Abelian Surfaces over totally real fields are Potentially Modular

Abstract: We show that abelian surfaces (and consequently curves of genus 2) over totally real fields are potentially modular. As a consequence, we obtain the expected meromorphic continuation and functional equations of their Hasse--Weil zeta functions. We furthermore show the modularity of infinitely many abelian surfaces A over Q with End_C(A)=Z. We also deduce modularity and potential modularity results for genus one curves over (not necessarily CM) quadratic extensions of totally real fields.

Overview of the Paper on Potential Modularity of Abelian Surfaces

The paper "Abelian Surfaces over Totally Real Fields are Potentially Modular" by Boxer, Calegari, Gee, and Pilloni addresses potential modularity of abelian surfaces, particularly those defined over totally real fields. The paper leverages deep insights from number theory and algebraic geometry, focusing on the $L$-functions and cohomological aspects of abelian surfaces, and employs techniques developed through modularity lifting and deformation theory.

Main Contributions and Theorems

1. Potential Modularity for Abelian Surfaces: The authors establish that abelian surfaces over totally real fields are potentially modular. This result extends the modularity principle known for elliptic curves, as shown by Wiles and others, to higher-dimensional abelian varieties.
2. Hasse--Weil Zeta Functions: The paper demonstrates that the expected meromorphic continuation and functional equations hold for the Hasse--Weil zeta functions of abelian surfaces, directly following the potential modularity result.
3. Infinitely Many Modularity Cases: A significant portion of the paper focuses on showing the modularity of infinitely many abelian surfaces $A$ over $\mathbb{Q}$, with $\End_{\mathbb{Q}}A = \mathbb{Z}$, using specialized techniques.
4. Potential Modularity for Genus One Curves: The authors extend their results to genus one curves over quadratic extensions of totally real fields, aligning with traditional results known for elliptic curves.

Technical Developments and Insights

• Modularity Lifting Theorems:

The paper employs modularity lifting techniques similar to those used in proving the Taniyama--Shimura conjecture for elliptic curves. This involves sophisticated patching arguments to handle the representation-theoretic aspects that underlie the transition from potential automorphy to modularity.

• Automorphic Forms:

The authors speculate on the future development of automorphic forms attached to $\mathrm{GSp}_4$, suggesting an intricate relationship between symmetric powers of automorphic forms and abelian varieties of general type.

• Use of Symplectic Representations:

By reducing the problem to symplectic representations, the authors utilize the duality theory between algebraic and automorphic structures, making significant use of the symplectic nature of the Galois representations associated to abelian surfaces.

Applications

The results have profound implications for number theory and the broader field of arithmetic geometry. The potential modularity of abelian surfaces indicates that their $L$-functions are closely linked to automorphic forms, offering insights that might extend to higher-dimensional varieties. Furthermore, understanding the modularity of these surfaces could have ramifications for the Langlands program and its conjectural framework linking Galois representations with automorphic forms.

Theoretical Implications

The findings push the boundaries of known results from the modularity of elliptic curves to higher dimensions in the field of abelian surfaces. This provides new opportunities to explore the arithmetic of surfaces and might help in formulating stronger conjectures on modularity in mathematics.

Conclusion

Boxer, Calegari, Gee, and Pilloni's paper enriches the understanding of modular properties in the arithmetic of abelian surfaces, proposing methodologies and results that could influence further research in algebraic number theory. Their approach and innovation in modularity lifting are crucial steps in potentially simplifying the concept of modularity for complex algebraic structures.