Curves over finite fields of large genus with many rational points have been of interest for both theoretical reasons and for applications. In the past, various methods have been employed for the construction of such curves. One such method is by means of explicit recursive equations and will be the emphasis of this talk.The first explicit examples were found by Garcia–Stichtenoth over quadratic finite fields in 1995. Afterwards followed the discovery of good towers over cubic finite fields and finally all nonprime finite fields in 2013 (B.–Beelen–Garcia–Stichtenoth). The recursive nature of these towers makes them very special and in fact all good examples have been shown to have a modular interpretation of some sort. The questions of finding good recursive towers over prime fields resisted all attempts for several decades and lead to the common belief that such towers might not exist. In this talk I will try to give an overview of the landscape of explicit recursive towers and present a recently discovered tower over all finite fields including prime fields, except $F_{2}$ and $F_{3}$.
This is joint work with Christophe Ritzenthaler.[-]

Let X and Y be two curves over a common base field k. Then we can consider the Jacobians Jac (X) and Jac (Y). On the level of principally polarized abelian varieties, we can form the product Jac (X) x Jac (Y). A logical question is then whether there exists a curve Z over k such that Jac (Z) is (possibly up to twist) isogenous to Jac (X) x Jac (Y).Frey and Kani considered the case where X and Y both have genus 1. The current talk will consider the case where X and Y have genus 1 and 2, respectively, which was considered in joint work with Jeroen Hanselman and Sam Schiavone for the case of gluing along 2-torsion.We will give criteria for the curve Z to exist, and methods to find an equation if it does. The first of these uses interpolation, and also determines the relevant twisting scalar. It can be used to find a Jacobian over QQ that admits a rational 70-torsion point. The second method is more geometrically inspired and exploits the geometry of the Kummer surface of Y. Applications will be discussed in passing.[-]

For a long time people have been interested in finding and constructing curves over finite fields with many points. For genus 1 and genus 2 curves, we know how to construct curves over any finite field of defect less than 1 or 3 (respectively), i.e. with a number of points at distance at most 1 or 3 to the upper bound given by the Hasse-Weil-Serre bound. The case of genus 3 is still open after more than 40 years of research. In this talk I will take a different approach based on the random matrix theory of Katz-Sarnak, that describe the distribution of the number of points, to prove the existence, for all $\epsilon>0$, of curves of genus $g$ over $\mathbb{F}_{q}$ with more than $1+q+(2 g-\epsilon) \sqrt{q}$ points for $q$ big enough. I will also discuss some explicit constructions as well as some details about the asymmetric of the distribution of the trace of the Frobenius for curves of genus 3 .This is a joint work with J. Bergström, E. Howe and C. Ritzenthaler.[-]

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 $E/\mathbb{F}_q(t)$.[-]

The Ceresa class is the image under a cycle class map of a canonical algebraic cycle associated to a curve in its Jacobian. This class vanishes for all hyperelliptic curves, and is known to be non-vanishing for the generic curve of genus at least 3. It is necessary for the Ceresa class to have infinite order for the Galois action on the fundamental group of a curve to have big image. We will present an algorithm for certifying that a curve over a number field has infinite order Ceresa class.

N.B. This is preliminary joint work with Jordan Ellenberg, Adam Logan and Akshay Venkatesh.[-]

Algebraic curves over a finite field $\mathbb{F}_{q}$ and their function fields have been a source of great fascination for number theorists and geometers alike, ever since the seminal work of Hasse and Weil in the 1930s and 1940s. Many important and fruitful ideas have arisen out of this area, where number theory and algebraic geometry meet. For a long time, the study of algebraic curves and their function fields was the province of pure mathematicians. But then, in a series of three papers in the period 1977-1982, Goppa found important applications of algebraic curves over finite fields to coding theory. The key point of Goppa's construction is that the code parameters are essentially expressed in terms of arithmetic and geometric features of the curve, such as the number $N_{q}$ of $\mathbb{F}_{q}$-rational points and the genus $g$. Goppa codes with good parameters are constructed from curves with large $N_{q}$ with respect to their genus $g$. Given a smooth projective, algebraic curve of genus $g$ over $\mathbb{F}_{q}$, an upper bound for $N_{q}$ is a corollary to the celebrated Hasse-Weil Theorem,$$N_{q} \leq q+1+2 g \sqrt{q} .$$Curves attaining this bound are called $\mathbb{F}_{q}$-maximal. The Hermitian curve $\mathcal{H}$, that is, the plane projective curve with equation$$X^{\sqrt{q}+1}+Y^{\sqrt{q}+1}+Z^{\sqrt{q}+1}=0,$$is a key example of an $\mathbb{F}_{q}$-maximal curve, as it is the unique curve, up to isomorphism, attaining the maximum possible genus $\sqrt{q}(\sqrt{q}-1) / 2$ of an $\mathbb{F}_{q^{-}}$ maximal curve. Other important examples of maximal curves are the Suzuki and the Ree curves. It is a result commonly attributed to Serre that any curve which is $\mathbb{F}_{q}$-covered by an $\mathbb{F}_{q}$-maximal curve is still $\mathbb{F}_{q}$-maximal. In particular, quotient curves of $\mathbb{F}_{q}$-maximal curves are $\mathbb{F}_{q}$-maximal. Many examples of $\mathbb{F}_{q}$-maximal curves have been constructed as quotient curves $\mathcal{X} / G$ of the Hermitian/Ree/Suzuki curve $\mathcal{X}$ under the action of subgroups $G$ of the full automorphism group of $\mathcal{X}$. It is a challenging problem to construct maximal curves that cannot be obtained in this way for some $G$. In this talk, we will describe our main contributions to both the theory of maximal curves over finite fields and to applications of algebraic curves with many points in coding theory. In particular, the following three topics will be discussed:
1. Construction of maximal curves
2. Weierstrass semigroups and points on maximal curves;
3. Algebraic curves with many rational points and coding theory.[-]