I‑AIMS Institute for Artificial Intelligence
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Isogenous components of Jacobian surfaces

Lubjana Beshaj, Artur Elezi, Tanush Shaska

I-AIMS paper 2017-2 · Published in Eur. J. Math. 6 (2020), 1276–1302 · 2020

Let C be a genus 2 curve defined over a field K, char K = p ≥ 0, and Jac(C, ι) its Jacobian, where ι is the principal polarization of Jac(C) attached to C. Assume that Jac(C) is (n,n)-geometrically reducible with E₁ and E₂ its elliptic components. We prove that there are only finitely many curves C (up to isomorphism) defined over K such that E₁ and E₂ are N-isogenous for n = 2 and N = 2, 3, 5, 7 with Aut(Jac C) ≅ V₄ or n = 2, N = 3, 5, 7 with Aut(Jac C) ≅ D₄. The same holds if n = 3 and N = 5. Furthermore, we determine the Kummer and the Shioda-Inose surfaces for the above Jac C and show how such results in positive characteristic p > 2 suggest nice applications in cryptography.

MSC classes: Primary 11G10; Secondary 14K02, 14H40

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