Equations of genus $4$ curves from their theta constants

Technical Summary: Equations of Genus 4 Curves from Their Theta Constants

Problem Statement
The primary objective of this work is to provide explicit, closed-formulas for recovering the equations of a generic genus 4 curve $C$ over an algebraically closed field $k$ (of characteristic 0 or sufficiently large $p$), given only the values of its algebraic theta constants. While the problem of reconstructing curves from theta constants has been solved for genus 2 (Rosenhain) and genus 3 (Aronhold–Weber), the genus 4 case presents unique difficulties. Unlike lower genera, the moduli space of genus 4 curves with a full level 2 structure is not known to be unirational, and not every principally polarized abelian variety (p.p.a.v.) of dimension 4 is a Jacobian (a condition governed by the vanishing of the Schottky modular form). Previous approaches, such as those by Lehavi, often required solving non-linear systems for all 255 non-trivial two-torsion points, rendering them computationally inefficient for explicit reconstruction.

Methodology
The authors employ the Schottky–Jung approach, leveraging the Prym construction and classical geometric results concerning cubic symmetroids. The method proceeds through the following logical steps:

1. Prym Canonical Map and Symmetroids: For a generic non-hyperelliptic genus 4 curve $C$ and a non-trivial 2-torsion point $\eta \in \text{Jac}(C)[2]$, the Prym canonical map $\phi_\eta: C \to \mathbb{P}^2$ maps $C$ to a singular plane sextic. The image of the canonical embedding of $C$ in $\mathbb{P}^3$ is recovered as the intersection of a quadric $Q$ and a cubic symmetroid $\Gamma_\eta$. Theorem 2.5 establishes a bijection between non-zero 2-torsion points and irreducible cubic symmetroids containing the canonical image.

2. Linear Maps and Milne's Bijection: The core of the reconstruction relies on a linear map $\phi: V_{C,\eta} \to S^2 H^0(C, \Omega_C(\eta))$, where $V_{C,\eta}$ is a specific subspace of quadratic forms. This map is linked to Milne's bijection (Theorem 2.10), which relates pairs of tritangent planes of $C$ differing by $\eta$ to bitangents of a genus 3 curve $X$ (the Prym variety).

   • The equation $\phi(HH') = \lambda \ell^2$ connects the product of tritangent planes $H, H'$ on $C$ to the square of a bitangent line $\ell$ on $X$.
   • The bitangents of $X$ are computed using classical Aronhold–Weber formulas applied to the theta constants of $X$, which are derived from those of $C$ via Schottky–Jung relations.

3. Explicit Computation via Theta Constants:

   • Tritangent Planes: The authors utilize a generalized Jacobi derivative identity (Theorem 2.15) to express the equations of tritangent planes purely in terms of theta constants. They select a specific auxiliary set of odd theta characteristics (Lemma 3.1) to ensure the Jacobian nullvalues can be expressed as polynomials in theta constants.
   • Solving for $\phi$: By constructing 10 pairs of tritangent planes $(H_i, H'_i)$ and their corresponding bitangents $\ell_i$, the authors set up a homogeneous linear system to solve for the unknown scalars $\lambda_i$. This allows for the explicit computation of the matrix representing $\phi$.
   • Recovering Equations:
     • The quadric $Q$ is found as the generator of the kernel of $\phi$ (Lemma 3.5).
     • The cubic $\Gamma_\eta$ is recovered by constructing a right-inverse $\psi$ of $\phi$ and computing a degree 6 form $G = \det(\Delta)$, where $\Delta$ is derived from $\psi$. The cubic $\Gamma_\eta$ is the square root of $G$ (Proposition 3.7).

Key Contributions and Results

• Explicit Formulas: The paper provides the first closed formulas for the equations of a generic genus 4 curve in terms of its theta constants (Theorem 4.2).
• Algebraic Complexity: The method requires only elementary arithmetic, square root extraction, and the solution of linear systems of equations. It avoids the non-linear algebraic systems required by previous methods (e.g., Lehavi's approach).
• Handling Special Cases: The framework accommodates the bielliptic case (where the Prym map factors through a degree 2 map to an elliptic curve) and cases where the curve $X$ is hyperelliptic (Remark 4.5), adjusting the geometric interpretation of the bitangents accordingly.
• Numerical Implementation: The authors demonstrate the feasibility of the method over finite fields (Example 5.1) and number fields (Examples 5.2 and 5.3). In the number field case, they utilize floating-point approximations of theta constants (derived from period matrices) combined with LLL lattice reduction to recover exact rational coefficients.

Significance and Applications
The authors position this work as a significant advancement in the Schottky problem, specifically for genus 4. The ability to reconstruct curves from theta constants enables the explicit construction of curves with special arithmetic properties, such as:

• Curves whose Jacobians are isogenous to products of lower-dimensional abelian varieties.
• Curves with modular Jacobians (e.g., associated with modular forms).
• Curves with specific endomorphism rings or Hodge tensors.

The paper notes that these formulas have already been used in conjunction with invariant theoretic techniques to discover new families of genus 4 curves of Mumford type, providing the first examples of Shimura families not of PEL-type. The method is presented as a practical tool for arithmetic and complex geometry, allowing for the exploration of examples previously inaccessible to existing numerical or symbolic methods.

Limitations and Future Work
The authors acknowledge that the method relies on the curve being generic; specific degenerations (e.g., vanishing denominators in the tritangent formulas or failure of the linear independence of products $H_i H'_i$) may require choosing a different 2-torsion point $\eta$. The paper concludes by discussing the generalization to higher genera ($g \geq 5$), noting that while the linear algebraic structure persists for $g=5$, the recovery of Jacobian nullvalues as rational functions of theta constants remains an open problem for $g \geq 6$.