## 2024

### Getaltheorie in het Vlakkeland – Arithmétique en Plat Pays

5 February, Leuven. See web site for details.### Intercity Number Theory Seminar

22 March, UvA and VU Amsterdam. Room 11A33 in the main building of the VU.- 11:00–12:00
- Duco van Straten Johannes Gutenberg-Universität Mainz, Siegel paramodular forms and pencils of Calabi-Yau threefolds
The search for the automorphic origin of Galois-representations arising from geometry is central to the Langlands program and arithmetic algebraic geometry. The understanding of the relation between elliptic curves and classical modular forms arose from the systematic study of examples and culminated in the modularity theorem of Wiles and others. Calabi-Yau 3-folds form a class of varieties with rich and intriguing properties that come next to elliptic curves and K3 surfaces. In the talk I will report on joint work with V. Golyshev and describe some examples of Calabi-Yau 3-folds Y for which the Galois representation
*H*^{3}(*Y*) is related to weight 3 Siegel paramodular modular forms. - 13:00–14:00
- Rob de Jeu Vrije Universiteit Amsterdam,
*K*_{2}of elliptic curves over non-Abelian cubic and quartic fieldsAfter a review of some earlier results on (mostly)*K*_{2}of curves, we give constructions of families of elliptic curves over certain cubic or quartic fields with three, respectively four, ‘integral’ elements in the kernel of the tame symbol on the curves. The fields are in general non-Abelian, and the elements linearly independent. For their integrality, we discuss a new criterion that does not ignore any torsion. We also verify Beilinson’s conjecture numerically for some of the curves. This is joint work with François Brunault, Liu Hang, and Fernando Rodriguez Villegas. - 14:15–15:15
- Pol van Hoften Vrije Universiteit Amsterdam, A new proof of the Eichler--Shimura congruence relation
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 Γ_{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". - 15:45–16:45
- Tian Wang MPIM Bonn, On the distribution of supersingular primes for abelian surfaces
In 1976, Lang and Trotter made a conjecture that predicts the number of primes
*p*up to*x*, for which the reduction of a non-CM elliptic curve*E*/mathbb*Q*at*p*is supersingular. Though the conjecture is still open, we now have unconditional upper and lower bounds thanks to the work of several mathematicians in the past few decades. However, less has been studied for the distribution of supersingular primes for abelian surfaces (even conjecturally). In this talk, I will present my recent work on unconditional upper bounds for the number of primes*p*up to*x*, for which the reduction of a fixed abelian surface at*p*is supersingular.

### Intercity Number Theory Seminar

19 April, Leiden. Snellius building, room 312- 11:00–12:00
- Mike Daas Leiden University, A p-adic analogue of a formula by Gross and Zagier
In their 1984 paper “On singular moduli”, Gross and Zagier proved an explicit factorisation formula for the norm of the difference between two CM-values of the classical
*j*-function. In 2022, it was conjectured by Giampietro and Darmon that the CM-values of certain*p*-adic theta-functions on Shimura curves should obey similar factorisation patterns. In this talk, we explore the classical result about the*j*-function, discuss its proofs and outline how the study of infinitesimal deformations of*p*-adic Hilbert Eisenstein series was used to settle the conjectures about the theta-function. This*p*-adic analytic approach bears resemblance to some of the newly developed methods in modern RM-theory. - 13:15–14:15
- Jesse Vogel Leiden University, An arithmetic-geometric correspondence for character stacks
Algebra and geometry over the complex numbers is strongly related to algebra and geometry over finite fields. This is reflected for instance in the Lefschetz principle, or in the relation between mixed Hodge structures and ℓ-adic cohomology. In this talk, based on arXiv:2309.15331, which is joint work with M. Hablicsek and Á. Gonzalez-Prieto, we describe how this correspondence between arithmetic and complex geometry can be used to study the representation varieties and character stacks of closed orientable surfaces, and how it unifies a number of known methods for studying these objects.
- 14:45–15:45
- Margherita Pagano Leiden University, The wild Brauer-Manin obstruction on K3 surfaces
A way to study rational points on a variety is by looking at their image in the
*p*-adic points. Some natural questions that arise are the following: are the rational points dense inside the*p*-adic points? If not, can we have control on the set of primes that cause the failure of the density of the rational points inside the product of the*p*-adic points? I will explain how primes of good reduction can play a role in the Brauer-Manin obstruction to weak approximation, with particular emphasis on the case of K3 surfaces. I will then explain how the reduction type (in particular, ordinary or non-ordinary good reduction) plays a role. - 16:00–17:00
- Emanuel Reinecke IAS, Relative Poincare duality in rigid geometry
While the etale cohomology of
*Z*/*p*-local systems on smooth*p*-adic rigid spaces is in general hard to control, it becomes more tractable when the spaces are proper. For example, in the proper case it is finite-dimensional and has recently been shown in work of Zavyalov and of Mann to satisfy Poincare duality. In my talk, I will explain a new proof of Poincare duality in this context, which also works for more general spaces and coefficients and in the relative setting. It relies on a new construction of trace maps for smooth morphisms of rigid spaces. Joint work in progress with Shizhang Li and Bogdan Zavyalov.

### Intercity Number Theory Seminar

10 May, Groningen. Bernoulliborg, room 105- 11:45–12:45
- Diane Maclagan Warwick, Tropical Vector Bundles
In this talk I will describe a new definition, joint with Bivas Khan, for a tropical vector bundle on a subvariety of a tropical toric variety. This builds on the tropicalizations of toric vector bundles. I will discuss when these bundles do and do not behave as in the classical setting.
- 14:00–15:00
- Matt Baker Georgia Tech, Band schemes, moduli spaces of matroids, and a theorem of Lafforgue
We introduce a generalization of commutative rings called bands, along with the corresponding geometric theory of band schemes. Among other things, band schemes provide a new viewpoint on tropical geometry and Berkovich analytifications, and they enable a partial explanation for phenomena observed by Jacques Tits concerning algebraic groups over the "field of one element". Band schemes also furnish a natural algebro-geometric setting for studying matroid theory; in particular, they allow us to construct a moduli space of matroids, and they provide new tools for studying realization spaces of matroids. In the latter vein, we will discuss an application to a generalization of Laurent Lafforgue's theorem on realization spaces of rigid matroids. This is joint work with Oliver Lorscheid and Tong Jin.
- 15:10–16:10
- Manoel Jarra Groningen, Matroids and Euler characteristics of quiver Grassmannians
We introduce morphisms of matroids with coefficients, which leads to a categorical framework for Baker-Bowler theory. Inspired by the idea that matroids are linear subspaces of F1-vector spaces, we construct quiver Grassmannians of matroids for quiver representations over F1. It turns out that in "nice" cases, the cardinality of F1-rational points (in a suitable sense) of a matroid quiver Grassmannian and the Euler characteristic of its associated complex variety are the same, which establishes an intimate link between cluster algebras and matroid theory. This is a joint work with Oliver Lorscheid and Eduardo Vital.
- 16:40–17:40
- David Holmes Leiden, Modular forms for universal abelian varieties
Modular forms can be seen as sections of line bundles on moduli spaces of elliptic curves. These generalise naturally to Siegel-Jacobi forms, which are sections of line bundles on the universal abelian variety over a moduli space of abelian varieties. Finite generation (or otherwise) of rings of these forms has various applications, such as to understanding projective embeddings of these moduli spaces. In this talk we will spend quite some time setting up the background, after which we will explain how techniques from tropical geometry can help determine the (non)finite generation of various rings of Siegel-Jacobi forms. This is joint work with José Burgos, Ana Maria Botero, and Robin de Jong.

### Intercity Number Theory Seminar

24 May, Utrecht. Room 2.02 in the Minnaert building.- 13:00–14:00
- Cécile Dartyge University of Lorraine, Reversible almost primes
The reverse of an n-bit integer
*a*= ∑_{j=0}^{n-1}*e*_{j}(*a*) 2^{j}, with*e*_{j}(*a*) ∈{0,1} for*j*= 0,...,*n*-1, is defined by*a*_{rev}= ∑_{j=0}^{n-1}*e*_{j}(*a*) 2^{n-1-j}. It is natural to expect that there exist infinitely reversible primes that is primes*p*such that the reverse*p*_{rev}is also a prime number. This question is still open. In this direction we have proved that there are infinitely many integers a with at most 8 prime factors such that*a*_{rev}has also 8 prime factors. The proof uses sieve methods and exponential sums mixing integers and their reverse. This is joint work with Bruno Martin, Joël Rivat, Igor E. Shparlinski and Cathy Swaenepoel. - 14:15–15:15
- Steve Fan MPIM Bonn, Counting shifted-prime divisors
A shifted prime is a number of the form
*p*-1, where*p*is prime. For every positive integer*n*, let ω*(*n*) denote the number of positive divisors of*n*that are shifted primes. First studied by K. Prachar in an influential paper from 70 years ago, the function ω* shares some interesting features with the divisor function tau and the prime-divisor function ω, but its distribution still remains a mystery. The study of this function was recently picked up by M. R. Murty and V. K. Murty who proved lower and upper bounds for the second moment of ω*, and shortly after their work, Y. Ding obtained a refinement of their lower bound. Jointly with V. Z. Guo and Y. Zhang, he also provided a possible asymptotic formula. In this talk, we continue this trend of study by investigating the higher moments of ω* and other related problems. This is based on joint work with Carl Pomerance. - 15:45–16:45
- Simon Rydin Myerson University of Warwick, Forms with real coefficients and differing degrees
We explore approaches to systems of forms with differing degrees which use the ‘repulsion’ technique. This allows for Diophantine inequalities with real coefficients to be studied in the general style of Browning and Heath-Brown (2017).
- 17:00–18:00
- Gunther Cornelissen Utrecht University, Asymptotic Mahler measure of some cyclotomic integers
We construct a sequence of cyclotomic integers (Gaussian periods) and study the asymptotics of their Mahler measure/height as a function of their conductor. The growth rate is the multidimensional Mahler measure of a family of explicit Calabi-Yau varieties of increasing dimension. In turn, their asymptotics (for prime dimension) relates to the problem of random flights. We compare some of the values obtained to the minimal Mahler measure of non-cyclotomic algebraic integers with cyclic Galois group of given order. The proofs use ideas from uniform distribution, discrepancy theory, and combinatorics of polytopes. (Joint ongoing work with David Hokken and Berend Ringeling.)

### Intercity Number Theory Seminar

7 June, Utrecht. Minnaertgebouw, room 2.02- 13:15–14:05
- Rosa Winter UniDistance Suisse, Weak weak approximation for del Pezzo surfaces of degree 2
Del Pezzo surfaces are classified by their degree
*d*, an integer between 1 and 9. The lower the degree, the more arithmetically complex these surfaces are. It is generally believed that, if a del Pezzo surface has one rational point, then it has many, and that they are well-distributed. After giving an overview of different notions of 'many’ rational points and what is known so far for del Pezzo surfaces, I will focus on joint work with Julian Demeio and Sam Streeter where we prove weak weak approximation for del Pezzo surfaces of degree 2 with a general point. - 14:15–15:05
- Julia Brandes Chalmers TU and University of Gothenburg, A minimalist approach to the circle method and Diophantine problems over thin sets
Even though additive problems such as Waring's problem are usually treated over the integers, there is also a substantial body of work studying additive questions over other sets. We confirm in quantitative terms the well-known heuristic that a mean value estimate and an estimate of Weyl type, together with suitable distribution properties of the underlying set over a set of admissible residue classes, are sufficient to implement the circle method. This allows us to give a rather general proof of Waring’s problem which is applicable to a range of sufficiently well-behaved thin sets.
- 15:35–16:25
- Judith Ortmann Leibniz University Hannover, Rational points of bounded height on a conic bundle surface over
*F*_{2}(*t*)We consider a conic bundle surface*C*in*P*^{2}_{K}×*A*^{1}_{K}with a morphism*C*→*A*^{1}_{K}over the global function field*K*=*F*_{2}((*t*)). We are interested in an asymptotic formula for the number of fibres of*A*^{1}_{K}of bounded height that have a rational point. To obtain such an asymptotic formula, the main idea is to use harmonic analysis to compute the height zeta function and then use a Tauberian theorem.