A modular construction of unramified p-extensions of $\Q(N^{1/p})$

Title:A modular construction of unramified p-extensions of $\Q(N^{1/p})$

Speaker:Jacky Lang (Philadelfia)

Organizer: Emmanuel Lecouturier (BIMSA)

Time:9:00-10:00, Nov. 19, 2021

Venue:BIMSA 1118

Zoom ID: 849 963 1368   Password: YMSC

Abstract:

In Mazur's seminal work on the Eisenstein ideal, he showed that when N and p > 3 are primes, there is a weight 2 cusp form of level N congruent to the unique weight 2 Eisenstein series of level N if and only N = 1 mod p. Calegari--Emerton, Merel, Lecouturier, and Wake--Wang-Erickson have work that relates these cuspidal-Eisenstein congruences to the p-part of the class group of $\Q(N^{1/p})$. Calegari observed that when N = -1 mod p, one can use Galois cohomology and some ideas of Wake--Wang-Erickson to show that p divides the class group of $\Q(N^{1/p})$. He asked whether there is a way to directly construct the relevant degree p everywhere unramified extension of $\Q(N^{1/p})$ in this case. After discussing some of this background, I will report of work with Preston Wake in which we give a positive answer to this question using cuspidal-Eisenstein congruences at prime-square level.