### Abstract

We prove that the jacobian of a hyperelliptic curve y^{2} = (x - t)h(x) has no non-trivial endomorphisms over an algebraic closure of the ground field K of characteristic zero if t ∈ K and the Galois group of the polynomial h(x) over K is an alternating or symmetric group on deg(h) letters and deg(h) is an even number greater than 8. (The case of odd deg(h) > 3 follows easily from previous results of the author.)

Original language | English (US) |
---|---|

Pages (from-to) | 24-54 |

Number of pages | 31 |

Journal | Proceedings of the London Mathematical Society |

Volume | 100 |

Issue number | 1 |

DOIs | |

State | Published - Jan 1 2010 |

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### All Science Journal Classification (ASJC) codes

- Mathematics(all)

### Cite this

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*Proceedings of the London Mathematical Society*, vol. 100, no. 1, pp. 24-54. https://doi.org/10.1112/plms/pdp020

**Families of absolutely simple hyperelliptic jacobians.** / Zarkhin, Yuriy G.

Research output: Contribution to journal › Article

TY - JOUR

T1 - Families of absolutely simple hyperelliptic jacobians

AU - Zarkhin, Yuriy G.

PY - 2010/1/1

Y1 - 2010/1/1

N2 - We prove that the jacobian of a hyperelliptic curve y2 = (x - t)h(x) has no non-trivial endomorphisms over an algebraic closure of the ground field K of characteristic zero if t ∈ K and the Galois group of the polynomial h(x) over K is an alternating or symmetric group on deg(h) letters and deg(h) is an even number greater than 8. (The case of odd deg(h) > 3 follows easily from previous results of the author.)

AB - We prove that the jacobian of a hyperelliptic curve y2 = (x - t)h(x) has no non-trivial endomorphisms over an algebraic closure of the ground field K of characteristic zero if t ∈ K and the Galois group of the polynomial h(x) over K is an alternating or symmetric group on deg(h) letters and deg(h) is an even number greater than 8. (The case of odd deg(h) > 3 follows easily from previous results of the author.)

UR - http://www.scopus.com/inward/record.url?scp=73649111385&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=73649111385&partnerID=8YFLogxK

U2 - 10.1112/plms/pdp020

DO - 10.1112/plms/pdp020

M3 - Article

VL - 100

SP - 24

EP - 54

JO - Proceedings of the London Mathematical Society

JF - Proceedings of the London Mathematical Society

SN - 0024-6115

IS - 1

ER -