Constructing genus 2 curves with given refined Humbert invariants

TL;DR

This work addresses the problem of realizing a given geometric positive definite ternary quadratic form $f$ as a refined Humbert invariant $q_{(A,\theta)}$ of a principally polarized abelian surface $(A,\theta)$, thereby yielding genus $2$ curves whose refined Humbert invariants match $f$. The authors devise explicit algorithms that, depending on whether $f$ is primitive or imprimitive, construct CM product surfaces $A=E\times E'$ with a principal polarization $\theta$ so that $f\sim q_{(A,\theta)}$. The core contributions are (i) a constructive lattice-based method to realize $A$ from $f$ via a suitable binary form $q$ (and, for imprimitive cases, a binary form $q_{I_1,I_2}$), (ii) a detailed procedure to produce $\theta$ as $\mathbf{D}(n,m,kh)$ and verify $q_{(A,\theta)}$, and (iii) a simplified practical implementation that relies on binary/ternary form reciprocity and, under GRH, the guaranteed existence of primes represented by forms. Together, these results enable explicit generation of genus $2$ curves whose refined Humbert invariants match prescribed ternary forms, with applications to studying CM points on Humbert surfaces and Kaplansky-type conjectures. The methods bridge arithmetic of quadratic forms with complex multiplication geometry, providing a usable route to construct Jacobians and divisorial representatives with targeted invariants. Practically, this advances the ability to produce and analyze genus $2$ curves with prescribed arithmetic-geometric features and to investigate CM-point intersections on Humbert surfaces.

Abstract

In 1994, Kani introduced an algebraic version of the Humbert invariant, known as the refined Humbert invariant. This invariant q_C is a positive definite quadratic form attached to a smooth curve C of genus 2. It serves as a vital tool, as many geometric properties of C are reflected in the arithmetic properties of q_C. When the Jacobian J_C of a genus 2 curve C is isogenous to a product of an elliptic curve with complex multiplication, the forms q_C have been completely classified recently. In this paper, building upon this classification, we present a constructive algorithm that produces J_C and a divisorial representative of a curve C of genus 2 such that its refined Humbert invariant q_C is equivalent to a given integral ternary quadratic form.

Theorems & Definitions (35)

• Definition 1.1
• Theorem 1: GRH
• Remark 1.2
• Proposition 2
• proof
• Proposition 3
• proof
• Theorem 4: Theorem 1 of refhum and Theorem 1.2 of harun_1
• Theorem 5: Theorem 2 of refhum (see Theorem 4.1.4 of kir2024curvesPHD)
• Proposition 6