BEGIN:VCALENDAR VERSION:2.0 PRODID:researchseminars.org CALSCALE:GREGORIAN X-WR-CALNAME:researchseminars.org BEGIN:VEVENT SUMMARY:Jaclyn Lang (Oxford) DTSTART:20201210T190000Z DTEND:20201210T200000Z DTSTAMP:20260423T024553Z UID:UCSBsga/3 DESCRIPTION:Title: Eisenstein congruences at prime-square level and an application to class n umbers\nby Jaclyn Lang (Oxford) as part of UCSB Seminar on Geometry an d Arithmetic\n\n\nAbstract\nIn Mazur's seminal work on the Eisenstein idea l\, 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 seri es of level $N$ if and only if $N = 1$ mod $p$. Calegari--Emerton\, Lecout urier\, and Wake--Wang-Erickson have work that relates these cuspidal-Eise nstein congruences to the $p$-part of the class group of $\\mathbb{Q}(N^{1 /p})$. Calegari observed that when $N = -1$ mod $p$\, one can use Galois c ohomology and some ideas of Wake--Wang-Erickson to show that $p$ divides t he class group of $\\mathbb{Q}(N^{1/p})$. He asked whether there is a way to directly construct the relevant degree $p$ everywhere unramified extens ion of $\\mathbb{Q}(N^{1/p})$ in this case. After discussing some of this background\, I will report of work in progress with Preston Wake in which we give a positive answer to this question using cuspidal-Eisenstein congr uences at prime-square level.\n LOCATION:https://researchseminars.org/talk/UCSBsga/3/ END:VEVENT END:VCALENDAR