BEGIN:VCALENDAR VERSION:2.0 PRODID:researchseminars.org CALSCALE:GREGORIAN X-WR-CALNAME:researchseminars.org BEGIN:VEVENT SUMMARY:Anna Medvedovsky DTSTART:20210526T150000Z DTEND:20210526T160000Z DTSTAMP:20260418T063718Z UID:LNTS/39 DESCRIPTION:Title: Co unting modular forms with fixed mod-p Galois representation and Atkin-Lehn er-at-p eigenvalue\nby Anna Medvedovsky as part of London number theor y seminar\n\n\nAbstract\nWork in progress joint with Samuele Anni and Alex andru Ghitza. For N prime to p\, we count the number of classical modular forms of level Np and weight k with fixed residual Galois representation a nd Atkin-Lehner-at-p sign\, generalizing both recent results of Martin gen eralizing work of Wakatsuki (no residual representation constraint) and th e rhobar-dimension-counting formulas of Bergdall-Pollack and Jochnowitz. T o resolve tension between working mod p and the need to invert p\, we use the trace formula to establish up-to-semisimplifcation isomorphisms betwee n certain mod-p Hecke\nmodules (namely\, refinements of the weight-filtrat ion graded pieces W_k) by exhibiting ever-deeper congruences between trace s of prime-power Hecke operators acting on characteristic-zero Hecke\nmodu les. This last technique is new\, purely algebraic\, and may be of indepen dent interest\; it relies on a combinatorial theorem whose proof benefited from a beautiful boost from Gessel.\n LOCATION:https://researchseminars.org/talk/LNTS/39/ END:VEVENT END:VCALENDAR