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.[-]

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.[-]