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.