Modularity of Abelian Surfaces

• Modularity of Abelian Surfaces is defined by the correspondence between a principally polarized abelian surface and a Siegel modular form whose L-function matches that of the surface.
• Key methodologies include paramodular lifting and the 2–3 switch, which transfer modularity from one residual Galois representation to another under specific local-global conditions.
• This topic impacts number theory and automorphic forms, offering insights into phenomena like the Birch–Swinnerton-Dyer conjecture and extending modularity results to new classes of abelian varieties.

The modularity of abelian surfaces is a central topic in arithmetic geometry, intertwining the study of automorphic forms, Galois representations, Shimura varieties, and the arithmetic of genus-2 curves. Modularity, in this context, concerns the correspondence between abelian surfaces (principally polarized, typically defined over ℚ or number fields) and certain Siegel modular forms, mirroring the now classical theorem for elliptic curves. The recent decades have witnessed the formulation and verification of modularity theorems—particularly in the paramodular case—for large families of abelian surfaces, culminating in strong theorems and conjectures supported by explicit examples and sophisticated modularity lifting arguments.

1. Fundamental Notions and Classical Background

The modularity of an abelian surface $A/\mathbb{Q}$ asserts the existence of a Siegel modular form $F$ of genus 2 whose associated standard (or spinor) $L$‑function matches the Hasse–Weil $L$‑function of $A$: $L(A, s) = D(F, s)$ where $D(F, s)$ is the standard (or spinor) zeta function attached to $F$, and $L(A, s)$ is the $L$‑function defined via the characteristic polynomials of Frobenius on the étale cohomology of $A$ (Yang, 11 Jul 2025). The modularity paradigm generalizes the elliptic case, where the modularity theorem (formerly Taniyama–Shimura–Weil) equates the $L$‑function of an elliptic curve to that of a modular eigenform of weight 2.

For abelian surfaces, modularity can be formulated in several equivalent ways (Yang, 11 Jul 2025):

• Existence of a Siegel cusp form $F$ with $L(A, s) = D(F, s)$.
• Existence of a nonconstant morphism from a (compactified) Siegel moduli space or its Jacobian to $A$.
• The existence, for every prime $\ell$, of a compatible system of $4$-dimensional Galois representations $(\rho_{A,\ell})$ matching those associated to $F$.

This framework extends to higher-dimensional simple abelian varieties, but for $g = 2$, the correspondence is with weight 3 Siegel cusp forms and representations on $\mathrm{GSp}_4$ (Yang, 11 Jul 2025, Gee, 3 Oct 2025).

2. Paramodular Conjecture, Modularity Lifting, and Main Theorems

A central direction in the modularity of abelian surfaces is the paramodular conjecture, predicting that absolutely simple, principally polarized abelian surfaces over $\mathbb{Q}$ with trivial endomorphism ring are paramodular: there exists a genus‑2 Siegel paramodular newform $f$ of weight 2 and level $N$ such that

$L(A, s) = L(f, s, \text{spin}).$

This has been affirmed for abelian surfaces of odd conductor under specific local–global conditions. The main theorem in (Gee, 3 Oct 2025, Boxer et al., 28 Feb 2025) establishes the modularity of a positive proportion of abelian surfaces $A/\mathbb{Q}$ subject to the following:

• $A$ is ordinary at 3.
• The residual mod 3 Galois representation $\bar\rho_{A,3}$ is "big" (i.e., surjective onto $\mathrm{GSp}_4(\mathbb{F}_3)$).
• Specific local restrictions on decomposition groups at 2 (unramifiedness and avoidance of degenerate characteristic polynomials).
• The modularity of the associated residual representation is established via a "2‑3 switch" mechanism analogous to Wiles's 3‑5 trick (Gee, 3 Oct 2025, Boxer et al., 28 Feb 2025).

The strategy combines:

• Construction of auxiliary abelian surfaces $B$ whose mod‑2 representation is rendered solvably modular (often via explicit geometric or group-theoretic properties of genus‑2 Jacobians).
• Switching the modularity to the mod‑3 representation, then leveraging powerful modularity lifting theorems (via patching, Taylor–Wiles, or Calegari–Geraghty methods) and new classicality theorems for $p$-adic Siegel modular forms to obtain the modularity of $A$ (Gee, 3 Oct 2025, Boxer et al., 28 Feb 2025).

3. Residual and Potential Modularity; The 2–3 Switch

A crucial device is the systematic study of residual Galois representations. For $A/\mathbb{Q}$,

$$\rho_{A,\ell} : \mathrm{Gal}_\mathbb{Q} \to \mathrm{GSp}_4(\mathbb{Q}_\ell)$$

is attached to the $\ell$-adic Tate module. The approach, as outlined in (Gee, 3 Oct 2025), involves:

• Selecting genus‑2 Jacobians with a rational Weierstrass point to produce abelian surfaces $B$ where $\rho_{B,2}$ factors through $S_5$, recognizable as "solvably modular" via symmetric cube lifts from $A_5$-valued representations (as analyzed in classical Galois theory).
• Demonstrating that for any $A$ satisfying the modularity theorem's hypotheses, there exists $B$ with $\rho_{B,3} \cong \rho_{A,3}$, permitting the transfer of modularity from the 2-adic to the 3-adic representation ("2–3 switch").
• This machinery crucially leverages modularity lifting theorems: given a known modularity in one residual characteristic and sufficient local and image conditions, modularity propagates to $A$.

Potential modularity results extend these ideas to abelian surfaces over totally real fields, asserting that every abelian surface over such a field becomes modular over some finite extension (Boxer et al., 2018).

4. Modularity Tests, Symmetry, and Galois Images

Verification of modularity for individual abelian surfaces often requires explicit control of residual representations and group-theoretic invariants. A key methodology is:

• Checking that the image of, say, $\rho_{A,3}$ is "big" (often characterized as being the full $\mathrm{GSp}_4(\mathbb{F}_3)$ or containing suitable large subgroups).
• Ensuring, via local calculations at $p = 2,3$, that the action of decomposition and inertia subgroups at bad and small primes avoid forbidden types (e.g., no extra isogenies or reduction degeneracies).
• Using invariants—such as the level of ramification, root discriminant bounds (e.g., Odlyzko's tables), and structure of finite flat group schemes—to restrict possibilities and ultimately to use the classification of modular abelian surfaces of low conductor (Verhoek, 2012, Brumer et al., 2010).

Matching of L-series at the local and global level, along with "torsion filter" theorems linking torsion group schemes and isogeny classes, further pin down the possibilities (Verhoek, 2012).

5. Theoretical and Computational Techniques

Modern modularity proofs for abelian surfaces synthesize diverse technical and computational ingredients:

• p-adic modular forms and classicality: For ordinary representations, the existence of overconvergent p-adic modular forms congruent to classical forms is established using spectral sequences, completed cohomology, and geometric Sen theory (Gee, 3 Oct 2025).
• Spectral sequences and the flag variety: Applying the Hodge–Tate period map to connect algebraic $(\mathfrak{g},B)$-modules to sheaves on the flag variety, facilitating the translation of representation-theoretic statements to properties of p-adic modular forms.
• Explicit Galois techniques: Analysis of mod-$\ell$ representations of $\mathrm{GSp}_4$, explicit calculation of the image of Frobenius and inertia, and determination of the associated groups (e.g., $S_5$, $A_5$, $S_6$) arising from the Galois theory of Weierstrass points on genus-2 curves.
• Modularity lifting and patching: Use of Taylor–Wiles, Calegari–Geraghty, and Hida theory to propagate modularity from residual representations to the full p-adic representation.

These methods are complemented by algorithmic and computational work to evaluate modular forms, Hecke eigenvalues, and Galois structures in explicit families (Kieffer, 2020, Brumer et al., 2018).

6. Broader Context: Automorphic Forms, Shimura Varieties, and L-functions

Modularity of abelian surfaces is deeply interconnected with the theory of automorphic forms for symplectic and spin groups. For instance:

• Functorial lifts: The existence of compatible automorphic and Galois lifts from Hilbert modular forms to $\mathrm{GSpin}_{2g+1}$ or $\mathrm{GSp}_4$ provides assurance of the modularity of abelian varieties not of $\mathrm{GL}_2$-type (Cunningham et al., 2017).
• Shimura varieties and moduli: The Siegel and Hilbert modular varieties, as moduli of abelian surfaces with various structures, serve as the natural geometric home for modularity conjectures and their verification.
• Birch–Swinnerton-Dyer implication: Proven modularity theorems for abelian surfaces allow interpretation and partial verification of the BSD conjecture via Euler systems and $p$-adic L-functions constructed from genus-2 Siegel modular forms (Loeffler et al., 2021).

7. Open Problems and Future Developments

While recent advances have established modularity for a wide range of abelian surfaces, several open problems remain:

• Removing ordinarity and "big image" assumptions, possibly via small slope or alternative modularity lifting arguments.
• Generalizing results from $\mathbb{Q}$ to totally real fields, leveraging Hilbert–Siegel Shimura varieties.
• Proving full analogues of Serre's conjecture for $\mathrm{GSp}_4$ and understanding the modularity of all (simple) abelian surfaces.
• Refining explicit analytic and cohomological tools for higher-genus and higher-dimensional automorphic forms (Gee, 3 Oct 2025).

Further research is expected to expand the scope of modularity theorems for abelian surfaces, encompassing non-principally polarized, non-generic, or exotic cases, and deepening the structural understanding of the connection between arithmetic geometry and automorphic representation theory.