BEGIN:VCALENDAR VERSION:2.0 PRODID:researchseminars.org CALSCALE:GREGORIAN X-WR-CALNAME:researchseminars.org BEGIN:VEVENT SUMMARY:Naomi Sweeting (Harvard University) DTSTART:20210204T220000Z DTEND:20210204T230000Z DTSTAMP:20260423T022805Z UID:UCSD_NTS/23 DESCRIPTION:Title: Kolyvagin's conjecture and higher congruences of modular forms\nby N aomi Sweeting (Harvard University) as part of UCSD number theory seminar\n \nLecture held in normally APM 7321\, currently online.\n\nAbstract\nGiven an elliptic curve E\, Kolyvagin used CM points on modular curves to cons truct a system of classes valued in the Galois cohomology of the torsion p oints of E. Under the conjecture that not all of these classes vanish\, h e gave a description for the Selmer group of E. This talk will report on recent work proving new cases of Kolyvagin's conjecture. The methods foll ow in the footsteps of Wei Zhang\, who used congruences between modular fo rms to prove Kolyvagin's conjecture under some technical hypotheses. We re move many of these hypotheses by considering congruences modulo higher po wers of p. The talk will explain the difficulties associated with higher congruences of modular forms and how they can be overcome. I will also pro vide an introduction to the conjecture and its consequences\, including a 'converse theorem': algebraic rank one implies analytic rank one.\n\npre-t alk at 1:30\n LOCATION:https://researchseminars.org/talk/UCSD_NTS/23/ END:VEVENT END:VCALENDAR