Department of Mathematics

Algebra

  •  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 (wakepres@msu.edu)

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.

 

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