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 discuss Arithmetic Statistics as a 'new' branch of number theory by briefly sketching its development in the last 50 years. The non-triviality of the meaning of `random behaviour' and the problematic absence of good probability measures on countably infinite sets are illustrated by the example of the 1983 Cohen-Lenstra heuristics for imaginary quadratic class groups. We then focus on the Negative Pell equation, of which the random behaviour in the case of fundamental discriminants (Stevenhagen's conjecture) has now been established after 30 years.
We explain the open conjecture for the general case, which is based on equidistribution results for units over residue classes that remain to be proved.[-]

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.[-]

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.[-]

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

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.[-]

The theme of the survey is how arithmetic theory of quatenion hermitian lattices can be applied to the theory of supersingular abelian varieties. Here the following geometric objects will be explained by arithmetics. Principal polarizations of superspecial abelian varieties, of supersingular surfaces, components of supersingular moduli and their configuration for small dimensions, their automorphims, fields of definition, and existence of maximal curves of genus three over $F_{p}^{2}$ for all odd primes $p$. Corresponding arithmetics are class numbers, type numbers, lattice automorphisms, and parahoric subgroups of quaternion hermitian groups, mass or trace formulas.[-]