Fall 2025

Basel Number Theory Seminar: Fall 2025

Time: 14.15 on 9 Oct, 2025

Place: Spiegelgasse 5, Seminarraum 05.002

Thu Hà Trieu (Hanoi University of Science and Technology / Oberwolfach Leibniz Fellow)

The Mahler measure of exact polynomials and L-function of motives

The Mahler measure of polynomials was introduced by Mahler in 1962 as a tool to study transcendental number theory. Over time, numerous connections have been discovered between Mahler measure and special values of L-functions. In this talk, we express the Mahler measure of an exact polynomial in arbitrarily many variables in terms of Deligne–Beilinson cohomology, and study its relationship with Beilinson regulators. As an application, we show that the Mahler measure of certain three-variable polynomials can be expressed in terms of special values of the L-functions of elliptic curves and the Bloch–Wigner dilogarithm. In the four-variable case, the Mahler measure can be written as a linear combination of special values of the L-functions of K3 surfaces and of the Riemann zeta function.

Time: 14.15 on 16 Oct, 2025

Place: Spiegelgasse 5, Seminarraum 05.001

Remke Kloosterman (University of Padova)

The average Mordell--Weil rank of elliptic surfaces over number fields

Let $K$ be a number field and let $n$ be a non-negative integer.In this talk we determine the average (arithmetic) Mordell--Weil rank of elliptic surfaces over $K$ with base curve $\mathbb{P}^1$ and geometric genus $n$.Hereby we prove a conjecture of Alex Cowan, which in turn was inspired by Nagao's conjecture and subsequent work.The proof consists of two parts, the first part relies on work by Andr\'e and Maulik--Poonen on the jump loci of the Picard Number in flat families. This is sufficient to prove that the average Mordell--Weil rank equals the generic Mordell--Weil rank.The second part of the proof uses an argument involving quadratic twists in order to show that the generic Mordell--Weil rank of elliptic surfaces over a number field with fixed topological equals zero.

Time: 14.15 on 20 Nov, 2025

Place: Spiegelgasse 5, Seminarraum 05.002

Marc Abboud (Université de Neuchâtel)

On the rigidity of periodic points for automorphisms of affine surfaces

I will discuss the following results. Let S be a complex affine surface and f,g be two automorphisms of positive entropy. If f and g have a Zariski dense set of periodic points in common then they have the same set of periodic points. The proof uses the dynamics at infinity of such automorphisms and the construction of their canonical Green functions and equilibrium measures both for archimedean places and non-archimedean ones. One of the main ingredients is the theorem of arithmetic equidistribution on adelic line bundles over quasiprojective varieties from Yuan and Zhang. I will also discuss examples of affine surfaces where I manage to show a stronger rigidity: having the same periodic points imply that the automorphisms share a common iterate. The examples are the affine plane and Markov surfaces which are related to the character variety of the punctured torus.

Time: 14.15 on 4 Dec, 2025

Place: Spiegelgasse 5, Seminarraum 05.002

Jerson Caro (Boston University)

Counting and Finding Rational Points on Surfaces

A celebrated result of Coleman gives an explicit version of Chabauty's theorem, bounding the number of rational points on curves over number fields via the study of zeros of p-adic analytic functions. While many developments have extended and refined this result, obtaining analogous explicit bounds for higher-dimensional subvarieties of abelian varieties remains a major challenge.In this talk, I will sketch the proof of such an explicit bound for surfaces contained in abelian varieties — a step toward a higher-dimensional Chabauty–Coleman method. This is joint work with Héctor Pastén.I will also describe an application of this method to a computational problem: determining an upper bound for the number of unexpected quadratic points on hyperelliptic curves of genus 3 defined over Q. I will illustrate the method through an explicit example where this set can be computed. This is joint work with Jennifer Balakrishnan.

Time: 14.15 on 10 Dec, 2025

Place: Alte Universität - Seminarraum -201

Alina Ostafe (UNSW Sydney)

Counting linear recurrence sequences with zeros and solvable S-unit equations

In this talk I will discuss recent joint work with Carl Pomerance and Igor Shparlinski where we obtain a tight upper bound on the number of integer linear recurrence sequences which attain a zero value. The argument is based on modular techniques combined with a classical idea of P. Erdos. Using similar ideas, we also show that only a rather small proportion of linear equations are solvable in elements of a fixed finitely generated subgroup of a multiplicative group of a number field.

Time: 14.15 on 11 Dec, 2025

Place: Spiegelgasse 5, Seminarraum 05.002

Shaoshi Chen (Academy of Mathematics and Systems Science, Chinese Academy of Sciences)

Arithmetic Dynamics around Linear Differential Equations

We will talk about two types of dynamical systems coming from linear differential equations. The first dynamical system is defined by the coefficient sequence of a linear differential equation. We present a Skolem-Mahler-Lech type theorem on this dynamical system. The second dynamical system is defined by iterated integration of solutions of a linear differential equation. We present some stability results on the order of linear differential equations satisfied by the iterated integrals of these solutions.

Date	Speaker	Title
9 Oct, 2025 14.15	Thu Hà Trieu (Hanoi University of Science and Technology / Oberwolfach Leibniz Fellow)	The Mahler measure of exact polynomials and L-function of motives
16 Oct, 2025 14.15	Remke Kloosterman (University of Padova)	The average Mordell--Weil rank of elliptic surfaces over number fields
7 Nov, 2025 14.15	Rhine Seminar on Transcendence Basel-Freiburg-Strasbourg, 8th edition (Link to program)
20 Nov, 2025 14.15	Marc Abboud (Université de Neuchâtel)	On the rigidity of periodic points for automorphisms of affine surfaces
4 Dec, 2025 14.15	Jerson Caro (Boston University)	Counting and Finding Rational Points on Surfaces
10 Dec, 2025 14.15	Alina Ostafe (UNSW Sydney)	Counting linear recurrence sequences with zeros and solvable S-unit equations
11 Dec, 2025 14.15	Shaoshi Chen (Academy of Mathematics and Systems Science, Chinese Academy of Sciences)	Arithmetic Dynamics around Linear Differential Equations