The endomorphism rings of jacobians of cyclic covers of the projective line

Details The endomorphism rings of jacobians of cyclic covers of the projective line The endomorphism rings of jacobians of cyclic covers of the projective line

2001-03-28

arxiv.org/abs/math/0103203v2

Mathematics

Algebraic Geometry

LaTeX2e, 14 pages. The paper will appear in Math. Proc. Cambridge Philos. Soc

Scientific paper

Suppose K is a field of characteristic 0, $K_a$ is its algebraic closure, p is an odd prime. Suppose, $f(x) \in K[x]$ is a polynomial of degree $n \ge 5$ without multiple roots. Let us consider a curve $C: y^p=f(x)$ and its jacobian J(C). It is known that the ring End(J(C)) of all $K_a$-endomorphisms of J(C) contains the ring $Z[\zeta_p]$ of integers in the pth cyclotomic field (generated by obvious automorphisms of C). We prove that $End(J(C))=Z[\zeta_p]$ if the Galois group of f over K is either the symmetric group $S_n$ or the alternating group $A_n$.

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