### Abstract

Let K be a field of characteristic zero, n ≥ 5 an integer, f(x) an irreducible polynomial over K of degree n, whose Galois group contains a doubly transitive simple non-abelian group. Let p be an odd prime, ℤ[ζ _{p}] the ring of integers in the pth cyclotomic field, C_{f, p} : y^{p} = f(x) the corresponding superelliptic curve and J(C _{f, p}) its jacobian. Assuming that either n = p + 1 or p does not divide n(n - 1), we prove that the ring of all endomorphisms of J(C _{f, p}) coincides with ℤ[ζ_{p}]. The same is true if n = 4, the Galois group of f(x) is the full symmetric group S_{4} and K contains a primitive pth root of unity.

Original language | English (US) |
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Pages (from-to) | 691-707 |

Number of pages | 17 |

Journal | Mathematische Zeitschrift |

Volume | 261 |

Issue number | 3 |

DOIs | |

State | Published - Mar 2009 |

### All Science Journal Classification (ASJC) codes

- Mathematics(all)

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## Cite this

Zarhin, Y. G. (2009). Endomorphisms of superelliptic jacobians.

*Mathematische Zeitschrift*,*261*(3), 691-707. https://doi.org/10.1007/s00209-008-0342-5