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Isogenous components of Jacobian surfaces. (English) Zbl 1469.11180

Summary: Let \(\mathscr{C}\) be a genus 2 curve defined over a field \(K\), \(\mathrm{char} K = p \geqslant 0\), and \(\mathrm{Jac}(\mathscr{C},\iota)\) its Jacobian, where \(\iota\) is the principal polarization of \(\mathrm{Jac}(\mathscr{C}\)) attached to \(\mathscr{C}\). Assume that \(\mathrm{Jac}(\mathscr{C}\)) is (\(n, n\))-geometrically reducible with \(E_1\) and \(E_2\) its elliptic components. We prove that there are only finitely many curves \(\mathscr{C}\) (up to isomorphism) defined over \(K\) such that \(E_1\) and \(E_2\) are \(N\)-isogenous for \(n = 2\) and \(N = 2,3,5,7\) with \(\operatorname{Aut}(\mathrm{Jac}(\mathscr{C})) \cong V_4\) or \(n = 2\), \(N = 3,5,7\) with \(\operatorname{Aut}(\mathrm{Jac}(\mathscr{C})) \cong D_4\). The same holds if \(n = 3\) and \(N = 5\). Furthermore, we determine the Kummer and Shioda-Inose surfaces for the above \(\mathrm{Jac}(\mathscr{C}\)) and show how such results in positive characteristic \(p > 2\) suggest nice applications in cryptography.

MSC:

11G10 Abelian varieties of dimension \(> 1\)
14K02 Isogeny
14H40 Jacobians, Prym varieties
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