Decomposing Jacobians Via Galois covers

Abstract

Let $\varphi :X\to Y$ be a (possibly ramified) cover, with X and Y of strictly positive genus. We develop tools to identify the Prym variety of $\varphi$, up to isogeny, as the Jacobian of a quotient curve C of the Galois closure of a composition $X\to Y\to \mathbb{P}{}^1$ of $\varphi$ with a well-chosen map $Y\to \mathbb{P}{}^1$ that identifies branch points of $\varphi$. To our knowledge, this method recovers all previously obtained descriptions of Prym varieties as Jacobians. It also finds new decompositions, and for some of these, including one where X has genus 3, Y has genus 1 and $\varphi$ is a degree 3 map totally ramified over 2 points, we find an algebraic equation of the curve C.

2010 Mathematics Subject Classification.:

• 14H40
• 14H45
• 14L30
• 11R32

Keywords:

• algebraic curves
• plane quartics
• low genus
• explicit methods
• Galois theory
• Jacobians
• decomposition

Acknowledgments

The fourth author was moreover supported by a Juniorprofessurenprogramm of the Science Ministry of Baden-Württemberg. Part of the work for this paper was carried out at the workshop “Arithmetic of Curves”, held in Baskerville Hall in August 2018. We would like to thank the organizers Alexander Betts, Tim Dokchitser, Vladimir Dokchitser, Céline Maistret and Beryl Stapleton, as well as the Baskerville Hall staff, for providing a great opportunity to concentrate on this project.

Declaration of Interest

No potential conflict of interest was reported by the author(s).

Additional information

Funding

We acknowledge support from the PICS JADERE (281036) from the French CNRS and the second and third authors acknowledge support from PHC Bosphorus 39652NB - TÜBİTAK 117F274.