Gluing curves of genus 1 and 2 along their 2-torsion
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 involves the use of hyperplane sections of the Kummer variety of $Y$ whose desingularization is isomorphic to $X$, whereas the second is based on interpolation methods involving numerical results over $\mathbb{C}$ that are proved to be correct over general fields a posteriori. As an application, we find a twist of a Jacobian over $\mathbb{Q}$ that admits a rational 70-torsion point.
- Publication:
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arXiv e-prints
- Pub Date:
- May 2020
- DOI:
- 10.48550/arXiv.2005.03587
- arXiv:
- arXiv:2005.03587
- Bibcode:
- 2020arXiv200503587H
- Keywords:
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- Mathematics - Algebraic Geometry;
- Mathematics - Number Theory;
- 14H40;
- 14H25
- E-Print:
- 41 pages