Let $E$ be an elliptic curve over the rationals, and let $\chi$ be a Dirichlet character of order $\ell$ for some odd prime $\ell$. Heuristics based on the distribution of modular symbols and random matrix theory have led to conjectures predicting that the vanishing of the twisted $L$-functions $L(E, \chi, s)$ at $s = 1$ is a very rare event (David-Fearnley-Kisilevsky and Mazur-Rubin). In particular, it is conjectured that there are only finitely many characters of order $\ell > 5$ such that $L(E, \chi, 1) = 0$ for a fixed curve $E$.
We investigate the case of elliptic curves over function fields. For Dirichlet $L$-functions over function fields, Li and Donepudi-Li have shown how to use the geometry to produce infinitely many characters of order $l \geq 2$ such that the Dirichlet $L$-function $L(\chi, s)$ vanishes at $s = 1/2$, contradicting (the function field analogue of) Chowla's conjecture. We show that their work can be generalized to constant curves $E/\mathbb{F}_q(t)$, and we show that if there is one Dirichlet character $\chi$ of order $\ell$ such that $L(E, \chi, 1) = 0$, then there are infinitely many, leading to some specific examples contradicting (the function field analogue of) the number field conjectures on the vanishing of twisted $L$-functions. Such a dichotomy does not seem to exist for general curves over $\mathbb{F}_q(t)$, and we produce empirical evidence which suggests that the conjectures over number fields also hold over function fields for non-constant $E/\mathbb{F}_q(t)$.[-]

Subrings of number fields, especially those that are of finite type, are worth being investigated from analgorithmic point of view. The question of how such rings are best represented in algorithms already presents several problems that challenge our theoretical understanding. In the lecture, both new results and open problems will be discussed. It is based on work that was done jointly with Daanvan Gent, Samuel Tiersma, and Jeroen Thuijs, all at Leiden.[-]

We determine the average size of the 3-torsion in class groups of $G$-extensions of a number field when $G$ is any transitive 2 -group containing a transposition, for $\theta$-xample $D_4$. It follows from the Cohen--Lenstra--Martinet heuristics that the average size of the $p$-torsion in class groups of $G$-extensions of a number field is conjecturally finite for any $G$ and most $p$ (including $p \nmid|G|$ ). Previously this conjecture had only been proven in the cases of $G=S_2$ with $p=3$ and $G=S_3$ with $p=2$. We also show that the average 3-torsion in a certain relative class group for these $G$-extensions is as predicted by Cohen and Martinet, proving new cases of the Cohen--Lenstra--Martinet heuristics. Our new method also works for many other permutation groups $G$ that are not 2-groups.[-]

Let $E / \mathbb{Q}$ be a non-CM elliptic curve and let $\mathcal{E}$ denote the collection of all elliptic curves geometrically isogenous to $E$. That is, for every $E^{\prime} \in \mathcal{E}$, there exists an isogeny $\varphi: E \rightarrow E^{\prime}$ defined over $\overline{\mathbb{Q}}$. Motivated by ties to Serre's Uniformity Conjecture, we will discuss the problem of identifying minimal torsion curves in $\mathcal{E}$, which are elliptic curves $E^{\prime} \in \mathcal{E}$ attaining a point of prime-power order in least possible degree. Using recent classification results of Rouse, Sutherland, and Zureick-Brown, we obtain an answer to this question in many cases, including a complete characterization for points of odd degree.

This is joint work with Nina Ryalls and Lori Watson.[-]

The main result of the talk by X. Guitart in this conference classifies the 92 geometric endomorphism algebras that arise among geometrically split abelian surfaces defined over $\mathbb{Q}$. In this talk, we will explain how only 54 of them arise as geometric endomorphism algebras of Jacobians of genus 2 curves defined over $\mathbb{Q}$, and how the remaining 38 do not. In particular, we exhibit 38 abelian surfaces defined over $\mathbb{Q}$ that are not isogenous over an algebraic closure of $\mathbb{Q}$ to any Jacobian of a genus 2 curve defined over $\mathbb{Q}$.

This is a joint work with X. Guitart and E. Florit, that builds on examples supplied by N. Elkies and C. Ritzenthaler, and uses F. Narbonne's thesis in an essential way.[-]

An abelian surface defined over $\mathbb{Q}$ is said to be geometrically split if its base change to the complex numbers is isogenous to a product of elliptic curves. In this talk we will determine the algebras that arise as geometric endomorphism algebras of geometrically split abelian surfaces defined over $\mathbb{Q}$. In particular, we will show that there are 92 of them. A key step is determining the set of imaginary quadratic fields $M$ for which there exists an abelian surface over $\mathbb{Q}$ which is geometrically isogenous to the square of an elliptic curve with CM by $M$.

This is joint work with Francesc Fité.[-]

In this talk we will see that there are only finitely many modular curves that admit a smooth plane model. Moreover, if the degree of the model is greater than or equal to 19, no such curve exists. For modular curves of Shimura type we will show that none can admit a smooth plane model of degree 5, 6 or 7. Further, if a modular curve of Shimura type admits a smooth plane model of degree 8 we will see that it must be a twist of one of four curves.

This is joint work with Samuele Anni and Eran Assaf.[-]

The study of rational points on modular curves has a long history in number theory. Mazur's 1970s papers that describe the possible torsion subgroups and isogeny degrees for rational elliptic curves rest on a computation of the rational points on $X_{0}(N)$ and $X_{1}(N)$, and a large body of work since then continues this tradition.

Modular curves are parameterized by open subgroups $H$ of $\mathrm{GL}_{2}(\hat{\mathbb{Z}})$, and correspondingly parameterize elliptic curves $E$ whose adelic Galois representation $\displaystyle \lim_{ \leftarrow }E[n]$ is contained in $H$. For general $H$, the story of when $X_{H}$ has non-cuspidal rational or low degree points (and thus when there exist elliptic curves with the corresponding level structure) becomes quite complicated, and one of the best approaches we have for understanding it is large-scale computation.

I will describe a new database of modular curves, including rational points, explicit models, and maps between models, along with some of the mathematical challenges faced along the way. The close connection between modular curves and finite groups also arises in other areas of number theory and arithmetic geometry. Most well known are Galois groups associated to field extensions, but one attaches automorphism groups to algebraic varieties and Sato-Tate groups to motives. Building on existing tables of groups, we have added a new finite groups section to the L-functions and modular forms database, which we hope will prove useful both to number theorists and to others who are using and studying finite groups.[-]

This is the introductory lecture of the course on class field theory in the Research school on Arithmetic Statistics in May 2023. It briefly reviews the necessary algebraic number theory, and presents class field theory as the analogue of the Kronecker-Weber theorem over number fields. In a similar way, the Chebotarev density theorem is treated as an analogue of the Dirichlet theorem on primes in arithmetic progressions.
Two further lectures dealt with idelic and cohomological reformulations of the main theorem of class field theory, and two more were devoted to power reciprocity laws and Redei reciprocity.[-]