Vol. 4, No. 1, 2020

Download this article
Download this article For screen
For printing
Recent Volumes
5: Gauge Theory and Low-Dimensional Topology
4: ANTS XIV
3: Hillman: Poincaré Duality
2: ANTS XIII
1: ANTS X
The Open Book Series
All Volumes
 
About the Series
Ethics Statement
Purchase Printed Copies
Author Index
 
ISSN (electronic): 2329-907X
ISSN (print): 2329-9061
 
MSP Books and Monographs
Other MSP Publications
Genus 3 hyperelliptic curves with CM via Shimura reciprocity

Bogdan Adrian Dina and Sorina Ionica

Vol. 4 (2020), No. 1, 161–178
Abstract

Up to isomorphism, every three-dimensional simple principally polarized abelian variety over is the Jacobian of a smooth projective curve of genus 3. Furthermore, this curve is either a hyperelliptic curve or a plane quartic. To define hyperelliptic class polynomials, we note that given a hyperelliptic Jacobian with CM, all principally polarized abelian varieties that are Galois conjugated to it are hyperelliptic. Using Shimura’s reciprocity law, we then compute approximations of the invariants of the initial curve, as well as their Galois conjugates. We show examples of class polynomials computed using this method for the Shioda and Rosenhain invariants.

Keywords
hyperelliptic curve, complex multiplication, theta constants, class field
Mathematical Subject Classification 2010
Primary: 11G10, 11G15, 11G30
Milestones
Received: 28 February 2020
Revised: 1 August 2020
Accepted: 29 August 2020
Published: 29 December 2020
Authors
Bogdan Adrian Dina
Institute of Theoretical Computer Science
Ulm University
Ulm
Germany
Sorina Ionica
Laboratoire Modélisation, Information & Systèmes
Université de Picardie Jules Verne
Amiens
France