BEGIN:VCALENDAR VERSION:2.0 PRODID:researchseminars.org CALSCALE:GREGORIAN X-WR-CALNAME:researchseminars.org BEGIN:VEVENT SUMMARY:Samuel Le Fourn (University of Grenoble) DTSTART:20260211T160000Z DTEND:20260211T170000Z DTSTAMP:20260528T081214Z UID:LNTS/183 DESCRIPTION:Title: I ntegrality of rational points of star quotients X_0(N)^* for N non-squaref ree\nby Samuel Le Fourn (University of Grenoble) as part of London num ber theory seminar\n\nLecture held in Room 505\, UCL Maths building\, 25 G ordon Street.\n\nAbstract\nModular curves X_0(N) parametrise elliptic curv es E endowed with a cyclic torsion subgroup of order N\, which makes thei r rational points related to Galois representation associated to torsion p oints of elliptic curves. A famous result of Mazur implies that X_0(N) has only "trivial" rational points for N>163 prime\, and it leads to solving the problem for a general N.\nFor N squarefree\, the method used by Mazur (based on formal immersions) cannot work as there should be (under BSD con jecture) no abelian subvariety of the jacobian of X_0(N)^* of Mordell-Weil rank zero. Surprisingly though\, when N is not squarefree\, there is jus t enough leeway to allow for such subvarieties to exist. In this talk base d on a joint work with Sachi Hashimoto and Timo Keller\, I will explain ho w to make this idea work in practice to prove integrality\n(i.e. potential ly good reduction of elliptic curves associated to non-cuspidal rational p oints) for almost all non-squarefree levels N\, and how some unexpected ra tional points have been found for some sporadic levels N."\n LOCATION:https://researchseminars.org/talk/LNTS/183/ END:VEVENT END:VCALENDAR