Non-isogenous superelliptic Jacobians

Research output: Contribution to journalArticle

4 Citations (Scopus)

Abstract

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: y = f(x) and C h,ℓ: y = 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.

Original languageEnglish (US)
Pages (from-to)537-554
Number of pages18
JournalMathematische Zeitschrift
Volume253
Issue number3
DOIs
StatePublished - Jul 1 2006

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Splitting Field
Roots of Unity
Galois group
Divisible
Disjoint
Closure
Linearly
Odd
Roots
Curve
Polynomial
Integer
Zero

All Science Journal Classification (ASJC) codes

  • Mathematics(all)

Cite this

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abstract = "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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Non-isogenous superelliptic Jacobians. / Zarhin, Yuri G.

In: Mathematische Zeitschrift, Vol. 253, No. 3, 01.07.2006, p. 537-554.

Research output: Contribution to journalArticle

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T1 - Non-isogenous superelliptic Jacobians

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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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