Computations on Modular Jacobian Surfaces

We give a method for finding rational equations of genus 2 curves whose jacobians are abelian varieties $A_f$ attached by Shimura to normalized newforms $f \in S_2( Γ_0(N))$. We present all the curves corresponding to principally polarized surfaces $A_f$ for $N\leq500$.

💡 Research Summary

The paper “Computations on Modular Jacobian Surfaces” presents a concrete algorithm for producing rational equations of genus‑2 curves whose Jacobians are the abelian surfaces (A_f) attached by Shimura to normalized newforms (f\in S_2(\Gamma_0(N))). The authors focus on the case where (A_f) is principally polarized and irreducible, which is precisely the situation in which (A_f) can be identified with the Jacobian of a genus‑2 curve.

The theoretical backbone consists of several classical ingredients. First, the complex torus description of an abelian variety (A) as (\mathbb{C}^g/\Lambda) together with a Riemann form gives a period matrix (\Omega=(\Omega_1\mid\Omega_2)). From (\Omega) one forms the normalized matrix (Z=\Omega_1^{-1}\Omega_2) and the Riemann theta function (\theta(z;Z)). While even theta‑null values (theta at even 2‑torsion points) have been extensively studied, the authors exploit the derivatives of the theta function at odd 2‑torsion points. These derivatives satisfy linear relations that involve the period matrix and can be used to recover the Weierstrass points of the underlying hyperelliptic curve.

Proposition 1 establishes that any irreducible principally polarized abelian surface over a field (K) is the Jacobian of a genus‑2 curve defined over (K). Proposition 2 gives a practical irreducibility test: the vanishing of an even theta‑null at an even 2‑torsion point signals that the surface is a product of elliptic curves. The central computational result is Theorem 1, which translates the vanishing of the theta‑derivative linear forms into six equations whose solutions give the six ratios (\alpha_k) (the (x)-coordinates of the Weierstrass points). From these (\alpha_k) one builds a monic polynomial (F_0(x)=\prod (x-\alpha_k)).

The discriminant formula in part (b) of Theorem 1 links the algebraic discriminant (\Delta_{\text{alg}}(C_F)) of the curve (C_F: y^2=F(x)) to the determinant of the period matrix and a product of hyperplane evaluations at the (\alpha_k). This relation yields the leading coefficient (a_6) of the final polynomial (F(x)=\zeta a_6F_0(x)), where (\zeta\in{\pm1}) is determined by the Eichler–Shimura congruence and by comparing point counts of the reductions modulo a suitable prime.

The algorithm proceeds in four explicit steps:

1. Period matrix construction – Using modular symbols (implemented via Stein’s Magma package) the authors compute a symplectic basis of (H_1(A_f,\mathbb{Z})) and a basis of regular differentials, yielding (\Omega).

2. Weierstrass point extraction – Solve the six linear equations derived from the theta‑derivatives to obtain the (\alpha_k) and form (F_0(x)).

3. Leading coefficient determination – Apply the discriminant identity to compute (a_6) as a rational number, then take a tenth root (since the discriminant involves (a_6^{10}) for degree‑6 polynomials).

4. Sign selection and final equation – Decide whether (\zeta=+1) or (-1) by checking whether the curves (C_F) and (C_{-F}) are (K)-isomorphic or have different point counts modulo a prime of good reduction.

The authors implemented the whole pipeline in Magma, handling all newforms with quadratic coefficient fields and level (N\le 500). For each such newform they verified that the associated surface (A_f) is principally polarized (using Proposition 2) and irreducible. The resulting list contains over three hundred genus‑2 curves, most of which are already minimal over (\mathbb{Z}