Decomposition of Jacobians by Prym Varieties

This monograph studies the decompositions of the Jacobian of a smooth projective curve into a product of abelian subvarieties, using Prym varieties of pairs of subcovers to give new proofs of classical constructions.

Format: Paperback / softback
Length: 251 pages
Publication date: 25 November 2022
Publisher: Springer International Publishing AG

This monograph delves into the intricate analysis of decompositions of the Jacobian of a smooth projective curve, driven by the action of a finite group. The authors present a comprehensive theorem that applies in various scenarios and is applied to several groups, including groups of small order and certain series of groups. These decompositions often involve Prym varieties of pairs of subcovers, resulting in novel proofs for classical constructions such as the bigonal and trigonal constructions. Furthermore, several isogenies between Prym varieties are discovered, shedding light on the connections between these mathematical structures.

Introduction:
The study of decompositions of the Jacobian of a smooth projective curve, induced by the action of a finite group, has been an active area of research in mathematics. These decompositions provide insights into the geometric properties and arithmetic behavior of the curve and its associated group. In this monograph, we focus on the analysis of such decompositions, particularly when the group is finite.

Theorem:
The main theorem of the monograph is a general theorem that describes how to decompose the Jacobian of a smooth projective curve into a product of abelian subvarieties. This theorem works in many cases and provides a framework for understanding the decomposition process. The authors then apply this theorem to several groups, including groups of small order and some series of groups.

Applications:
The authors demonstrate the usefulness of their theorem by applying it to several groups. For example, they show how to decompose the Jacobian of a smooth projective curve over a finite field into a product of Prym varieties of pairs of subcovers. This decomposition has several advantages, including the ability to generalize to more general situations. Additionally, the authors obtain new proofs for classical constructions, such as the bigonal and trigonal constructions, which have the advantage of generalizing to more general situations.

Isogenies:
As a result of the decomposition process, several isogenies between Prym varieties are discovered. These isogenies provide new insights into the relationships between different components of the Jacobian and can be used to study the geometry of the curve and its associated group.

Conclusion:
This monograph provides a comprehensive study of decompositions of the Jacobian of a smooth projective curve, induced by the action of a finite group. The theorem presented is a general framework that works in many cases and is applied to several groups. The applications demonstrate the usefulness of the theorem and the discovery of isogenies between Prym varieties. The study of decompositions of the Jacobian continues to be an active area of research, with many open questions and potential applications.

Weight: 415g
Dimension: 235 x 155 (mm)
ISBN-13: 9783031101441
Edition number: 1st ed. 2022