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.[-]

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.[-]

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.[-]

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 modular curve $Y^1(N)$ parametrises pairs $(E,P)$, where $E$ is an elliptic curve and $P$ is a point of order $N$ on $E$, up to isomorphism. A unit on the affine curve $Y^1(N)$ is a holomorphic function that is nowhere zero and I will mention some applications of the group of units in the talk.
The main result is a way of generating generators (sic) of this group using a recurrence relation. The generators are essentially the defining equations of $Y^1(N)$ for $n < (N + 3)/2$. This result proves a conjecture of Maarten Derickx and Mark van Hoeij.[-]