Let `$N$`

be a prime number `$>3$`

. Mazur has defined, from the theory of modular forms, a unit `$u$`

in `$\mathbb{Z}/N$`

. This unit turned out to be, up to a `$6$`

-th 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 `$\mathbb{Q}(\sqrt{-N})$`

.
For every prime number `$p$`

dividing `$N-1$`

, It is important to determine when `$u$`

is a `$p$`

-th power in `$(\mathbb{Z}/N)^*$`

.
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 `$\mathbb{Z}_p$`

(the original motivation of Mazur). The unit plays an important role in the developments around the conjecture of Harris and Venkatesh.

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Last updated: 17 May 2024