Speaker: Loïc Merel
Date: October 12, 2023
Affiliation: Institut de Mathematiques de Jussieu, Paris, France

Abstract: Let $N$ be a prime number $>3$. Mazur has defined, from the theory of modular forms, a unit $u$ in $Z/N$. This unit turned out to be, up to a 6th root of unity, $\prod_{k=1}^{(N-1)/2}k^k$. In this talk we will describe how the unit is connected to various objects in number theory. For instance: `The unit $u$ can be understood as a derivative of the zeta function at $-1$, (despite living in a finite field)'. Lecouturier has shown that this unit is the discriminant of the Hasse polynomial: $\sum_{i=0}^{(N-1)/2}a_i X^i$ modulo $N$, where $a_i$ is the square of the $i$-th binomial coefficient in degree $N$. `Calegari and Emerton have related $u$ to the class group of the quadratic field $Q(\sqrt{-N})$. For every prime number $p$ dividing $N-1$, it is important to determine when $u$ is a $p$-th power in $(Z/N)^*$. If time allows, I will describe the connections to modular forms and Galois representations, and the general theory that Lecouturier has developed from this unit. For instance, when $u$ is not a $p$th power, a certain Hecke algebra acting on modular forms is of rank 1 over the ring of $p$-adic integers $Z_p$ (the original motivation of Mazur). The unit plays an important role in the developments around the conjecture of Harris and Venkatesh.