Arithmetic geometry

Speaker : Martin LÜDTKE (University of Oldenburg,Germany)

This will be a session introducing students to the basics of the computer algebra system MAGMA, which will be used later in some courses. The session will mostly be interactive, consisting of a short introduction followed by an exercise sheet for the students to work through.

Speaker : Diana MOCANU (Max Planck Institute for Mathematics Bonn,Germany)

This course is divided into two parts. In the first part, we review fundamental aspects of the theory of elliptic curves and introduce their residual Galois representations. In the second part, we apply this knowledge to the study of generalized Fermat equations. To this end, we outline the main steps of the modular method, with an emphasis on the construction of the so-called Frey elliptic curves. Throughout, we will use basic MAGMA commands to help with necessary computations

Speaker : Yukako KEZUKA (Kanazawa University,Japan)

Iwasawa theory is one of the main branches of modern number theory, providing a powerful framework to study arithmetic objects over infinite extensions, with applications ranging from class groups of number fields to elliptic curves and L-functions. Originating in Iwasawa’s pioneering work on cyclotomic fields, the subject has since become central to understanding p-adic properties of zeta and L-functions, Selmer groups of elliptic curves, and major conjectures such as the Birch and Swinnerton–Dyer conjecture. This course introduces the basic tools and classical results of Iwasawa theory, while developing intuition and laying the groundwork for more advanced topics.

Speaker : Céline MAISTRET (University of Bristol,United Kingdom)

Elliptic curves are abelian varieties of dimension 1. The set of their rational points carries an abelian group structure, and by a famous theorem of Mordell, this group is finitely generated. The rank of this group is therefore a crucial invariant to compute when trying to understand rational points on these curves. This is a hard problem and a milestone in modern number theory. A solution to this problem is given by the Birch and Swinnerton-Dyer conjecture (BSD), one of the Clay Mathematical Institute Millennium Prize Problems. Although the question of proving BSD in general is still open, we can already provide theoretical evidence that it holds by showing that it correctly predicts the parity of the rank. This is known as proving the parity conjecture. The aim of this course is to prove the parity conjecture for elliptic curves under some conditions. To do so, we will discuss the arithmetic of elliptic curves over local and global fields, present BSD, understand how one can give a formula for the parity of the rank and prove that this formula is compatible with the parity predicted by BSD.

Speaker : Lei ZHANG (National University of Singapore,Singapore)

This minicourse offers an introduction to the theory of modular forms, designed for participants who may have some background in complex analysis and algebra but little or no prior exposure to the subject. Modular forms stand at the intersection of analysis, algebra, and number theory, and play a central role in modern mathematics, from the proof of Fermat’s Last Theorem to the Langlands program. We begin by motivating the theory through classical examples: elliptic functions and the modular group. After a review of the action of SL_2(Z) on the upper half-plane, we define modular forms and cusp forms, and highlight their key analytic and transformation properties. Emphasis will be placed on accessible examples, such as Eisenstein series and the discriminant modular form Delta(z), which illustrate the deep arithmetic encoded in Fourier expansions. Next, we explore the algebraic structure of modular forms: vector spaces of forms of a given weight, the role of Hecke operators, and the emergence of eigenforms with rich arithmetic significance. Even within a brief time frame, students will see how modular forms provide a unifying language for congruences, partition functions, and L-functions. The final segment will connect these ideas to broader perspectives: the modularity of elliptic curves, the relationship to Galois representations, and glimpses into how modular forms interface with the Langlands program. While we cannot cover every detail in this minicourse, the minicourse aims to provide a roadmap of the subject, highlight key ideas and results, and point toward deeper references for further study.

Given an elliptic curve E over the rational numbers Q, one can consider its reduction modulo a given prime p. Except in finitely many cases, this reduction is again smooth and there an elliptic curve over F_p, so that one can meaningfully consider its number of points #E(F_p), as well as the quantity a_p = p + 1 - E(F_p). It then turns out that there exists a cusp form f = \sum_{n in N} a_n q^n whose p-th coefficient is given by said quantity for all primes p at which E has good reduction. This statement, though "well-known" in some sense, is far from trivial, being the modularity theorem by Wiles and his collaborators. In the current course, we ask ourselves what happens when we instead consider a curve X over Q of genus 2, which is so to say the next step after elliptic curves. Most of these results are conjectural and can be seen as special cases of the general Langlands philosophy. Unsurprisingly, this situation is somewhat more complicated, and we get a zoo of possibilities; the corresponding classification essentially depends on the endomorphism ring of the Jacobian of X. We explore what this classification looks like, what consequences this has for modularity (that is, what it means at all that one can once again associate a modular form to X), and which of these conjectural statements are in range of being proved, either in given specific cases or in general.

Speaker : Valentijn KAREMAKER (University of Amsterdam,Netherlands)

In this course we will introduce abelian varieties and study their arithmetic and geometric properties. Abelian varieties exist in any dimension >=1: in dimension 1 they are elliptic curves, in higher dimensions they have some analogous properties to elliptic curves but also exhibit interesting new behaviour. We will study abelian varieties over different ground fields, in particular over number fields and finite fields. As a concrete source of examples we will use Jacobians of curves (where elliptic curves are their own Jacobians). Rational points on an abelian variety form an abelian group as well as a Galois module, which will lead us to study their Galois representations, Tate modules and p-divisible groups.

Speaker : Marta PIEROPAN (Utrecht University,Netherlands)

Given a variety over a field that is not algebraically closed, for example a number field, it is natural to ask whether it has points defined over that field, and if so, how many there are. If the set of points over a number field is infinite, one can measure their distribution very precisely using height functions. A conjecture of Manin predicts a relation between the asymptotic distribution of rational points of bounded height and some geometric invariants of the variety. This minicourse introduces the concepts of rational points and heights, and some procedures for counting points of bounded height on projective varieties over number fields. The tools involved include torsor parameterizations, Moebius inversion, and a variety of lattice point counting techniques. The theory will be complemented by working examples and exercises to become familiar with the new concepts and techniques. These include a modern treatment of Schanuel’s result for projective spaces. We will then move on to products of projective spaces and to other explicit toric varieties. The minicourse will end with an overview of results and open problems related to Manin’s conjecture.

Speaker : Detchat SAMART (Burapha University,Thailand)

Mahler measure forms a bridge between analysis, algebra, arithmetic geometry, and beyond. Originating in the study of polynomials as a height function, it has revealed surprising connections with the arithmetic of algebraic varieties through special values of L-functions. This minicourse will introduce its classical properties and explore generalizations including multivariable Mahler measures and connections to regulators. Through a mix of lectures and problem sessions, participants will gain both a conceptual and computational understanding of Mahler measure and its role in arithmetic geometry.