Potential Modularity: Theorems and Applications

• Potential Modularity Theorems define the scenario in which arithmetic objects acquire automorphic properties after a finite field extension, linking Galois representations with modular forms.
• They utilize techniques like automorphy lifting, patching, and deformation theory to overcome local and global obstructions in establishing modularity.
• This framework has led to advances such as the meromorphic continuation of L-functions and proofs of conjectures like Sato–Tate for various arithmetic varieties.

Potential Modularity Theorems constitute a foundational framework in the paper of the relationship between arithmetic objects—principally elliptic curves, general abelian varieties, and higher-dimensional varieties—and automorphic representations. These theorems assert that, under certain structural and local conditions, one can always pass to a finite extension of the base number field over which the object in question becomes modular, i.e., corresponds to an automorphic form. In particular, potential modularity results have been essential for the establishment of meromorphic continuation and functional equations for Hasse–Weil $L$-functions, the proof of the Sato–Tate conjecture for various classes of varieties, and the development of automorphy lifting techniques far beyond classical contexts.

1. Conceptual Definition and Core Scope

Potential modularity theorems state that for a wide class of arithmetic objects—such as elliptic curves, abelian surfaces, and K3 surfaces over number fields—there exists a finite extension of the base field over which their associated Galois representations become modular or automorphic. More precisely, a variety $X/F$ is called potentially modular if, after base-changing to some finite extension $F'/F$, its relevant étale cohomology groups yield Galois representations arising from algebraic cuspidal automorphic representations of a suitable reductive group (e.g., $\mathrm{GL}_n$, $\mathrm{GSp}_{2g}$) over $F'$ (Gu, 4 Dec 2025, Gee, 3 Oct 2025, Calegari et al., 2012).

The paradigm is significantly broader than "strong" modularity theorems (such as the modularity of all elliptic curves over $\Q$), allowing flexible passage to extensions and thus circumventing local or global obstructions that might not be resolvable over the original base field (Buzzard, 2010).

2. Historical Evolution and Key Milestones

The origins of potential modularity lie in the approach to the modularity of semistable elliptic curves employed by Wiles and Taylor–Wiles, with subsequent generalization to all elliptic curves over $\Q$ and, later, totally real fields (Buzzard, 2010). The ideas were dramatically expanded by:

• Taylor, who first proved that every elliptic curve over a totally real field is potentially modular using the "p–$\ell$ trick" and Moret–Bailly's theorem on the density of rational points on moduli (Buzzard, 2010);
• Kisin, Skinner–Wiles, Diamond, and Breuil–Conrad–Diamond–Taylor, who developed powerful automorphy lifting theorems accommodating increasingly general local conditions;
• Calegari–Geraghty, who substantially extended modularity lifting beyond the Taylor–Wiles framework to include objects whose associated automorphic forms do not appear only in the middle degree of Shimura variety cohomology (Calegari et al., 2012, Calegari et al., 2012).

Recent advances have pushed potential modularity and automorphy to higher-dimensional abelian varieties and K3 surfaces, with explicit theorems for abelian surfaces (Gee, 3 Oct 2025) and K3s of high Picard rank (Gu, 4 Dec 2025).

3. Foundational Techniques and Theorem Statements

Potential modularity theorems integrate several core algebraic and arithmetic ingredients:

• Automorphy Lifting Theorems: These establish that, under certain residual irreducibility and local deformation conditions, a $p$-adic Galois representation is automorphic provided that a "minimal" base change exists over which the residual representation is automorphic. The Taylor–Wiles and Kisin mechanisms employ delicate deformation theory, local–global compatibility, and patching of modular or cohomological structures (Gee, 2022).
• p–$\ell$ Trick and Moret–Bailly's Theorem: Taylor's method uses auxiliary primes and moduli of Hilbert–Blumenthal (or more general) abelian varieties, together with Moret–Bailly's theorem on the density of rational points in families over number fields with prescribed splitting conditions, to construct global points satisfying specific level structures (Buzzard, 2010).
• Patching and Cohomological Techniques: Calegari–Geraghty and others introduced patching complexes of cohomological objects, permitting control beyond the middle degree and dramatically enlarging the setting for automorphy lifting arguments (Calegari et al., 2012, Calegari et al., 2012).

A canonical result for elliptic curves is: For any elliptic curve $E$ over a number field $F$, there exists a finite extension $F'/F$ such that $E$ is modular over $F'$—i.e., the $L$-function $L(E/F', s)$ matches that of a cuspidal automorphic representation of $\mathrm{GL}_2(\A_{F'})$ (Calegari et al., 2012, Buzzard, 2010).

For abelian surfaces over totally real fields, the theorem is: For every abelian surface $A$ over $F$, there is a finite totally real extension $F'/F$ and a regular algebraic cuspidal automorphic representation $\pi$ of $\mathrm{GL}_4(\A_{F'})$ or $\mathrm{GSp}_4(\A_{F'})$ such that the $\ell$-adic realizations of $A$ correspond to the compatible system attached to $\pi$ (Gee, 3 Oct 2025).

For K3 surfaces of geometric Picard rank $\rho \geq 17$ over totally real $F$, after base-change to a totally real $F'$, the transcendental motives become automorphic, via lifting to a GSp$_4$-type abelian variety (Gu, 4 Dec 2025).

4. Structural Hypotheses and Local Properties

Potential modularity theorems require substantial local and global input. Common hypotheses include:

• Residual Representations: For Galois representations attached to the object (e.g., $E[F]$ or $A[\ell]$), the reduction mod $\ell$ must be absolutely irreducible, often even after restriction to $F(\zeta_\ell)$. This condition is necessary for the Taylor–Wiles patching machinery and its variants (Buzzard, 2010, Calegari et al., 2012).
• Local Deformation Conditions: At primes dividing $\ell$, the representations must admit ordinary, potentially Barsotti–Tate, or Fontaine–Laffaille lifts (depending on the context). For higher rank, ordinariness or positive density of distinguished primes is often required (Gee, 3 Oct 2025, Gu, 4 Dec 2025).
• Big Image/“Adequacy”: The image of the residual Galois representation must satisfy a big image or "adequacy" property (in the sense of Clozel–Harris–Taylor), ensuring sufficient flexibility for patching arguments (Gee, 3 Oct 2025).

For certain modularity theorems over cyclotomic extensions, fine Iwasawa-theoretic control is also critical, as in the modularity of elliptic curves over the $\mathbb{Z}_p$-extension of a real quadratic field (Zhang, 2022).

5. Influence, Applications, and Further Directions

Potential modularity theorems have direct arithmetic consequences:

• Meromorphic Continuation and Functional Equation of $L$-functions: By establishing automorphy of Galois representations attached to varieties, potential modularity permits application of the Langlands–Shahidi method (or converse theorems) to deduce analytic continuation, functional equations, and often compatibility with conjectural functoriality (Gee, 3 Oct 2025, Gu, 4 Dec 2025, Buzzard, 2010).
• Sato–Tate Conjecture: Once automorphy is achieved for all symmetric powers of an elliptic curve (or higher-dimensional variety after base change), the equidistribution of Frobenius eigenvalues (Sato–Tate law) follows from analytic properties of the associated $L$-functions (Calegari et al., 2012).
• Resolution of Birch–Swinnerton–Dyer-type Questions: For (potentially) modular abelian surfaces, the appropriate analytic properties can be established, initiating further paper of special-value formulae and $p$-adic heights (Gee, 3 Oct 2025).
• New Frameworks for Modularity Lifting: Potential modularity is both a target of modularity lifting arguments and a tool for proving the modularity of more complex or twisted objects by base change and descent (Calegari et al., 2012, Gu, 4 Dec 2025).

6. Generalizations and Open Problems

Several major directions remain for potential modularity:

• Beyond Ordinary or Small-Slope: Full modularity beyond the ordinary or small-slope settings for higher-dimensional cases (e.g., abelian varieties of dimension $g \geq 3$) remains open, with only partial results currently available (Gee, 3 Oct 2025).
• Serre-Type Conjectures in Higher Rank: The formulation and proof of Serre-style potential modularity conjectures for $\mathrm{GSp}_4$ or higher symplectic groups are unresolved, but would tightly connect mod $p$ Galois representations to automorphy (Gee, 3 Oct 2025).
• Conjectural Galois–Automorphic Correspondence for Torsion: Progress toward resolving the full scope of Conjecture AA (existence of Galois representations attached to torsion cohomology of locally symmetric spaces) would make many current results unconditional (Calegari et al., 2012, Calegari et al., 2012).
• Potential Modularity in Non-Totally Real Settings: While fully general results over CM fields exist for conjugate–self–dual representations, extending these to arbitrary varieties or motives remains a frontier area.

The theory of potential modularity constitutes the backbone of the modern approach to automorphy, Galois representations, and the global paper of arithmetic varieties, providing a flexible template for reducing Diophantine and analytic problems to questions about automorphic forms. The ongoing refinement of these theorems continues to shape the arithmetic geometry landscape (Calegari et al., 2012, Gee, 3 Oct 2025, Gu, 4 Dec 2025, Buzzard, 2010, Gee, 2022, Calegari et al., 2012).