## 2022

### Intercity number theory seminar

13 May, Utrecht. Morning: Buys Ballotgebouw room 209; afternoon: Koningsbergergebouw room 1.50 PANGEA- 11:00–12:00
- Florian Wilsch IST Austria, Integral points of bounded height on a certain toric variety
- 13:15–14:15
- Miriam Kaesberg Goettingen, On Artin’s Conjecture: Pairs of Additive Forms
- 14:30–15:30
- Nirvana Coppola VU, Coleman integrals over number fields: a computational approach
- 15:45–16:45
- Vladimir Mitankin Hannover, Semi-integral points on quadrics

### Intercity number theory seminar

3 June, VU Amsterdam. Main building (Hoofdgebouw), room HG–06A32- 11:00–12:00
- Anne Baanen VU, Introduction to formalizing number theory
Libraries of formalized mathematics are covering more and more definitions and theorems. This talk introduces the essential concepts of formalizing mathematics, with a special focus on number theory, from the way computer languages can represent proofs, via the current state of formal libraries, to the big questions of the field of formalization. No previous knowledge of formalization or foundations of mathematics is required for this talk.
- 13:15–14:15
- Manuel Eberl Innsbruck, How to avoid bad points in contour integration, rigorously
In this talk I will first give an overview over recent efforts to formalise the foundations of analytic number theory in the proof assistant Isabelle/HOL: Dirichlet series, the Prime Number Theorem, elliptic functions, modular functions and modular forms, etc. In particular, I will highlight some of the difficulties we encountered in the process of formalising this material. For the main part of the talk, I will focus on one particular issue that is always glossed over on paper but that is at first glance very painful in formalisation: the deformation of an integration contour to avoid singularities.
- 14:30–15:30
- Alex Best VU, Formalization in number theory, past, present and future
I’ll give a broad overview of many past and ongoing projects, by various people, that formalize results in and around number theory. This will focus on some of the lessons learned throughout these projects that will inform future progress in the area, and mention some goals and open problems that we foresee ahead.
- 16:00–17:00
- Johan Commelin Freiburg, Liquid Tensor Experiment
In December 2020, Peter Scholze posed a challenge to formally verify the main theorem on liquid ℝ-vector spaces, which is part of his joint work with Dustin Clausen on condensed mathematics. I took up this challenge with a team of mathematicians to verify the theorem in the Lean proof assistant. Half a year later, we reached a major milestone, and our expectation is that shortly we will have completed the full challenge. In this talk I will give a brief motivation for condensed/liquid mathematics, a demonstration of the Lean proof assistant, and discuss our experiences formalizing state-of-the-art research in mathematics.

### Intercity seminar

1 September, Groningen. Linked to the conference Curves over finite fields and arithmetic of K3 surfaces for Jaap Top's 62nd birthday – please register here by 20 August if you plan to attend!- 11:30–12:30
- Matthias Schütt Hannover, Explicit RM for K3 surfaces
While the CM case is fairly well understood, the RM case (with a field different from ℚ), remains quite mysterious, with no explicit examples exceeding one-dimensional families. I will report on joint work in progress with Bert van Geemen which aims to push our knowledge much further. One of our two main approaches is partly based on work of our birthday boy.
- 13:30–14:30
- Cecília Salgado RUG, Non-thin rank jumps for elliptic K3 surfaces
We discuss recent progress on the variation of the Mordell–Weil rank in families of elliptic curves over number fields. In the case of elliptic K3 surfaces, we show, under certain conditions, that the set of fibres for which the Mordell–Weil rank is strictly larger than the generic rank is not thin, as a subset of the base of the fibration. This is based on joint work with Hector Pasten.
- 15:00–16:00
- Tim Dokchitser Bristol, Models of hyperelliptic curves
It is an important question how to find a regular model of a curve
*C*/*K*with respect to some valuation on*K*. Motivated by the Birch–Swinnerton-Dyer conjecture, in 1972 Tate gave an algorithm for elliptic curves, based on the Kodaira–Neron classification (“Tate’s algorithm”). Later, Liu gave an algorithm in genus 2, based on the Namikawa–Ueno classification. There has been a lot of recent activity to extend these results to higher genus, and in June 2022 Simone Muselli gave a general algorithm for hyperelliptic curves of arbitrary genus in residue characteristic ≠2. It is very much in the spirit of Tate’s algorithm, and is also as explicit, uses minimal machinery, and works over any discrete valuation ring. In this talk, I will give an overview of his work and some related results. - 16:15–17:15
- Irene Bouw Ulm, Computing Weil representations of superelliptic curves
Superelliptic curves are curves
*Y*that admit a map*f*:*Y*→ ℙ^{n}that becomes cyclic after an extension of scalars. We consider superelliptic curves defined over a*p*-adic field*K*with potentially good reduction to characteristic*p*. In an earlier paper we determined a Galois extension*L*/*K*together with a smooth model of*Y*over*L*. Building on work of Dokchitser–Dokchitser, we explain how to compute the Weil representation of*Y*from the reduction of*Y*. A key step is point counting on suitable twists of the reduction of*Y*in characteristic*p*. This is joint work with with Do and Wewers.

### Intercity / Getaltheorie in het Vlakke Land

23 September, Utrecht. Zaal Wit, Ruppertgebouw. Attention: we require that you**register**for this event via the GVL/APP website here: https://www.mathconf.org/app-gvl-autumn2022 (in Dutch or French) before 22 Sept.

- 13:15–14:15
- Laurent Berger ENS Lyon, Unlikely intersections for formal groups
If two elliptic curves over an algebraic closure have infinitely many torsion points in common, they are isomorphic (Bogomolov and Tschinkel). I will discuss a
*p*-adic version of this statement, where elliptic curves are replaced by formal groups. This result belongs to the theory of*p*-adic dynamical systems. - 14:15–15:15
- Jungwon Lee Warwick, Another view of the Washington Theorem
We reprove the main equidistribution instance in the Washington proof of the vanishing of cyclotomic Iwasawa mu-invariant, based on the ergodicity of a certain
*p*-adic skew extension dynamical system that can be identified with Bernoulli Shift (joint with Bharathwaj Palvannan). - 15:45–16:45
- Joseph Silverman Brown, Finite orbits of points on tri-involutive K3 surfaces
The classical Hurwitz equation
*M*:*x*^{2}+*y*^{2}+*z*^{2}=*axyz*+*b*admits three non-commuting involutions coming from the three double covers from*M*to A^{2}. There has recently been considerable interest in studying the orbit structure of the points on*M*over a finite field under the action of the involutions. In this talk I will discuss some of this history, and then describe analogous results and conjectures on K3 surfaces*W*in P^{1}x P^{1}x P^{1}given by the vanishing of a (2,2,2) form. Just as with the Hurwitz surface, the three projections from*W*to P^{1}x P^{1}are double covers that induce three non-commuting involutions on*W*. We let*G*be the group of automorphisms of*W*generated by these involutions and investigate the*G*-orbit structure of*W*. In particular, we study*G*-orbital components of the points of*W*over a finite field and finite*G*-orbits in the points of*W*over the complex numbers. (Joint work with Elena Fuchs, Matthew Litman, and Austin Tran) - 16:45–17:45
- Lola Thompson Utrecht, Salem numbers and short geodesics
We will discuss how Mahler measure and related concepts are connected to problems about lengths of geodesics on arithmetic hyperbolic orbifolds. As a result, by solving problems using tools from number theory, we are able to answer quantitative questions in spectral geometry. This talk is based on joint work with Misha Belolipetsky, Matilde Lalin, and Plinio G.P. Murillo; and with Benjamin Linowitz, D.B. McReynolds, and Paul Pollack.

### Belgian-Dutch Algebraic Geometry Day

30 September, Leuven. Please register before Monday 26 September. See here for details and registration form.### Intercity seminar

28 October, Leiden. Snellius building, room 407/409- 13:30–14:30
- Marta Pieropan Utrecht, Hyperbola method and Campana points on toric varieties
In joint work with Damaris Schindler we develop a new version of the hyperbola method for counting rational points of bounded height that generalizes the work of Blomer and Brüdern for products of projective spaces. The hyperbola method transforms a counting problem into an optimization problem on certain polytopes. For rational points on toric varieties the polytopes have a geometric meaning that reflects Manin's conjecture, and the same holds for counts of Campana points of bounded height. I will present our results as well as some general heuristics.
- 15:00–16:00
- Maarten Derickx Leiden, The uniform boundedness of isogenies with unramified galois representation
This talk is about some recent joint work with Barinder Banwait. Let
*E*be an elliptic curve over a number field*K*, then it is a well known result of Merel that the order of torsion points in*E*(*K*) can be bounded in terms of only the degree of the number field*K*. This begs the question whether a similar statement also holds for isogenies. In this talk I will shortly discuss what is expected to be true, as well as prove some partial statements in this direction regarding isogenies of prime order*p*. In particular we prove uniform boundedness for*p*isogenies whose associated isogeny character is unramified at all primes lying above*p*. Our results are a strengthening of the already known uniform boundedness of torsion, which is a special case of our results. - 16:15–17:15
- Misja Steinmetz Leiden, Explicit Serre weights for GL
_{2}via Kummer theoryIn this talk I will speak about a new explicit method for computing the set of Serre weights attached to a 2-dimensional mod*p*Galois representation. I will start by giving a general introduction and brief historical overview of (Serre) weights attached to mod*p*Galois representations. Then I will talk about new results which give the set of Serre weights attached to such a representation explicitly. In this new work we use Kummer theory and the Artin–Hasse exponential to arrive at the set of weights avoiding some combinatorial difficulties found in other approaches. This is joint work with Robin Bartlett.