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

Abstract: Over the complex field, we show that the principally polarised abelian surfaces with a given refined Humbert invariant of rank 3 are all Galois conjugated.

• The paper presents the classification of rank 3 refined Humbert loci as single Galois orbits of CM abelian surfaces.
• It employs lattice isometry and Hermitian matrix analysis to connect refined invariants with ideal class computations.
• The results unify modular and arithmetic perspectives, offering new tools for explicit CM and moduli space calculations.

Refined Humbert Invariants and the Geometry of Rank 3 Subvarieties in $\mathcal{A}_2(\mathbb{C})$

Introduction and Context

The paper "Refined Humbert invariants and subvarieties of $\mathcal{A}_2(\mathbb{C})$: the rank 3 case" (2607.03173) investigates the geometry and arithmetic of Humbert loci within the moduli space $\mathcal{A}_2(\mathbb{C})$ of principally polarized abelian surfaces, with a focus on the loci defined by refined Humbert invariants of rank 3. Classical Humbert surfaces correspond to divisors in $\mathcal{A}_2$ characterized by the existence of a divisor with a fixed intersection form—equivalently, a Humbert invariant of rank 1. This work examines the significantly more intricate rank 3 case, developing both a geometric description and a Galois-theoretic characterization of the corresponding subvarieties.

Refined Humbert Invariants and the Humbert Locus

Given a principally polarized abelian surface $(A,\lambda)$ over $\mathbb{C}$, the Néron-Severi group $\mathrm{NS}(A)$ encodes algebraic cycle classes. The refined Humbert invariant $q_{(A,\lambda)}$ is a positive-definite integral quadratic form defined on the group $$\mathrm{NS}(A,\theta):=\mathrm{NS}(A)/\mathbb{Z}\theta$$, where $\theta$ is a polarization. The locus $\mathcal{A}_2(\mathbb{C})$0 in the moduli space is then the set of isomorphism classes of principally polarized abelian surfaces for which the associated refined Humbert invariant primitively represents the quadratic form $\mathcal{A}_2(\mathbb{C})$1.

For $\mathcal{A}_2(\mathbb{C})$2 of rank 1, $\mathcal{A}_2(\mathbb{C})$3 reduces to the classical Humbert surface $\mathcal{A}_2(\mathbb{C})$4. For higher rank— notably rank 2 and rank 3—these loci reflect deeper relationships between the geometry of intersecting Humbert surfaces and the arithmetic of associated quadratic forms. Prior work by Kani, Runge, Rotger, Guo-Yang, among others, has developed structural and computational results for the lower rank settings, with limited results available for the general rank 3 case.

Main Theorem: The Rank 3 Case

The principal result rigorously classifies the locus $\mathcal{A}_2(\mathbb{C})$5 when $\mathcal{A}_2(\mathbb{C})$6 is a ternary (rank 3) quadratic form and $\mathcal{A}_2(\mathbb{C})$7. Specifically, when the discriminant of $\mathcal{A}_2(\mathbb{C})$8 is $\mathcal{A}_2(\mathbb{C})$9, where $\mathcal{A}_2(\mathbb{C})$0 is a fundamental discriminant, the locus $\mathcal{A}_2(\mathbb{C})$1 consists of all Galois conjugates under $\mathcal{A}_2(\mathbb{C})$2 of a principally polarized abelian surface isomorphic to a product of two CM elliptic curves with maximal order in the imaginary quadratic field $\mathcal{A}_2(\mathbb{C})$3. Thus, the moduli-theoretic points corresponding to prescribed refined Humbert invariants of rank 3 are organized into single Galois orbits, possibly with multiplicities depending on automorphism groups.

This assertion clarifies structural questions about irreducibility/connectedness and provides an explicit relationship between modular and Galois-theoretic descriptions of these loci. The proof draws on the interplay between the theory of complex multiplication for elliptic curves, ideal class group computations, and isogeny modules, utilizing Kani's categorical equivalence between principally polarized products of CM elliptic curves and certain module-theoretic classes.

A notable claim is that all principally polarized surfaces with a given refined Humbert invariant of rank 3 are Galois conjugate, up to isomorphism, under the appropriate field of complex multiplication. The cardinality of $\mathcal{A}_2(\mathbb{C})$4 is shown to be at most $\mathcal{A}_2(\mathbb{C})$5, with explicit identification of cases where this bound is not sharp due to polarization equivalences or automorphisms.

Methods and Technical Contributions

The constructive and theoretical aspects of the proof hinge on the explicit description of $\mathcal{A}_2(\mathbb{C})$6 as Galois orbits, informed by several key ingredients:

• Albert's classification and Shioda-Mitani's theorem, which ensure that the relevant abelian surface $\mathcal{A}_2(\mathbb{C})$7 is always isomorphic to a product $\mathcal{A}_2(\mathbb{C})$8 of CM elliptic curves with fixed CM field.
• The translation of refined Humbert invariants to lattices of Hermitian matrices, identifying $\mathcal{A}_2(\mathbb{C})$9 with the lattice $\mathcal{A}_2$0 where $\mathcal{A}_2$1 is a maximal order, and the analysis of isometries via explicit matrix algebra.
• The use of the Artin reciprocity map to connect the Galois action on CM elliptic curves to ideal class arithmetic and module structure.
• The double coset decomposition of the group of positive isometries acting on these lattices, yielding a quotient of cardinality $\mathcal{A}_2$2, and showing that only the $\mathcal{A}_2$3-factor is essential for the moduli problem due to the trivial action of certain involutions.

A new notion of twisted content for isometry matrices is used to track ideal-theoretic data across module isomorphisms, demonstrating injectivity and completeness for the class group parametrization.

Experimental confirmation is provided using explicit computations for fundamental discriminants up to $\mathcal{A}_2$4, supporting the main theorem's claim even in the presence of nontrivial automorphism groups and equivalences.

Implications, Applications, and Future Directions

The identification of rank 3 refined Humbert loci as single Galois orbits fundamentally simplifies the structure of the intersection of higher-dimensional Humbert loci within the Siegel moduli space. This plays a critical role in understanding the interplay between moduli of abelian varieties, CM theory, and the arithmetic of ideal class groups.

Potential applications include:

• Explicit determination of CM points and their fields of moduli, with implications for class field theory and the construction of high-genus curves with prescribed geometric and arithmetic properties.
• Algorithmic applications to computing moduli spaces, class polynomials, and explicit CM constructions relevant to both arithmetic geometry and cryptography.
• Deepening the theoretical understanding of the interaction between polarization types, endomorphism rings, and the stratification of $\mathcal{A}_2$5 by arithmetic invariants.

The techniques introduced, such as the use of Hermitian lattice isometries and twisted content, may have further impact on the general study of moduli spaces of abelian varieties with additional structure, and in the parametric study of automorphisms and endomorphism rings.

Further work is suggested in:

• Dropping the maximal order condition in the statement and proof, potentially extending the theorem to non-maximal CM orders.
• Connecting these results to explicit arithmetic models of Shimura curves and explicit singular relations as studied in related literature.
• Extending the analysis to analogous situations in higher genus or with more general polarization types.

Conclusion

This paper resolves a natural and long-standing question concerning the moduli-theoretic and Galois-theoretic structure of subvarieties defined by rank 3 refined Humbert invariants in $\mathcal{A}_2$6 (2607.03173). By exhibiting that these loci correspond precisely to Galois orbits of products of CM elliptic curves with prescribed polarization data, it clarifies the relationship between modular forms, quadratic form arithmetic, and the geometry of the Siegel modular variety. The techniques and results extend and unify prior approaches to loci defined by Humbert invariants, offering both theoretical insight and a toolkit for explicit computation in arithmetic geometry.