Department of Mathematics


  •  Anna Medvedovsky, Boston University
  •  Counting modular forms with fixed mod-p Galois representation and Atkin-Lehner-at-p eigenvalue
  •  11/18/2020
  •  4:00 PM - 5:00 PM
  •  Online (virtual meeting)
  •  Preston Wake (

Work in progress joint with Samuele Anni and Alexandru 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 and Atkin-Lehner-at-p sign, generalizing both recent results of Martin (no residual representation constraint) and rhobar-dimension-counting formulas of Jochnowitz and Bergdall-Pollack. One challenge is the tension between working modulo p and the need to invert p when working with the Atkin-Lehner involution. To address this, we use the trace formula to establish up-to-semisimplifcation isomorphisms between certain mod-p Hecke modules (namely, refinements of weight-graded pieces of spaces of mod-p forms) by exhibiting ever-deeper congruences between traces of prime-power Hecke operators acting on characteristic-zero Hecke modules. This last technique is new and combinatorial in nature; it relies on a theorem discovered by the authors and beautifully proved by Gessel, and may be of independent interest.



Department of Mathematics
Michigan State University
619 Red Cedar Road
C212 Wells Hall
East Lansing, MI 48824

Phone: (517) 353-0844
Fax: (517) 432-1562

College of Natural Science