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BIMSA-YMSC Tsinghua Number Theory Seminar
A modular construction of unramified p-extensions of \Q(N^{1/p})
A modular construction of unramified p-extensions of \Q(N^{1/p})
Organizers
Yueke Hu
, Emmanuel Lecouturier
,
Cezar Lupu
Speaker
Jaclyn Lang
Time
Friday, November 19, 2021 9:00 AM - 10:00 AM
Venue
1118
Online
Zoom 849 963 1368
(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.