Cyclic covers of the projective line, their jacobians and endomorphisms (Q2781430)

We study the endomorphism ring \(\text{End} (J(C))\) 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 coefficients 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 \(\text{End} (J(C))\) is the ring of integers \(\mathbb{Z} [\zeta_p]\) in the \(p\)-th cyclotomic field \(\mathbb{Q} [\zeta_p]\) if \(p\) is a Fermat prime (e.g., \(p=3, 5, 17, 257)\). Notice that recently the author extended this result to the case of an arbitrary odd prime \(p\) [\textit{Yu. G. Zarkhin}, ``The endomorphism rings of jacobians of cyclic covers of the projective line'' (\url{http://xxx.lanl.gov/abs/math.AG/0103203})].NEWLINENEWLINENEWLINEThe case of positive characteristic \(\neq p\) (under additional assumptions that \(n>8\) and \(p\mid n)\) is discussed in another one of the author's papers [\textit{Yu. G. Zarkhin}, ``Endomorphism rings of certain jacobians'' (to appear)] there he proves that \(\mathbb{Z} [\zeta_p]\) coincides with its own centralizer in the endomorphism ring of the jacobian.