Endomorphisms of ordinary superelliptic jacobians

TL;DR

This paper investigates the endomorphism rings of superelliptic Jacobians over fields of prime characteristic, showing they are as small as possible under certain conditions, which advances understanding of their algebraic structure.

Contribution

It proves that the endomorphism ring of the Jacobian of certain superelliptic curves is exactly the ring of cyclotomic integers, extending knowledge of their algebraic properties in prime characteristic.

Findings

Endomorphism ring equals Z[ζ_l] for ordinary Jacobians

Results exclude the case (l,n) = (5,5)

Provides conditions under which the endomorphism ring is determined

Abstract

Let $K$ be a field of prime characteristic $p$, $n>4 $ an integer, $f(x)$ an irreducible polynomial over $K$ of degree $n$, whose Galois group is either the full symmetric group $S_n$ or the alternating group $A_n$. Let $l$ be an odd prime different from $p$, $Z[\zeta_l]$ the ring of integers in the $l$th cyclotomic field, $C_{f,l}:y^l=f(x)$ the corresponding superelliptic curve and $J(C_{f,l})$ its jacobian. We prove that the ring of all endomorphisms of $J(C_{f,l})$ coincides with $Z[\zeta_l]$ if $J(C_{f,l})$ is an ordinary abelian variety and $(l,n)\ne (5,5)$.