Readme for reconstructing curves of genus 4

This repository contains code to reconstruct a genus 4 curve from its theta constants. See https://arxiv.org/abs/2402.03160 for the accompanying article.

Structure of the repository

The repo contains four folders:

• magma
• examples
• Cext
• tests

In the Magma folder we have the folowing files:

• auxiliary.m Contains a few auxiliary methods.
• azygetic.m Contains the method that takes a system of four odd azygetic characteristics and extends it to a fundamental system.
• fast_theta.m An adaption of Labrande's fast theta code to compute Theta null values.
• FlintWrapper.m ThetaFlint(z, tau : ord): theta functions (and their derivatives in z) for all characteristics, computed rigorously by FLINT's acb_theta through the external program in Cext/ (see below).
• gluingfuncs.m Contains an array of maximal isotropic subgroups of F_2^4 and a few methods that are helpful for gluing.
• good-coordinates.m Contains tools for arithmetic construction from small period matrices
• igusa_quartic.m Reconstruct a genus 2 curve from the fourth powers of the thetas.
• reconstruction.m The main algorithm to reconstruct a genus 4 curve from its theta constants.
• rosenhain.m Compute the Rosenhain invariants of a genus 4 curve from the squares of its theta constants.
• schottky.m Computes the value of the Schottky modular form at a given period matrix tau.
• signs.m Code that uses Riemann's formula to correct the signs of the thetas that has been lost due to taking squares of thetas.
• theta.m Contains methods that compute Theta functions and the Siegel reduction of a period matrix.

The examples folder has two subfolders. In paper-examples are the computations from the article:

• Example-Gluing.m Glues the Jacobians of two genus 2 curves along their 2-torsion and reconstructs a genus 4 curve over Q with the glued Jacobian.
• Example-Modular.m Reconstructs a genus 4 curve over Q from the big period matrix (hard-coded, 300 digits) of the modular abelian fourfold attached to the newform 778.2.a.a; 778.2.a.a.m is the LMFDB download of that newform, used to verify the curve.
• QMexample.m, RMexample.m Search the Igusa quartic for genus 2 curves that glue to a genus 4 curve whose Jacobian has quaternionic, resp. real, multiplication. Galois.m (Galois action and cocycle on the Igusa coordinates) and g2-rmlmfdb.m (genus 2 curves with RM from the LMFDB) are loaded by these.
• Finite-Field-example.m Runs the reconstruction from tritangent data over F_37.

In construct-examples, construct-Lemma-Section-3.2-proofs.m constructs a plane quartic from an Aronhold system of bitangents and Aronhold-check.m verifies the identities used in the proofs of Section 3.2.

In the Cext folder, acb_theta_cli.c is a small C program linked against FLINT that evaluates theta functions and their derivatives; ThetaFlint in magma/FlintWrapper.m calls it. It replaces the earlier python-flint route (CHIMP's Theta.magma) and needs nothing but libflint >= 3.1.

In the tests folder:

• theta_flint_vs_chimp.m compares ThetaFlint with CHIMP's python-flint values, the pure-Magma theta series, parity of theta constants and finite differences.
• run_example_gluing.m, run_example_modular.m run the two paper examples that go through the theta code and verify the results independently (see below).

Building the FLINT helper

cd Cext
make          # Linux with libflint-dev >= 3.1
make mac      # macOS: brew install flint mpfr gmp
make test     # self-test

FLINT 3.1 or later is required (acb_theta was added in 3.1); Ubuntu 24.04's libflint-dev is 3.0.1 and is not enough. On such a machine, or on a server without admin rights, build a private FLINT once and link statically against it:

cd Cext
sh build_deps.sh   # GMP + MPFR + FLINT into ~/.local/acb_theta, 10-30 minutes
make server
make test

See Cext/README.md for details. The Magma code finds Cext/acb_theta_cli automatically; to use a binary elsewhere set THETA_FLINT_BIN=/path/to/acb_theta_cli. If CHIMP is attached as well, note that both packages define ThetaFlint(z, tau); attach this package last so that the FLINT-only version is used.

How to run the examples

All examples need Magma and the compiled FLINT helper (see the previous section). Attach the package with

AttachSpec("magma/spec");

adjusting the path as needed; the example files do this themselves with paths relative to examples/paper-examples, so run them from that directory.

Gluing (needs nothing beyond this repository; about a second):

cd examples/paper-examples
magma Example-Gluing.m

This computes the period matrices of the two genus 2 curves at 200 digits, the maximal isotropic subgroup V given by the identification of their 2-torsion, the glued big period matrix Q, and finally Eqs := RationalReconstructCurveG4(Q), a quadric and a cubic in P^3 over Q. To also check the result, run the test driver instead:

magma ../../tests/run_example_gluing.m

It verifies that the Frobenius traces of the recovered curve agree with those of Jac(X1) x Jac(X2) at all good primes up to 200 (up to the sign of a quadratic character, which it identifies: the isogeny is defined over Q(sqrt(-730)); since a non-hyperelliptic curve has no quadratic twists, this is the only curve over Q in the picture) and, if Thomas Bouchet's Genus-4 and GL-Equivalence packages are found in ~/github or ~, that its invariants agree with the model recorded in QMexample.m.

Modular (needs nothing beyond this repository; a couple of seconds, plus about 20 s for the verification):

cd examples/paper-examples
magma ../../tests/run_example_modular.m

This reads the 300-digit big period matrix from Example-Modular.m, runs RationalReconstructCurveG4 on it and ReconstructCurveG4 over C on the Siegel-reduced small period matrix, and then checks, at every good prime p <= 60, that the L-polynomial of the recovered curve is the product over the four embeddings of the Hecke field of 1 - a_p T + p T^2, using the eigenvalues a_p of 778.2.a.a stored in 778.2.a.a.m. Example-Modular.m itself can also be run directly (magma Example-Modular.m; it expects GL-Equivalence and Genus-4 to be checked out next to this repository). Its own verification against the newform up to p = 1000, which rebuilds the newform with ModularForms, is switched off by default: it is guarded by the flag verify_with_newform := false near the top of the file, and the example prints a reminder when it is skipped. Set the flag to true to run it; expect it to take much longer than the reconstruction.

QM and RM (QMexample.m, RMexample.m) are searches rather than single computations: they loop over rational points on the Igusa quartic and test each candidate gluing, which takes hours. RMexample.m also needs the EndomorphismAlgebra intrinsic from the endomorphisms package (included in CHIMP). Run them from examples/paper-examples with magma QMexample.m or magma RMexample.m; QMexample.m writes its log to Output.txt. The pair of genus 2 curves found by the QM search is the one glued in Example-Gluing.m.

Finite field and construct-examples are self-contained: magma Finite-Field-example.m in examples/paper-examples, and magma Aronhold-check.m in examples/construct-examples.

Testing the theta code after building or updating FLINT:

cd Cext && make test && cd ..
magma tests/theta_flint_vs_chimp.m

The second command compares ThetaFlint against the pure-Magma theta series, checks the parity and numbering of the characteristics, and checks FLINT's derivatives against finite differences; if a python3 with python-flint is available (or CHIMP's own venv exists), it also compares against the values CHIMP's python-flint wrapper produces.