I‑AIMS Institute for Artificial Intelligence
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Degrees of Genus-Two Split-Jacobian Loci and Humbert-Form Reconstruction

Tanush Shaska

I-AIMS preprint sh-128 · 2026

Let H_(n²) ⊂ 𝒜₂ be the Humbert surface of discriminant n², G_(n²) be the Siegel modular form of level one with divisor H_(n²), and k(H_(n²)) be its weight. Under the Torelli map, H_(n²) is the closure of the locus ℒₙ ⊂ ℳ₂ of genus-two curves admitting a maximal degree-n elliptic subcover, the zero locus of an irreducible weighted-homogeneous Fₙ ∈ ℤ[J₂, J₄, J₆, J₁₀]. Let ν(n) = deg(X₁(n) → X(1)). We prove that the meromorphic form Fₙ(τ) obtained from Fₙ has a pole of order exactly ν(n) along the product locus, that G_(n²) is a constant multiple of χ₁₀^(ν(n)) Fₙ(τ), and that deg_w Fₙ = k(H_(n²)) − 10 ν(n) for every n ≥ 2, even or odd. We determine the restriction of G_(n²) to the product locus as a product of modular polynomials, its leading Fourier–Jacobi coefficient as a product of theta functions over the points of exact order n, and we characterize G_(n²) up to scalar by vanishing on a single torsion divisor. These data convert the computation of G_(n²) into a linear problem of the prescribed size, which we carry out for n = 5, producing the exact Humbert form G₂₅ in the generators ψ₄, ψ₆, χ₁₀, χ₁₂ and the equation F₅ over ℚ.

MSC classes: Primary 14H40, 11F46; Secondary 11F50, 14H10, 14Q05

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