Gluing curves of genus 1 and 2 along their 2-torsion

Hanselman, Jeroen; Schiavone, Sam; Sijsling, Jeroen

doi:10.1090/mcom/3627

Gluing curves of genus 1 and 2 along their 2-torsion
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by Jeroen Hanselman, Sam Schiavone and Jeroen Sijsling HTML | PDF

Math. Comp. 90 (2021), 2333-2379 Request permission

Abstract:

Let $X$ (resp. $Y$) be a curve of genus $1$ (resp. $2$) over a base field $k$ whose characteristic does not equal $2$. We give criteria for the existence of a curve $Z$ over $k$ whose Jacobian is up to twist $(2,2,2)$-isogenous to the products of the Jacobians of $X$ and $Y$. Moreover, we give algorithms to construct the curve $Z$ once equations for $X$ and $Y$ are given. The first of these is based on interpolation methods involving numerical results over $\mathbb {C}$ that are proved to be correct over general fields a posteriori, whereas the second involves the use of hyperplane sections of the Kummer variety of $Y$ whose desingularization is isomorphic to $X$. As an application, we find a twist of a Jacobian over $\mathbb {Q}$ that admits a rational $70$-torsion point.

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