# Invited Talks

**On smooth plane models of modular curves - Samuele Anni (Aix-Marseille Université) **

In this talk we will prove 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 Eran Assaf and Elisa Lorenzo Garcìa.

**Mock rational points in quadratic Chabauty - Jennifer Balakrishnan (Boston University) **

We discuss the computation of rational points on curves via the quadratic Chabauty method. We also consider the mock rational points (those finitely many points that are not rational) that may arise in the quadratic Chabauty set, in view of a conjecture of Kim. We give conditions under which one can find algebraic mock rational points and present a number of examples, primarily in the case of punctured elliptic curves, produced in ongoing joint work with Betts.

**Computing the endomorphism ring of an elliptic curve over a number field - John Cremona (University of Warwick) **

I will describe joint work with Andrew Sutherland, giving deterministic and probabilistic algorithms to decide whether a given monic irreducible polynomial H in Z[X] is a Hilbert class polynomial, and if so, which one. These algorithms can be used to determine whether a given algebraic integer is the j-invariant of an elliptic curve with complex multiplication (CM), and if so, the associated CM discriminant. Our algorithms admit simple implementations that are asymptotically and in practice faster than existing approaches.

**Conjectures on ranks of modular jacobians and linear bounds on prime order torsion on elliptic curves - Maarten Derickx**

Let S(d) denote the set of primes p such that there exists an elliptic curve over a number field of degree d with a point of order p. Oesterlé showed that the largest prime in S(d) is smaller than $(3^{d/2}+1)^2$, giving an exponential upperbound. However, explicit calculations for small values of d have shown that this bound is far from sharp. In this talk we will present some conjectures on the Mordell-Weil ranks of modular jacobians that imply a linear bound on the largest prime in S(d). We provide numerical evidence for these conjectures by verifying them for p < 100,000.

This is joint work with Michael Stoll

**Local arithmetic of hyperelliptic curves and applications - Céline Maistret (University of Bristol) **

In this talk, I will consider hyperelliptic curves over local fields of odd characteristics. I will first survey the "Cluster Picture" approach to understanding various arithmetic invariants of the curves and their Jacobians. Then, I will present two recent applications: computing Euler factors of curves (joint with A. Sutherland) and computing conductor exponents of family of Frey hyperelliptic curves (joint with M. Azon, M. Curco Iranzo, M. Khawaja and D. Mocanu).

**p-adic Arakelov theory and quadratic Chabauty - Steffen Müller (University of Groningen)**

The quadratic Chabauty method for the computation of rational points on certain curves, due to Balakrishnan and Dogra, is the simplest nontrivial instance of Kim's nonabelian Chabauty. Based on p-adic Hodge theory, it produces a p-adic function F and a finite set T such that F maps the rational points into T. While T can be shown to be trivial for several interesting modular curves, its computation is in general a difficult problem.

I will present an alternative approach to quadratic Chabauty that is based on p-adic Arakelov theory. Time permitting, I will also discuss work in progress that uses this to obtain an algorithm to compute the set T. This is joint work with Amnon Besser and Padmavathi Srinivasan.

**Quadratic points on modular curves - Filip Najman (University of Zagreb) **

In this talk I will survey results, old and new, on quadratic points on modular curves, as well as the arithmetic consequences of these results. I will also describe the techniques used to determine all the quadratic points on modular curves. The talk is based partially on joint work with Adžaga, Bruin, Keller, Michaud-Jacobs, Ozman and Vukorepa.

**Models for modular curves and some applications - Pïerre Parent (Université de Bordeaux)**

If one wants to study rational points of an algebraic curve over Q, say, it may help a lot to be able to "reduce the situation modulo prime numbers". But that implies to define and compute good models for those objects over the integers. In that talk, I will present recent results about semistable or regular models for modular curves associated with maximal subgroups of $GL_{2}$ in prime level. Then I will try to give diophantine applications, such as effective versions of the Bogomolov conjecture in some new cases.

**The rational cuspidal subgroup of $J_0(N)$ - Hwajong Yoo (Seoul National University)**

We introduce several questions related to generalized Ogg's conjecture. One important object to study is the rational cuspidal subgroup of $J_0(N)$, which we still do not know in general. We discuss a certain strategy to compute this group, and prove that it works when the prime divisors of N are at most two odd primes.

This is joint with M. Yu.

**Serre's open image theorem and families of modular curves - David Zywina (Cornell University)**

Consider a non-CM elliptic curve $E/\mathbb{Q}$. The Galois action on the torsion points of $E(\overline{\mathbb{Q}})$ can be expressed in terms of a Galois representation $\rho_E : Gal(\overline{\mathbb{Q}}/\mathbb{Q})\to GL_2(\widehat{\mathbb{Z}})$. A famous theorem of Serre say that the image of $\rho_E$ is an open, and hence finite index, subgroup of $GL_2(\widehat{\mathbb{Z}})$.

We shall describe ideas that allows us to explicitly compute the image in Serre's theorem for any non-CM $E/\mathbb{Q}$. When viewed in terms of modular curves, this seems very difficult at first since there are infinitely many curves that need to be considered. However, it becomes tractable when we put the relevant modular curves in suitable families of twists.