Refined Humbert invariants and subvarieties of $\Ac_2(\C)$: the rank 3 case

Technical Summary: Refined Humbert Invariants and Subvarieties of $\mathcal{A}_2(\mathbb{C})$: The Rank 3 Case

Problem Statement
The paper investigates the geometry of the Humbert locus $H(q)$ within the moduli space $\mathcal{A}_2(\mathbb{C})$ of principally polarized abelian surfaces. Specifically, it addresses the case where $q$ is a ternary (rank 3) integral quadratic form. The Humbert locus $H(q)$ is defined as the set of principally polarized abelian surfaces $(A, \lambda)$ such that the refined Humbert invariant $q(A, \lambda)$ primitively represents the quadratic form $q$. While the rank 1 case (classical Humbert surfaces $H_\Delta$) and various rank 2 intersections have been studied extensively, a complete geometric characterization of the rank 3 locus in $\mathcal{A}_2(\mathbb{C})$ had not been previously established. The central question is to describe the structure and cardinality of $H(q)$ when it is non-empty.

Methodology
The authors employ a synthesis of arithmetic geometry, the theory of complex multiplication (CM), and the algebraic structure of endomorphism rings.

1. Structural Reduction: Utilizing Albert's classification and results by Shioda and Mitani, the authors establish that if $H(q) \neq \emptyset$ for a rank 3 form $q$, any surface $(A, \lambda) \in H(q)$ is isogenous to a product of two CM elliptic curves, $A \simeq E_1 \times E_2$, sharing the same CM field $K = \mathbb{Q}(\sqrt{d})$.
2. Galois Conjugacy (Proposition 2.1): The proof begins by showing that if two surfaces $(A, \lambda)$ and $(A', \lambda')$ share the same refined Humbert invariant, their underlying abelian varieties $A$ and $A'$ must be Galois conjugates. This is achieved by analyzing the equivalence of the associated binary quadratic forms describing the isogenies between the CM factors, linking them to ideal classes in the endomorphism ring via the Artin map.
3. Lattice and Matrix Representation: The authors explicitly model the Néron-Severi group $NS(A)$ as a lattice of "twisted" Hermitian matrices, denoted $Hermtwist_2(\mathcal{O})$, where $\mathcal{O}$ is the maximal order of $K$. The refined Humbert invariant corresponds to the determinant on this lattice.
4. Group Theoretic Analysis: The problem is reduced to classifying principal polarizations on a fixed $A$ that satisfy the invariant condition. This involves analyzing the quotient group $G_A / H_A$, where $G_A$ is the group of isometries of $(NS(A), q_A)$ preserving the ample cone, and $H_A$ is the image of the geometric automorphism group $\text{Aut}(A)$.
5. Twisted Content: A novel tool, the "twisted content" of a matrix $M \in GL_2(K)$, is introduced to characterize the action of isometries on the lattice. This allows the authors to construct explicit representatives for the cosets of $G_A / H_A$ and relate them to the 2-torsion subgroup of the ideal class group, $\text{Cl}(\mathcal{O})[2]$.

Key Contributions and Results

• Theorem 1.2 (Main Result): The authors prove that if $q$ is a ternary quadratic form with discriminant $-16d$ (where $d < 0$ is a fundamental discriminant) and $H(q) \neq \emptyset$, then $H(q)$ consists precisely of the $\text{Gal}(\bar{K}/K)$-orbit of a single principally polarized abelian surface $(E_1 \times E_2, \lambda)$. Here, $E_1$ and $E_2$ are elliptic curves with CM by the maximal order $\mathcal{O}$ of the imaginary quadratic field $K = \mathbb{Q}(\sqrt{d})$.
• Classification of Polarizations: The paper establishes a bijection between the set of isomorphism classes of principal polarizations on $A$ with a given refined Humbert invariant and the set $G_A / H_A$.
• Cardinality and Structure: It is proven that $\#(G_A / H_A) = 2 \cdot \#\text{Cl}(\mathcal{O})[2]$. The authors construct explicit representatives for these cosets using matrices derived from ideal classes in $\text{Cl}(\mathcal{O})[2]$ and an involution $\tau$.
• Geometric Interpretation: The action of the $\text{Cl}(\mathcal{O})[2]$ component on the polarizations is interpreted geometrically as the pullback of the polarization on a Galois conjugate surface. The action of the $\mathbb{Z}/2\mathbb{Z}$ component (generated by $\tau$) is shown to be trivial in terms of producing new isomorphism classes of polarizations.
• Experimental Verification: The authors note that their theoretical results have been experimentally confirmed for fundamental discriminants up to $d = -199$ using existing code and results from Narayanan (2024).

Significance
The paper provides the first geometric characterization of the rank 3 Humbert locus in $\mathcal{A}_2(\mathbb{C})$. It demonstrates that, contrary to potentially complex moduli descriptions, these loci are finite sets of Galois conjugates of specific CM products. The work bridges the gap between the algebraic properties of refined Humbert invariants and the arithmetic of CM elliptic curves. By introducing the "twisted content" and explicitly describing the isometry group action on the Néron-Severi lattice, the authors offer a concrete framework for manipulating these loci. The results refine the understanding of how intersections of Humbert surfaces behave in the rank 3 case, showing they are governed by the 2-torsion of the class group of the underlying imaginary quadratic field. The authors acknowledge that while the condition on the maximal order in Theorem 1.2 might be relaxed with further work, the current framework successfully resolves the geometric description for the maximal order case.