reason why it is hard to find examples: If the homology of a DGA A is polynomial over a perfect F_p-algebra and the underlying ring spectrum of A refines to an E_3-ring spectrum, then A must already be trivial in a certain sense.

On a characterisation of perfectoid fields by Iwasawa theory
2024-07-12 13:30 - 15:00 | Heidelberg, Mathematikon, SR A and Livestream
Gautier Ponsinet (IHES, Université Paris Saclay)
With a p-adic representation of the Galois group of a p-adic field are associated the Bloch-Kato groups defined via p-adic Hodge theory.
Iwasawa theory motivates the study of these Bloch-Kato groups over infinite algebraic extensions of the field of p-adic numbers.
Over perfectoid fields, several results (by Coates-Greenberg, Perrin-Riou, Berger, P. …) state that the Bloch-Kato groups admit a simple description.
In this talk, we will present a reciprocal statement: the structure of the Bloch-Kato groups associated with certain crystalline representations characterises the algebraic extensions of the field of p-adic numbers whose completion are perfectoid fields. In particular, we will recover, via a different method, results by Coates and Greenberg for abelian varieties, and by Bondarko for p-divisible groups.

Theta correspondence via C*-algebras
2024-07-16 16:00 - 17:00 | Zoom
International Seminar on Automorphic Forms
Haluk Sengun (University of Sheffield):Theta correspondence via C*-algebras
Abstract: The local theta correspondence sets up a bijection between certain subsets of admissible duals of suitable pairs of reductive groups. There are two special cases in which the correspondence is known to enjoy extra features, the ‘equal rank’ case where temperedness is preserved and the ‘stable range’ case where unitarity is preserved. In joint work with Bram Mesland (Leiden), we show that in these special cases, the local theta correspondence is actually given by a Morita equivalence of suitable
C*-algebras. I will try to expose this result and, time permitting, some applications.
https://tu-darmstadt.zoom.us/j/68048280736

Certain families of K3 surfaces and their modularity
2024-07-18 14:15 - 15:15 | Mainz, Hilbertraum (05-432)
Yui, Noriko (Queen’s University)
Abstract: We start with a double sextic family of K3 surfaces with four parameters with Picard number 16 defined over Q. Then by geometric reduction (top-to-bottom) processes, we obtain three, two and one parameter families of K3 surfaces of Picard number 17, 18, and 19 respectively. All these families turn out to be of hypergeometric type in the sense that their Picard–Fuch differential equations are given by hypergeometric or Heun functions. We will study the geometry of two parameter families in detail.
We will then prove, after suitable specializations of parameters, these K3 surfaces will have CM (complex multiplication), and will become modular, i.e., the Galois representations of dimensions less than or equal to 6 associated to the transcendental lattices are all induced from 1-dimensional representations. Thus, these K3 surfaces will be determined by modular forms of various weights. This is done starting with one-parameter family and then applying arithmetic induction (bottom-to-top) processes.
Our final goal is to determine explicit modular forms that determine the L-functions of these K3 families
at special fibers.
This is a joint work with A. Clingher, S. Kim and A. Malmendier.

Seminar on Arithmetic Geometry
2024-07-19 15:30 - 17:00 | Darmstadt, Room 401 and Zoom, Schlossgartenstraße 7, Darmstadt, 64289, Germany
Konstantin Jakob (TU Darmstadt): Counting absolutely indecomposable G-bundles
Zoom (635 7328 0984, Kenncode: kleinste sechsstellige Primzahl

Initial degenerations of homogeneous varieties
2024-09-17 16:00 - 18:00 | Frankfurt, Rober-Mayer-Str. 10, Raum 711 klein
Victoria Schleis (Durham University):
Abstract: We study the geometry of initial degenerations of homogeneous subvarieties of (multi-)projective space using polyhedral geometry. In 2017, Corey characterized initial degenerations of the Grassmannian through limits of strata in a matroid stratification of the Grassmannian corresponding to the weight vector of the initial degeneration. Since then, similar results have been proven for the flag variety and Spinor varieties. In my talk, I will discuss recent work in progress extending this study beyond specific examples to homogeneous subvarieties of (multi)-projective space. The talk is based on joint work in progress with George Balla, Dan Corey, and Igor Mahklin.

Seminar on Arithmetic Geometry
2024-10-18 15:30 - 17:00 | Darmstadt, Room 401 and Zoom, Schlossgartenstraße 7, Darmstadt, 64289, Germany
Joakim Faergeman (Yale University): Motivicity of rigid G-local systems on curves
Abstract: A natural problem in the study of local systems on complex varieties is to characterize those that arise in a family of varieties. We refer to such local systems as motivic. While a classification of motivic local systems is evidently out of reach, Simpson conjectured that for a reductive group G, rigid G-local systems with suitable finiteness conditions at infinity are motivic. This was proven for curves when G=GL_n by Katz who classified such rigid local systems. In this talk, we discuss our generalization of Katz’ theorem to a general reductive group. Our proof goes through the (tamely ramified) categorical geometric Langlands program in characteristic zero.
Zoom (635 7328 0984, Kenncode: kleinste sechsstellige Primzahl

On the cohomology of $SL(n,mathbb Z)$ beyond the "stable range"
2024-10-22 16:00 - 17:00 | Zoom
International Seminar on Automorphic Forms
Harald Grobner (University of Vienna): On the cohomology of $SL(n,mathbb Z)$ beyond the “stable range”
The cohomology of the group $SL(n,mathbb{Z}), n>1$, plays a fundamental role in geometry, topology and representation theory, while yielding many number theoretical applications: For instance, Borel used his description of $H^*(SL(n,mathbb Z))$ to compute the algebraic K-theory of the integers; whereas the (non-)vanishing of $H^*(SL(n,mathbb Z))$ tells a lot about the existence of certain automorphic forms. In this talk we will study the cohomology of $SL(n,mathbb Z)$, „right outside“ of what one calls the stable range. More precisely, we will show new non-vanishing results in degrees n−1 and n. As a byproduct, we will also answer a question, recently asked by F. Brown for n=6 and explain a phenomenon for n=8, which has been considered by A. Ash. (This is joint work with N. Grbac.)
https://tu-darmstadt.zoom.us/j/68048280736

Oberseminar Algebra und Geometrie
2024-10-23 16:00 - 17:00 | Frankfurt, RM-Str. 6-8, R. 308
Marius Leonhardt (Universität Frankfurt): Affine abelian non-abelian Chabauty
Abstract: The central question of this talk is how to find all integral points on affine hyperbolic curves. For example, which integers x,y satisfy y^2 = x^3 + x^2 + x + 1?
We approach this problem by introducing Kim’s non-abelian Chabauty method, which constructs p-adic analytic functions that have the integral points among their zeroes. In joint work with M. Lütdke and J.S. Müller, we showed that the abelian version of this method succeeds if the curve satisfies a certain inequality involving its genus and the Mordell-Weil rank of its Jacobian.
This talk is an introduction to the Chabauty–Kim method. I will present the above result and report on work in progress with M. Lüdtke that turns it into an algorithm determining the integral points on the curve.

On the algebraic K-theory of algebraic tori
2024-10-24 14:15 - 15:15 | Mainz, Hilbertraum (05-432)
Florian Riedel (Kopenhagen)
Abstract:
I will describe work in progress joint with Bai, Carmeli and Juran. A classical computation by Quillen expresses the algebraic K-theory spectrum of the ring of Laurent polynomials as the group ring of S^1 over the K-theory of the base field. We generalize this by showing that the algebraic K-theory spectrum of a not-necessarily split algebraic torus is given by the group ring of the delooping of the character lattice of the torus, thus showing that a version of Cartier duality between algebraic and topological tori holds on the level of K-theory. We do this by constructing a motivic Fourier transform which is of independent interest and also recover an explicit formula of Merkujev-Panin for K_0 in a straightforward way.

A moduli-theoretic approach to Sen theory
2024-10-25 13:30 - 15:00 | Heidelberg, Mathematikon, SR A and Livestream
Ben Heuer (Universität Frankfurt)
Classical Sen theory describes C_p-semilinear representations of Galois groups of p-adic fields in terms of linear algebra data called Sen modules. I will first explain how this theory can be reinterpreted geometrically in terms of v-vector bundles, creating a relation to the p-adic Simpson correspondence. I will then explain how one can use this to upgrade Sen’s Theorem to a comparison of moduli spaces of Galois representations and Sen modules: This explains known subtleties in classical Sen theory in a geometric fashion. Finally, I will describe how we can use this to study moduli spaces of (phi,Gamma)-modules. This is joint work-in-progress with Eugen Hellmann.

Exact formulae for ranks of partitions
2024-10-29 16:00 - 17:00 | Zoom
International Seminar on Automorphic Forms
Qihang Sun (University of Lille): Exact formulae for ranks of partitions
Dyson’s ranks provided a new understanding of the integer partition function, especially of its congruence properties. In 2009, Bringmann used the circle method to prove an asymptotic formula for the Fourier coefficients of rank generating functions. In this talk, we will prove that the asymptotic formula, when summing up to infinity, converges and becomes a Rademacher-type exact formula for the rank of partitions.

https://tu-darmstadt.zoom.us/j/68048280736

Moduli of twisted maps to smooth pairs
2024-10-29 16:15 - 17:15 | Heidelberg, MATHEMATIKON, SR 5, INF 205, Heidelberg, Germany
Robert Crumplin (Heidelberg)

On the K-theory of curves over number fields.
2024-10-31 14:15 - 15:15 | Mainz, Hilbertraum (05-432)
Rob de Jeu (Amsterdam)
Abstract: Borel defined regulators for the odd degree higher K-groups of a number field k and proved a relation between these and the values of the zeta-function of k at 2, 3, 4, …, generalising the classical relation between its residue at s=1 and the regulator of the unit group of the ring of integers. Similar results were proved and/or conjectured by Bloch and Beilinson for the K-groups of varieties over number fields. After a review of the background, we discuss some recent joint work with François Brunault, Liu Hang, and Fernando Rodriguez Villegas on K_2 of elliptic curves over certain cubic or quartic number fields, and, time permitting, how one can try to describe the K_4 of curves over number fields.

Seminar on Arithmetic Geometry
2024-11-01 15:30 - 17:00 | Darmstadt, Room 401 and Zoom, Schlossgartenstraße 7, Darmstadt, 64289, Germany
Pol van Hoften (VU Amsterdam): Igusa stacks and the cohomology of Shimura varieties
Associated to a modular form f is a two-dimensional Galois representation whose Frobenius eigenvalues can be expressed in terms of the Fourier coefficients of f, using a formula known as the Eichler–Shimura congruence relation. This relation was proved by Eichler–Shimura and Deligne by analyzing the mod p (bad) reduction of the modular curve of level ?0(p). In this talk, I will discuss joint work with Patrick Daniels, Dongryul Kim and Mingjia Zhang, where we give a new proof of this congruence relation that happens “entirely on the generic fibre”. More precisely, we prove a compatibility result between the cohomology of Shimura varieties of Hodge type and the Fargues?Scholze semisimple local Langlands correspondence, generalizing the Eichler–Shimura relation of Blasius–Rogawski. Our proof makes crucial use of the Igusa stacks that we construct, generalizing earlier work of Zhang in the PEL case.
Zoom (635 7328 0984, Kenncode: kleinste sechsstellige Primzahl

Counting characters on algebraic tori according to their Langlands L-functions
2024-11-05 16:00 - 17:00 | Zoom
International Seminar on Automorphic Forms
Ian Petrow (UCL): Counting characters on algebraic tori according to their Langlands L-functions
Given a connected reductive group G over a global field, Langlands introduced the automorphic L-function L(s, π, r) of a cuspidal automorphic representation π of G and a complex representation r of the L-group of G. While in general very little is known about Langlands L-functions, if G = T is a torus the properties of these L-functions can be obtained from class field theory and one can attempt to study analytic problems pertaining to them. In this talk I will describe some analytic results on automorphic characters of tori with respect to the analytic conductor of L(s, π, r), attempting to focus on the interplay of analytic and algebraic ideas that arise in the proofs.
https://tu-darmstadt.zoom.us/j/68048280736
The password is the first Fourier coefficient of the modular j-function (as digits)

From SL(2) to SO(2): Rank-2 Vafa-Witten invariants
2024-11-08 11:15 - 12:15 | Heidelberg, MATHEMATIKON, SR 11, INF 205, Heidelberg, Germany
Simon Schirren (Rom)

Rigid cocycles and geodesics on Shimura curves
2024-11-08 13:30 - 14:30 | Heidelberg, Mathematikon, SR A and Livestream
Håvard Damm-Johnsen (University of Oxford /MPIM Bonn)
Darmon and Vonk’s theory of rigid meromorphic cocycles gives a conjectural description of abelian extensions of real quadratic fields akin to the classical theory of complex multiplication. While their conjectures currently seem out of reach, the work of Darmon-Pozzi-Vonk uses p-adic families of Hilbert modular forms to give unconditional results in this direction. I will explain how a reinterpretation in terms of the so-called Kudla programme suggests an approach to extending their results. This point of view also gives a new proof of a theorem of Rickards on intersections of geodesics on Shimura curves, which is related to forthcoming work of Darmon-Vonk.
