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Curves of genus 2 with (n,n)-decomposable Jacobians

Tanush Shaska

I-AIMS paper 2000-1 · Published in J. Symbolic Comput. 31 (2001), 603–617 · 2001

Let C be a curve of genus 2 and ψ₁: C → E₁ a map of degree n, from C to an elliptic curve E₁, both curves defined over ℂ. This map induces a degree n map ϕ₁: ℙ¹ → ℙ¹ which we call a Frey-Kani covering. We determine all possible ramifications for ϕ₁. If ψ₁: C → E₁ is maximal then there exists a maximal map ψ₂: C → E₂, of degree n, to some elliptic curve E₂ such that there is an isogeny of degree n² from the Jacobian J_C to E₁ × E₂. We say that J_C is (n,n)-decomposable. If the degree n is odd the pair (ψ₂, E₂) is canonically determined. For n = 3, 5, and 7, we give arithmetic examples of curves whose Jacobians are (n,n)-decomposable.

MSC classes: Primary 14H40; Secondary 14H52

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