In 1938, the Hungarian mathematician Redei defined a ternary Jacobi-like symbol that he used in the study of quadratic 2-class groups, and proved a certain reciprocity law for these symbols. The symbol, which is currently used in answering distribution questions for quadratic 2-class groups, found its ultimate definition only in the 21st century. Its modern definition gives rise to a particularly elegant reciprocity law that we will explain in this talk.

We  present  in   this talks  results about  the first   moment of cubic  twists of  Dirichlet L-functions over the function field $\mathbb{F}_q(T)$, when $q$ is congruent to $1$ mod $3$ (the Kummer case). The case of number fields was considered in previous work, but never for the full family of cubic twists over a field containing the third roots of unity.
We first explain the history of the problem, starting with the moments of the Riemann zeta function, the standard conjectures for moments of L-functions and presenting the previous results, over number fields and function fields.

This is joint work with A. Florea and M. Lalin.

For $k=\mathbb{Q}(\sqrt{-3})$ and an algebraic integer $c\in \mathfrak{O}_k$, let $\zeta_{k(c^{1/3})}(s)$ be the Dedekind zeta function of the cubic field $k(c^{1/3})$ and $\zeta_k(s)$ be the Dedekind zeta function of $k$. For fixed real $\sigma>1/2$, we study the asymptotic distribution of the values of the logarithms and the logarithmic derivatives of the Artin $L$-functions \begin{equation*} L_c(\sigma)= \frac{\zeta_{k(c^{1/3})}(\sigma)}{\zeta_k(\sigma)}, \end{equation*} as $c$ varies.

This is joint work with Alia Hamieh (University of Northern British-Columbia).

Mirror symmetry predicts surprising geometric correspondences between distinct families of algebraic varieties. In some cases, these correspondences have arithmetic consequences. Among the arithmetic correspondences predicted by mirror symmetry are correspondences between point counts over finite fields, and more generally between factors of their Zeta functions. In this talk, we will introduce these topics and describe these correspondences. Finally, we will discuss how all of this relates to hypergeometric functions and hypergeometric motives.

This is joint work with: Charles Doran (University of Alberta, Canada), Tyler Kelly (University of Cambridge, UK), Steven Sperber (University of Minnesota, USA), John Voight (Dartmouth College, USA), and Ursula Whitcher (American Mathematical Society, USA).

The analog of the Riemann Hypothesis for zeta functions of curves over finite fields has been proved by Hasse-Weil and implies strong bounds for the number of rational points on such curves. However, as noticed by Ihara, for large genera these bounds can be improved significantly. With the aim of understanding the maximal possible number of rational points on high genus curves a variety of methods (class field towers, modular curves of various type, explicit equations) have been used, giving in each case curves of high genus with many rational points. In this talk I will give a short overview of these methods and show how special curves on Drinfeld modular varieties of higher dimension can be used in the construction of such curves over all non-prime finite fields. This is joint work with Beelen, Garcia, Stichtenoth

A non hyperelliptic genus 3 curves over a number field can reduce modulo certain primes to a hyperelliptic curve of genus 3. We give a characterization of these primes in terms of the valuations of Dixmier-Ohno invariants of the corresponding plane quartic.