Construction of Gaussian test functions
2025-06-03 16:00 - 17:00 (Europe/Berlin) | Zoom
International Seminar on Automorphic Forms
Andreas Mihatsch (Zhejiang University)
Construction of Gaussian test functions
The relative trace formula comparison of Jacquet–Rallis relates two trace formulas: one for general linear groups and one for unitary groups. In this context, one considers the transfer of test functions between the two sides. At the archimedean place, the Gaussian for the positive definite unitary group provides a distinguished test function that often comes up in arithmetic settings. Accordingly, it is of interest to understand its transfers to the general linear side. In my talk, I will explain a direct construction of such transfers which is based on Kudla–Millson theory. This is joint work with Siddarth Sankaran and Tonghai Yang.
https://tu-darmstadt.zoom.us/j/68048280736
The password is the first Fourier coefficient of the modular j-function (as digits)

The arithmetic of Fourier coefficients of Gan-Gurevich lifts on G2
2025-06-10 16:00 - 17:00 | Zoom
International Seminar on Automorphic Forms
Naomi Sweeting (Princeton University)
The arithmetic of Fourier coefficients of Gan-Gurevich lifts on G2
The arithmetic of Fourier coefficients of Gan-Gurevich lifts on G2 Abstract: Modular forms on exceptional groups carry a surprisingly rich arithmetic structure. For instance, modular forms on G2 have a theory of Fourier expansions, in which the coefficients are indexed by cubic rings (e.g. rings of integers in cubic field extensions of Q). This talk is about the Gan-Gurevich lifts, which are modular forms on G2 arising by Langlands functoriality from classical modular forms on PGL2. Gross conjectured in 2000 that the norm squared of the Fourier coefficients of a Gan-Gurevich lift encode the cubic-twisted L values of the corresponding classical cusp form (echoing Waldspurger’s work on Fourier coefficients of half-integral weight modular forms). We prove this conjecture for a large class of Gan-Gurevich lifts coming from CM forms, thus giving the first complete examples of Gross’s conjecture. Based on joint work in progress with Petar Bakic, Alex Horawa, and Siyan Daniel Li-Huerta.
https://tu-darmstadt.zoom.us/j/68048280736
The password is the first Fourier coefficient of the modular j-function (as digits)

Critical values of Hecke L-funktions
2025-06-12 14:15 - 15:15 | Mainz, Hilbertraum (05-432)
Han-Ung Kufner (Universität Regensburg)
A conjecture of Deligne from 1977 relates the critical values of a motivic
L-function with certain periods of the motive. The goal of this talk is to
explain Deligne’s conjecture and to discuss a proof in the case of Hecke
L-functions. This generalizes a result of Blasius for Hecke characters of CMfields.
For the proof we use the recently constructed Eisenstein-Kronecker
classes of Kings-Sprang and combine them with ideas from Blasius’ proof.
1

Quantitative level lowering for modular forms
2025-06-13 13:30 - 14:30 | Heidelberg, Mathematikon, SR A and Livestream
Quantitative level lowering for modular forms
Mohamed Moakher (University of Pittsburgh)
Given a Hilbert modular form f of weight two over a totally real field F, we can associate to it a finite module Phi(f) known as the congruence module for f, which measures the congruences that f satisfies with other forms. When f is transferred to a quaternionic modular form f_D over a quaternion algebra D via the Jacquet-Langlands correspondence, we can similarly define a congruence module Phi(f_D) for f_D. Pollack and Weston proposed a quantitative relationship between the sizes of Phi(f) and Phi(f_D), expressed in terms of invariants associated to f and D. In this talk, I will outline the ideas underlying the proof of this relationship. The approach combines a method of Ribet and Takahashi with new techniques introduced by Böckle, Khare, and Manning.

Seminar on Arithmetic Geometry
2025-06-13 15:30 - 17:00 | Darmstadt, Room 401 and Zoom, Schlossgartenstraße 7, Darmstadt, 64289, Germany
Emanuel Reinecke (IHES): Poincare duality for proper morphisms in rigid geometry
While the Z/p-etale cohomology of rigid-analytic varieties is in general hard to control, it becomes more tractable when the varieties are proper. In my talk, I will explain a relative Poincare duality statement for etale cohomology with finite coefficients which applies to any proper morphism of rigid-analytic varieties over nonarchimedean fields of mixed characteristic, confirming an expectation of Bhatt-Hansen. A key ingredient in the proof will be a construction of trace maps for proper morphisms. Joint work with Shizhang Li and Bogdan Zavyalov.
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Seminar on Arithmetic Geometry
2025-06-20 15:30 - 17:00 | Darmstadt, Room 401 and Zoom, Schlossgartenstraße 7, Darmstadt, 64289, Germany
Thiago Landim (IMJ): Weights and motives on stacks
The existence of a motivic t-structure is an old problem in the center of many conjectures related with algebraic cycles. Inspired by Deligne, Bondarko defined a dual notion, now called weight structure, and proved Beilinson motives (and later integral cdh-motives) on nice schemes admit weight structures. In this talks, we are going to prove the category of K-motives (modules of genuine K-theory inside motivic spectra) on tame quotient stacks, as defined by Hoyois, admits a well-behaved category of geometric motives and prove the existence of bounded weight structure on them. If time allows, we are going to explain how this behaves better for Kan extended theories, e.g. cdh-motives, and how étale sheaves behaves even better.
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Bogomolov property for Galois representations with big local image
2025-06-27 13:30 - 14:30 | Heidelberg, Mathematikon, SR A and Livestream
Andrea Conti (Heidelberg)
An algebraic extension of the rational numbers is said to have the Bogomolov property if the absolute logarithmic Weil height of its non-torsion elements is uniformly bounded from below. Given a continuous representation $rho$ of the absolute Galois group $G_{mathbb Q}$ of $mathbb Q$, one can ask whether the field fixed by $mathrm{ker}(rho)$ has the Bogomolov property (in short, we say that $rho$ has (B)). In a joint work with Lea Terracini, we prove that, if $rhocolon G_{mathbb Q}tomathrm{GL}_N(mathbb Z_p)$ maps an inertia subgroup at $p$ surjectively onto an open subgroup of $mathrm{GL}_N(mathbb Z_p)$, then $rho$ has (B). More generally, we show that if the image of a decomposition group at $p$ is open in the image of $G_Q$, plus a certain condition on the center of the image is satisfied, then $rho$ has (B). In particular, no assumption on the modularity of $rho$ is needed, contrary to previous work of Habegger and Amoroso—Terracini.

Seminar on Arithmetic Geometry
2025-06-27 15:30 - 17:00 | Darmstadt, Room 401 and Zoom, Schlossgartenstraße 7, Darmstadt, 64289, Germany
Tianyi Feng (University Bonn): Metaplectic Satake with Ring Coefficients
In this talk we explain the statement and proof strategy of the geometric Satake equivalence for topological (aka metaplectic) coverings of reductive groups in DVR coefficients. This is joint work in progress with Yifei Zhao.
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Equivariant aspects of Hochschild homology
2025-07-03 09:00 - 10:00 | Mainz, Poissonraum (04-220), Staudingerweg 9, Rheinland-Pfalz - Mainz, 55128, Germany
Zhouhang Mao (University of Amsterdam)
Abstract: Many localizing invariants, after being applied to schemes, are equipped with a motivic filtration whose associated graded pieces are given by cohomology theories of schemes. In this talk, we give an equivariant aspects of two localizing invariants proposed by Kaledin, which correspond to non-Hodge-completed derived de Rham cohomology and de Rham–Witt complex respectively. Our description adapts to prismatic cohomology as well. If time permits, we also give an unexpected application of these considerations to prismatic logarithm.

Algebraic K-theory and the universal localising invariant
2025-07-04 13:30 - 14:30 | Heidelberg, Mathematikon, SR A and Livestream
Algebraic K-theory and the universal localising invariant
Christoph Winges (Universität Regensburg)
Essentially by construction, the abelian group K_0 is the target of the universal rank function for various types of objects, including finitely generated projective modules and perfect chain complexes. Over the last couple of decades, it has become possible to formulate and prove a similar universal property for higher algebraic K-theory in the sense of Quillen and Waldhausen. A closer inspection of various localisation phenomena in algebraic K-theory leads to the notion of a localising invariant, among which algebraic K-theory enjoys a similar universal property due to work of Blumberg, Gepner and Tabuada. I will survey these results and, as time allows, discuss an alternative perspective on parts of this story that I obtained in recent joint work with Ramzi and Sosnilo.

Seminar on Arithmetic Geometry
2025-07-04 15:30 - 17:00 | Darmstadt, Room 401 and Zoom, Schlossgartenstraße 7, Darmstadt, 64289, Germany
Matteo Tamiozzo (Université Sorbonne Paris Nord): Towards semi-global plectic conjectures
Known results on the Birch and Swinnerton-Dyer conjecture for elliptic curves of analytic rank at most one over totally real fields rely on CM points on Shimura curves. After recalling this, I will illustrate how an attempt to go beyond rank one leads to the plectic conjectures of Nekovář-Scholl for higher-dimensional quaternionic Shimura varieties. Finally, I will present joint work in progress with Tony Feng and Mingjia Zhang aimed at proving a “semi-global” version of these conjectures.
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The kernel of the adjoint exponential in Anderson $t$-modules
2025-07-11 13:30 - 14:30 | Heidelberg, Mathematikon, SR A and Livestream
The kernel of the adjoint exponential in Anderson $t$-modules
Giacomo H. Ferraro (Universität Heidelberg)

Given an algebraically closed complete valued field $K$ over $mathbb{F}_q$, an Anderson $t$-module of dimension $d$ is given by the topological $mathbb{F}_q$-vector space $K^d$, endowed with an $mathbb{F}_q$-linear action $phi_t=sum_{igeq0}T_itau^iin M_{dtimes d}(K)[tau]$, where $tau:K^dto K^d$ sends $(v_1,dots,v_d)$ to $(v_1^q,dots,v_d^q)$.
In analogy with complex abelian varieties, there is an analytic map $exp=sum_{igeq0}E_itau^i: K^dto K^d$—which is not necessarily surjective—such that $phi_texp=exp T_0$.
The adjoint exponential, defined as the series $exp^*:=sum_{igeq0}tau^{-i}E_i^T$, determines a (non-analytic) continuous map $K^dto K^d$. Using the factorization properties of $K[![x]!]$, Poonen proved that there is a perfect duality of topological $mathbb{F}_q$-vector spaces $ker(exp)timesker(exp^*)tomathbb{F}_q$ under the condition $d=1$.
In this talk, I explain that for an arbitrary textit{abelian} Anderson $t$-module, we have a collection of perfect pairings $ker(phi_{t^n})timesker(phi^*_{t^n})tomathbb{F}_q$, and that we can use them to obtain a canonical generating series $(F_phi)_cin M_{dtimes d}(K)[![tau^{-1},tau]!]$ for all $cinmathbb{F}_q(!(t^{-1})!)/mathbb{F}_q(t)$. The study of the properties of $F_phi$ allows us to prove that, if $exp$ is surjective, $ker(exp^*)$ is compact and isomorphic to the Pontryagin dual of $ker(exp)$. Moreover, we deduce an alternative explicit description of the Hartl–Juschka pairing, obtained by Gazda and Maurischat in a recent preprint.

Der syntomische Logarithmus
2025-07-18 13:30 - 14:30 | Heidelberg, Mathematikon, SR A and Livestream
Der syntomische Logarithmus
Matthias Flach (Caltech/USA)
In Gemeinschaftsarbeit mit A. Krause und B. Morin geben wir mit Hilfe von prismatischer Kohomologie eine neue Konstruktion der Bloch-Kato Logarithmusabbildung. Als Anwendung beweisen wir die Vermutung C_{EP} von Fontaine und Perrin-Riou für Tate-Motive über beliebigen lokalen Körpern der Charakteristik null.

Seminar on Arithmetic Geometry
2025-07-18 15:30 - 17:00 | Darmstadt, Room 401 and Zoom, Schlossgartenstraße 7, Darmstadt, 64289, Germany
Abhishek Oswal (University Freiburg): p-adic hyperbolicity of the moduli space of abelian varieties
By a theorem of Borel, any holomorphic map from a complex
algebraic variety to the moduli space of abelian varieties (and more
generally to an arithmetic variety) is in fact algebraic. A key input
is to prove that any holomorphic map from a product of punctured disks
to such an arithmetic variety does not have any essential
singularities. In this talk, I’ll discuss a p-adic analogue of these
results. This is joint work with Ananth Shankar and Xinwen Zhu (with an
appendix by Anand Patel).
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tba
2025-09-04 16:00 - 17:00 | Zoom
International Seminar on Automorphic Forms
Ziqi Guo ( Peking University) tba

https://tu-darmstadt.zoom.us/j/68048280736
The password is the first Fourier coefficient of the modular j-function (as digits)

Solid Locally Analytic Representations in Mixed Characteristic
2025-10-17 13:30 - 15:00 | Heidelberg, Mathematikon, SR A and Livestream
Dr. Gal Porat (Einstein Institute of Mathematics, Hebrew University of Jerusalem)
Abstract:
Locally analytic representations of p-adic Lie groups with Q_p coefficients are powerful tools in p-adic Hodge theory and the p-adic Langlands program. This perspective reveals important differential structures, such as the Sen and Casimir operators.
A few years ago, Rodrigues Jacinto and Rodriguez Camargo developed a “solid” version of this theory using the language of condensed mathematics, which provides more robust homological tools (comparison theorems, spectral sequences…) for studying these representations.
This talk will present ongoing work that extends this solid theory to a much broader class of mixed characteristic coefficients, such as F_p((X)) or Z_p[[X]]<p/x>, as well as semilinear representations. I will conclude by exploring how these ideas relate to mixed characteristic phenomena in p-adic Hodge theory and the Langlands program.

Some cases of cube sum problem
2025-10-24 13:30 - 15:00 | Heidelberg, Mathematikon, SR A and Livestream
Dipramit Majumdar (IIT Madras / Universität Heidelberg)
Abstract: An integer n is said to be a rational cube sum or simply a cube sum if n can be written as a sum of cubes of two rational numbers. For example, 6 = (17/21)^3+ (37/21)^3. A cube-free integer n > 2 is a cube sum if and only if the elliptic curve y^2=x^3-432n^2 has infinitely many solutions over the rational numbers. A recent result of Alpöge-Bhargava-Shnidman-Burungale-Skinner shows that a positive proportion of integers are cube sums and a positive proportion of integers are not.
We will discuss the cube sum problem for some special family of integers. This talk is based on joint works with De, Jha, Mondal, Shingavekar and Sury.

Seminar on Arithmetic Geometry
2025-10-24 15:30 - 17:00 | Darmstadt, Room 401 and Zoom, Schlossgartenstraße 7, Darmstadt, 64289, Germany
Michael Rapoport (University Bonn): Toric schemes and integral models of Shimura varieties
The modular curve with Gamma0(p)$-level structure has a beautiful integral model over BZp, and generalizations of this model to any Shimura variety with p-parahoric level have been constructed in the last 30 years. When passing from Gamma0(p)-level to Gamma1(p)-level, the situation changes drastically. In the talk I will explain a general method that potentially allows to deal with such level structures. The method is based on the construction of torus embeddings of the maximal torus of a reductive group scheme. Joint work with G. Pappas.
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Seminar on Arithmetic Geometry
2025-11-07 15:30 - 17:00 | Darmstadt, Room 401 and Zoom, Schlossgartenstraße 7, Darmstadt, 64289, Germany
Fei Ren (Bergische Universität Wuppertal): Coherent Six-Functor Formalisms: Pro vs Solid
In the classical theory for coherent sheaves, the only missing piece in the Grothendieck
six-functor formalism picture is j! for an open immersion j. Towards fixing this gap, Deligne
provided a construction of j! by extending the sheaf class to pro sheaves, while Clausen-
Scholze provided another solution by extending the sheaf class to solid modules.
In this talk, I will explain how Deligne’s construction coincides with the Clausen-Scholze construction via a natural functor, whose restriction to the full subcategory of Mittag-Leffler pro-systems is fully faithful.
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The Gross-Zagier formula on singular moduli for Shimura curves
2025-11-18 15:00 - 16:00 | Zoom
International Seminar on Automorphic Forms
Andrew Phillips (College of Idaho): The Gross-Zagier formula on singular moduli for Shimura curves
The Gross-Zagier formula on singular moduli, which gives a formula for the prime factorization of differences of j-values, can be seen as a calculation of the intersection multiplicity of two CM divisors on the integral model of a modular curve. We will discuss a generalization of this result to a Shimura curve.
https://tu-darmstadt.zoom.us/j/68048280736
The password is the first Fourier coefficient of the modular j-function (as digits)

Seminar on Arithmetic Geometry
2025-11-21 15:30 - 17:00 | Darmstadt, Room 401 and Zoom, Schlossgartenstraße 7, Darmstadt, 64289, Germany
Ioannis Zachos (University Münster): On integral models for some ramified unitary Shimura varieties
Local models of Shimura varieties are projective flat schemes over the spectrum of a discrete valuation ring. The importance of local models lies in the fact that under some assumptions they model the singularities that arise in the reduction modulo p of Shimura varieties. In this talk, we will first introduce the notion of local models and certain variants of them for some ramified unitary Shimura varieties. Building on this, we will resolve the singularities of these models, leading to regular integral models for the corresponding Shimura varieties. In the end we will discuss the Bruhat-Tits stratification of the associated Rapoport-Zink spaces.
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Refined equidistribution of Hecke points and cryptography
2025-11-25 15:00 - 16:00 | Zoom
International Seminar on Automorphic Forms
Radu Toma ( Institut de mathémathiques de Jussieu): Refined equidistribution of Hecke points and cryptography
A classic theorem states that, fixing a Euclidean lattice L, its sublattices of large index equidistribute in the space of lattices. The literature leaves open the question: how does the rate of equidistribution depend on L? In joint work with de Boer, Page, and Wesolowski, we answer this using automorphic theory and geometry of numbers. Motivated by lattice-based cryptography, we apply the result to show that a computational problem called SIVP is as hard for Haar random module lattices as it is in the worst case.
https://tu-darmstadt.zoom.us/j/68048280736
The password is the first Fourier coefficient of the modular j-function (as digits)

Seminar on Arithmetic Geometry
2025-11-28 15:30 - 17:00 | Darmstadt, Room 401 and Zoom, Schlossgartenstraße 7, Darmstadt, 64289, Germany
Juan Esteban Rodríguez Camargo (MPI Bonn): Cartier duality for gerbes of vector bundles
Using the theory of abstract six functor formalisms of
analytic stacks, I will explain a general Cartier duality for gerbes
of vector bundles in different algebraic and analytic setups. As an
application, admitting some foundational aspects of the theory of the
analytic Hodge-Tate stack (joint with Anschütz, Le Bras and Scholze),
one can deduce a Cartier duality between the categories of
quasi-coherent sheaves of the analytic
Hodge-Tate stack and weight one modules on the Simpson gerbe of Bhatt-Zhang.
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From Asai to Triple Product: Euler Systems and p-adic L-functions
2025-12-02 16:00 - 17:00 | Zoom
International Seminar on Automorphic Forms
Giada Grossi (Paris 13) : From Asai to Triple Product: Euler Systems and p-adic L-functions
https://tu-darmstadt.zoom.us/j/68048280736
The password is the first Fourier coefficient of the modular j-function (as digits)

Oberseminar Algebra und Geometrie
2025-12-03 16:00 - 17:00 | Frankfurt, Robert-Mayer-Str. 6-8, Raum 309
Kay Rülling (Universität Wuppertal): Tame cohomology of the structure sheaf in mixed characteristic
Abstract:
This is joint work in progress with Alberto Merici and Shuji Saito.
Let R be a complete discrete valuation ring of mixed characteristic with fraction field K.
We show that the tame cohomology of Hübner-Schmidt on smooth K-schemes relative to R of (a twist of) the structure sheaf is A^1-invariant and is a finite R-module up to bounded torsion.
This induces a canonical R-lattice in the cohomology of the structure sheaf of smooth proper K-schemes.
If X has a regular model over R and resolutions of singularities in mixed characteristic hold, then this lattice would be the cohomology of this regular model. The interesting point is that we get the existence of such a lattice also in case X has no regular model and without using resolutions.
To this end we use classical results by Bartenwerfer and van der Put in rigid geometry.

On the distribution of supersingular primes of abelian varieties and K3 surfaces
2025-12-05 13:30 - 15:00 | Heidelberg, MATHEMATIKON, SR A, Deutschland
Prof. Dr. Chun Yin Hui (The University of Hongkong)
Abstract: Let X be an abelian variety or a K3 surface defined over a number field K. We prove that the density of the supersingular primes of X/K is zero if X is non-CM. By applying an effective Chebotarev density theorem of Serre, we obtain asymptotic upper bounds of the counting function for these supersingular primes.

Seminar on Arithmetic Geometry
2025-12-05 15:30 - 17:00 | Darmstadt, Room 401 and Zoom, Schlossgartenstraße 7, Darmstadt, 64289, Germany
Felix Röhrle (Universität Tübingen): Quadratically enriched plane curve counting via tropical geometry
Consider the classical problem in enumerative geometry of counting rational plane curves through a fixed configuration of points. The problem may be considered over any base field and the point conditions might be scheme theoretic points. Recently, Kass–Levine–Solomon–Wickelgren have used techniques from $mathbb{A}^1$-homotopy theory to define an enumerative invariant for this problem which is defined over a large class of possible base fields. This new theory generalizes Gromov-Witten invariants (base field = complex numbers) and Welschinger invariants (base field = real numbers) simultaneously. In this talk I will present a tropical correspondence theorem, which allows to effectively compute these new invariants. This is joint work with Andrès Jaramillo-Puentes, Hannah Markwig, and Sabrina Pauli.
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Kudla's conjecture in cohomology for unitary Shimura varieties
2025-12-09 15:00 - 16:00 | Zoom
International Seminar on Automorphic Forms
François Greer (Michigan State University): Kudla’s conjecture in cohomology for unitary Shimura varieties
The generating series for special cycles in a unitary Shimura variety $X$ associated to a Hermitian lattice of signature $(1,n)$ is a modular form. Such a Shimura variety has a unique toroidal compactification, and one can consider the closures of the special cycles there. We prove that for codimension up to the middle, the generating series for these closures is quasi-modular, and explain how to make boundary corrections to restore modularity, answering a question of Kudla. This is based on joint work with Salim Ta
https://tu-darmstadt.zoom.us/j/68048280736
The password is the first Fourier coefficient of the modular j-function (as digits)

Oberseminar Algebra und Geometrie
2025-12-11 15:30 - 16:30 | Frankfurt, RM-Str. 6-8, R. 308
Annalisa Grossi (Università di Bologna): In search of maximal branes on Hyper-Kähler manifolds
Abstract: Given a holomorphic or anti-holomorphic involution on a complex variety, the Smith-Thom inequality says that the total mathbb{F}_2-Betti number of the fixed locus is no greater than the total mathbb{F}_2-Betti number of the ambient variety. The involution is called maximal when the equality is achieved. On a Hyper-Kähler manifold X a holomorphic or a anti-holomorphic involution is referred to as a brane involution. While examples of non-compact hyper-Kähler manifolds admitting maximal branes are known, the compact case is more intriguing. In particular, although there exist some K3 surfaces admitting maximal brane involutions, the main result that I will show you is the non-existence of maximal branes on Hyper-Kähler manifolds deformation equivalent to the Hilbert scheme of points on a K3 surface. This talk is based on a joint work and on a joint work in progress with S. Billi, L. Fu and V. Kharlamov.

Algebraicity of critical Hecke L-values
2025-12-12 13:30 - 15:00 | Heidelberg, Mathematikon, SR A and Livestream
Prof. Dr. Johannes Sprang (Universität Duisburg-Essen)
Euler’s beautiful formula for the values of the Riemann zeta function at the positive even integers can be seen as the starting point of the investigation of special values of L-functions.
In particular, Euler’s result shows that all critical zeta values are rational up to multiplication with a particular period, here the period is a power of 2πi. Conjecturally this is expected to hold for all critical L-values of motives. In this talk, I will explain a joint result with Guido Kings on the algebraicity of critical Hecke L-values up to explicit periods for totally imaginary fields. As an application, I will discuss the construction of p-adic L-functions for such fields.
