Curves of genus 2 with (n, n)--decomposable Jacobians
Let C be a curve of genus 2 and ψ1: C − → E 1 a map of degree n, from C to an elliptic curveE1 , both curves defined over C. This map induces a degree n map φ1:P1 − → P 1 which we call a Frey–Kani covering. We determine all possible ramifications for φ1. If ψ1:C − → E 1 is maximal then there exists a maximal map ψ2: C − → E 2 , of degree n, to some elliptic curveE2 such that there is an isogeny of degree n2from the JacobianJC to E1 × E2. We say thatJC 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.
Shaska, T. (2001). Curves of genus 2 with (N, N) decomposable Jacobians. Journal of Symbolic Computation, 31(5), 603-617. https://doi.org/10.1006/jsco.2001.0439
Genus two curves
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