Loïc Merel
Institut de Mathematiques de Jussieu, Paris, France
October 12, 2023
About the unit $1^1 \cdot 2^2 \cdot 3^3 \cdot 4^4 \cdots ((N-1)/2)^{(N-1)/2}$ modulo a prime number $N$: 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.
Institut de Mathematiques de Jussieu, Paris, France
October 12, 2023
About the unit $1^1 \cdot 2^2 \cdot 3^3 \cdot 4^4 \cdots ((N-1)/2)^{(N-1)/2}$ modulo a prime number $N$: 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.