I‑AIMS Institute for Artificial Intelligence
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Weighted Theta Heights and Kummer Heights on Genus-Two Jacobians

Sajad Salami, Tanush Shaska

I-AIMS preprint sh-115 · 2026

Let J be the Jacobian of a genus-two curve over a number field whose Weierstrass points are rational. Canonaco’s presentation of the theta ring embeds J into ℙ(1,2,2,2,3,3,3,3,3,4,4) and gives a weighted Weil height for the principal polarization Θ. Choosing at each place a normal form whose branch points have maximal pairwise distance, we define an adelically reduced weighted theta height h_(Θ,red) and reduced Kummer height h_(K,red) and prove that 0 ≤ h_(Θ,red) − 1/2 h_(K,red) ≤ 1/6 log 40. The seven generators outside the |2Θ| subsystem satisfy monic square relations over it, which force the weighted theta norm to equal the square root of the Kummer norm at every finite place. Hence the weighted theta model carries no local height information beyond the Kummer surface. Using the integrality of Flynn’s duplication forms on the reduced models, an exact scalar Λᵥ⁹ relating them to a global rigidification, and Stoll’s height constants, we prove the Silverman-type bound -11/3 H_(diam) − c₁ ≤ ĥ_M − h_(K,red) ≤ 3 H_(diam) + c₂ with absolute c₁, c₂, where H_(diam) is the diameter height of the branch configuration, comparable with the weighted moduli height of the curve. The Kummer subsystem of Canonaco’s ring is identified with the Cassels–Flynn coordinates, and the local functions of Müller–Stoll are transported to the weighted theta metric.

MSC classes: 11G50, 14G40, 14K15, 14G05

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