Cyclic covers of the projective line, their jacobians and endomorphisms

Details Cyclic covers of the projective line, their jacobians and endomorphisms Cyclic covers of the projective line, their jacobians and endomorphisms

2000-08-16

arxiv.org/abs/math/0008134v3

Mathematics

Algebraic Geometry

LaTeX2e, 17 pages Some typos were corrected

Scientific paper

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.

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