This is the GitHub repository containing code related to the paper Generic models for genus 2 curves with real multiplication by Alex Cowan, Sam Frengley, and Kimball Martin arXiv:2403.03191.

Most of the code in this repository is written in Magma though many of the files should be written in such a way as to be acessible to most computer algebra systems.

Using the intrinsics in models

To be able to use the provided Magma intrinsics within the models directory the user should first run setup.py with Python 3 (tested on Python 3.8.10 and Python 3.10.12). This short script edits a few absolute paths so that the relevant spec files can be attached in a directory independent fashion.

Alternatively you can simply edit the first lines of each of the .m files in models to contain the correct absolute path.

Directory structure

• minimize contains code for our minimization algorithm as described in Section 5 of the paper. Notable files within this directory are minimization.m which contains the minimization code and ICmestre.m which contains the IC and RM simplified Mestre conics. The main intrinsic we provide is:

  • MinimizationSearch which runs Algorithm 5.5.

• models contains the equations for the varieties in the paper (be they the genus 2 curves, the conics, or the 3-folds). Further information about where one can find which equations can be found in the README models/README.md. Notable intrinsics provided by the models/spec specification file are documented at models/README.md:

  • RMWeierstrassEquationRational
  • RMWeierstrassEquation

• proofs verifies computational claims in the paper and stores the requisite transformations to prove Theorems 1.1 and 1.5.

• tests contains several files testing the functionality provided by this repository:

  • We give examples for using MinimizationSearch
  • We give some sanity checks (using AnalyticJacobian) for the existence of an endomorphism of $\mathrm{Jac}(C)$ with discriminant $D$.