Non-isogenous superelliptic Jacobians |
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Authors: | Yuri G Zarhin |
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Institution: | (1) Department of Mathematics, Pennsylvania State University, University Park, PA 16802, USA |
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Abstract: | Let ℓ be an odd prime. Let K be a field of characteristic zero with algebraic closure Ka. Let n, m ≥ 4 be integers that are not divisible by ℓ. Let f(x), h(x) ∈ Kx] 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 of roots of f and that Gal(h) acts doubly transitively on as well. Let J(Cf,ℓ) and J(Ch,ℓ) be the Jacobians of the superelliptic curves Cf,ℓ:yℓ=f(x) and Ch,ℓ:yℓ=h(x) respectively. We prove that J(Cf,ℓ) and J(Ch,ℓ) are not isogenous over Ka if the splitting fields of f and h are linearly disjoint over K and K contains a primitive ℓth root of unity. |
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Keywords: | Superelliptic Jacobians Homomorphisms of abelian varieties Permutational representations |
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