Elliptic Curves over Real Quadratic Fields are Modular (1310.7088v4)

Abstract: We prove that all elliptic curves defined over real quadratic fields are modular.

Modularity of Elliptic Curves Over Real Quadratic Fields

The paper "Elliptic Curves over Real Quadratic Fields are Modular," authored by Nuno Freitas, Bao V. Le Hung, and Samir Siksek, addresses the complex problem of establishing the modularity of elliptic curves over real quadratic fields. This work represents an extension of the well-established modularity of elliptic curves over the rational numbers, an achievement that played a pivotal role in Wiles' proof of Fermat’s Last Theorem.

Main Result and Strategy

The principal result of the paper is the proof that all elliptic curves defined over real quadratic fields are modular. This generalization of the previously known results to the setting of real quadratic fields is achieved through a series of intricate arguments and techniques primarily involving the paper of Galois representations and modular forms.

The proof strategy for this result is analogous to the approach taken in the case of elliptic curves over the rationals. It involves:

1. Modularity Lifting Theorems: The authors utilize modularity lifting theorems, which under suitable conditions allow for the lifting of modularity from Galois representations to the modularity of elliptic curves themselves.
2. Langlands–Tunnell Theorem: This theorem plays a foundational role in establishing the modularity of elliptic curves with certain irreducibility assumptions on their residual representations.
3. 3–5 and 3–7 Modularity Switching: These strategies are implemented to switch the focus between different primes to more tractable forms while maintaining modularity.
4. Handling Exception Cases: Similar to the approach over the rationals, exceptional cases where the image sizes of Galois representations are small are dealt with by an enumeration of points over modular curves.

Galois Representations and Modularity

A significant portion of the paper is devoted to examining the Galois representations associated with elliptic curves over real quadratic fields. The modularity of these representations is shown to imply the modularity of the curves themselves. Using the properties of these representations (e.g., the way they behave under restriction and lifting), the authors argue for their compatibility with the modular forms framework over real quadratic fields.

Computational Aspects

The authors make extensive use of computational techniques and various software tools, such as the Magma algebra system, to verify numerous aspects of their arguments. This computational component is essential for enumerating rational points on modular curves and verifying the finiteness results critical for proving modularity.

Implications and Future Directions

The implications of this work are far-reaching both in theoretical and practical terms. Theoretical advancements in understanding the structure of elliptic curves and their associated Galois representations could lead to significant progress in number theory, particularly concerning Diophantine equations over various fields.

Practically, these results open avenues for further research into the modularity of elliptic curves over broader fields, and potentially, higher-dimensional generalizations. Moreover, the techniques developed and utilized in this paper provide a framework and methodological foundation that could be adapted to other contexts within algebraic geometry and arithmetic.

Conclusion

The authors' contribution to the field provides a crucial confirmation of the modularity conjecture for elliptic curves over real quadratic fields, broadening the scope of modularity results and enhancing our understanding of elliptic curves. This work is a testament to the sophisticated interplay between number theory, algebraic geometry, and computational mathematics. The accomplishments detailed in the paper could serve as a catalyst for future research aimed at unraveling further complexities surrounding modularity in more generalized settings.