### Abstract

Lei ℓ be an odd prime. Let K be a field of characteristic zero with algebraic closure K_{a}. Let n, in ≥ 4 be integers that are not divisible by ℓ Let f(x), h(x) ∈ K [x] be irreducible separable polynomials of degree n and m respectively. Suppose that the Galois group Gal(f) of f acts doubly transitively on the set R_{f} of roots of f and that Gal(/z) acts doubly transitively on R_{h} as well. Let J(C _{f,ℓ}) and J(C_{h, ℓ}) be the Jacobians of the superelliptic curves C_{f,ℓ}: y_{ℓ} = f(x) and C _{h,ℓ}: y^{ℓ} = h(x) respectively. We prove that J(C_{f,ℓ}) and J(C_{hℓ}) are not isogenous over K _{a} if the splitting fields of / and h are linearly disjoint over K and K contains a primitive ℓth root of unity.

Original language | English (US) |
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Pages (from-to) | 537-554 |

Number of pages | 18 |

Journal | Mathematische Zeitschrift |

Volume | 253 |

Issue number | 3 |

DOIs | |

State | Published - Jul 1 2006 |

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### All Science Journal Classification (ASJC) codes

- Mathematics(all)

### Cite this

*Mathematische Zeitschrift*,

*253*(3), 537-554. https://doi.org/10.1007/s00209-005-0921-7

}

*Mathematische Zeitschrift*, vol. 253, no. 3, pp. 537-554. https://doi.org/10.1007/s00209-005-0921-7

**Non-isogenous superelliptic Jacobians.** / Zarhin, Yuri G.

Research output: Contribution to journal › Article

TY - JOUR

T1 - Non-isogenous superelliptic Jacobians

AU - Zarhin, Yuri G.

PY - 2006/7/1

Y1 - 2006/7/1

N2 - Lei ℓ be an odd prime. Let K be a field of characteristic zero with algebraic closure Ka. Let n, in ≥ 4 be integers that are not divisible by ℓ Let f(x), h(x) ∈ K [x] be irreducible separable polynomials of degree n and m respectively. Suppose that the Galois group Gal(f) of f acts doubly transitively on the set Rf of roots of f and that Gal(/z) acts doubly transitively on Rh as well. Let J(C f,ℓ) and J(Ch, ℓ) be the Jacobians of the superelliptic curves Cf,ℓ: yℓ = f(x) and C h,ℓ: yℓ = h(x) respectively. We prove that J(Cf,ℓ) and J(Chℓ) are not isogenous over K a if the splitting fields of / and h are linearly disjoint over K and K contains a primitive ℓth root of unity.

AB - Lei ℓ be an odd prime. Let K be a field of characteristic zero with algebraic closure Ka. Let n, in ≥ 4 be integers that are not divisible by ℓ Let f(x), h(x) ∈ K [x] be irreducible separable polynomials of degree n and m respectively. Suppose that the Galois group Gal(f) of f acts doubly transitively on the set Rf of roots of f and that Gal(/z) acts doubly transitively on Rh as well. Let J(C f,ℓ) and J(Ch, ℓ) be the Jacobians of the superelliptic curves Cf,ℓ: yℓ = f(x) and C h,ℓ: yℓ = h(x) respectively. We prove that J(Cf,ℓ) and J(Chℓ) are not isogenous over K a if the splitting fields of / and h are linearly disjoint over K and K contains a primitive ℓth root of unity.

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U2 - 10.1007/s00209-005-0921-7

DO - 10.1007/s00209-005-0921-7

M3 - Article

AN - SCOPUS:33646478149

VL - 253

SP - 537

EP - 554

JO - Mathematische Zeitschrift

JF - Mathematische Zeitschrift

SN - 0025-5874

IS - 3

ER -