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- W4384652303 abstract "For $q$ a power of a prime $p$, we study gamma factors of representations of $GL_n(mathbb{F}_q)$ over an algebraically closed field $k$ of positive characteristic $ell neq p$. We show that the reduction mod $ell$ of the gamma factor defined in characteristic zero fails to satisfy the analogue of the local converse theorem of Piatetski-Shapiro. To remedy this, we construct gamma factors valued in arbitrary $mathbb{Z}[1/p, zeta_p]$-algebras $A$, where $zeta_p$ is a primitive $p$-th root of unity, for Whittaker-type representations $rho$ and $pi$ of $GL_n(mathbb{F}_q)$ and $GL_m(mathbb{F}_q)$ over $A$. We let $P(pi)$ be the projective envelope of $pi$ and let $R(pi)$ be its endomorphism ring and define new gamma factors $widetildegamma(rho times pi) = gamma((rhootimes_kR(pi)) times P(pi))$, which take values in the local Artinian $k$-algebra $R(pi)$. We prove a converse theorem for cuspidal representations using the new gamma factors. When $n=2$ and $m=1$ we construct a different ``new'' gamma factor $gamma^{ell}(rho,pi)$, which takes values in $k$ and satisfies a converse theorem." @default.
- W4384652303 created "2023-07-19" @default.
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- W4384652303 date "2023-07-14" @default.
- W4384652303 modified "2023-09-26" @default.
- W4384652303 title "Mod $ell$ gamma factors and a converse theorem for finite general linear groups" @default.
- W4384652303 doi "https://doi.org/10.48550/arxiv.2307.07593" @default.
- W4384652303 hasPublicationYear "2023" @default.
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