AAG Seminar: "On $\ell$-torsion of superelliptic Jacobians over finite fields"

Abstract: For a prime $\ell\geq 3$, we study the $\ell$-torsion subgroup of Jacobians $J$ of curves $y^\ell = f(t)$ over a finite field $\mathbf{F}_q$. When $f(t)$ is a monic irreducible polynomial and $q$ and $d:=\deg(f)$ are both coprime to $\ell$, we give an upper bound on the $\ell$-rank of $J(\mathbf{F}_q)$ that depends only on $\ell$, $q$ and $d$. We also prove lower bounds using tools from Galois cohomology, and we find congruence conditions that can often be used to determine the $\ell$-rank exactly when the upper bound alone is not sufficient. This is joint work with Wanlin Li and Eric Stubley.

Host: Wanlin Li