Cyclic covers of the projective line, their jacobians and endomorphisms

TL;DR

This paper investigates the endomorphism rings of Jacobians of cyclic covers of the projective line, establishing conditions under which these rings are the integers in cyclotomic fields, especially for Fermat primes.

Contribution

It proves that for certain cyclic covers with specific Galois groups, the Jacobian's endomorphism ring is the ring of integers in the pth cyclotomic field, extending previous results to new cases.

Findings

Endomorphism ring is the ring of integers in the pth cyclotomic field for Fermat primes.

Results apply to curves with Galois group S_n or A_n over a field K.

Similar results are extended from hyperelliptic to more general cyclic covers.

Abstract

We study the endomorphism ring $End(J(C))$ of the complex jacobian $J(C)$ of a curve $y^p=f(x)$ where $p$ is an odd prime and $f(x)$ is a polynomial with complex coefficiens of degree $n>4$ and without multiple roots. Assume that all the coefficients of $f$ lie in a (sub)field $K$ and the Galois group of $f$ over $K$ is either the full symmetric group $S_n$ or the alternating group $A_n$. Then we prove that $End(J(C))$ is the ring of integers in the in the $p$th cyclotomic field, if $p$ is a Fermat prime (e.g., $p=3,5,17,257$). Similar results for $p=2$ (the case of hyperelliptic curves) were obtained by the author in Math. Res. Lett. 7(2000), 123--132.