How can one describe all irreducible representations of a finite group? One usually first learns about the character of a representation, which attaches a number to each conjugacy class. Can we tell apart representations using even fewer invariants? For the groups $GL_n$ over a finite field, the local converse theorem of Piatetski-Shapiro says that $\Gamma$ factors are such invariants by fully determining complex representations. But what if the representations are valued in a field of positive characteristic? For fields of characteristic $\ell$ coprime to $q$, I will report on the development of a mod $\ell$ Gamma factor. After showing that the naive reduction modulo $\ell$ of the $\Gamma$ factor is not always complete invariant, I will construct a candidate for a replacement, and describe some ongoing extensions to $GL_n(F_q)$. This is joint work in progress with Jacksyn Bakeberg, Heidi Goodson, Ashwin Iyengar, Gil Moss, and Robin Zhang.
Towards a converse theorem for mod $\ell$ gamma factors. (prerecorded)
Mathilde Gerbelli-Gauthier, IASAuthors: Jacksyn Bakeberg, Mathilde Gerbelli-Gauthier, Heidi Goodson, Ashwin Iyengar, Gil Moss, Robin Zhang
2022 AWM Research Symposium
Rethinking Number Theory