Courses

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.

The set of rational points of Elliptic curves carries an abelian group structure, and by a famous theorem of Mordell, 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 problem 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.

Abstract: 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 = ∑ 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.

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.