- The paper proves that the normaliser of the non-split Cartan subgroup at level 27 cannot be the 3-adic Galois image for non-CM elliptic curves.
- It employs quadratic Chabauty over number fields to overcome computational challenges and determine all 8 rational points on the modular curve Xₙₛ⁺(27)(ℚ).
- The study finalizes the classification of 3-adic images and expands computational methods for analyzing modular curves with high Mordell–Weil ranks.
Rational Points on the Non-Split Cartan Modular Curve of Level 27 and Quadratic Chabauty Over Number Fields
This paper addresses a longstanding open issue in the classification of $3$-adic Galois images of non-CM elliptic curves over $\Q$. Specifically, it focuses on the classification within the $3$-adic context, resolving the remaining question by proving that the normaliser of the non-split Cartan subgroup at level 27 does not occur as an image of a $3$-adic Galois representation. This result completes the classification of 3-adic images, a task previously left incomplete due to computational challenges posed by the curve $X_{\ns}^+(27)$.
Key Results and Methodology
The authors utilize a novel approach involving quadratic Chabauty methods over number fields to overcome the computational limitations posed by the large Mordell–Weil rank of the Jacobian associated with the relevant curve. The primary numerical results are as follows:
- It is shown that the normaliser of the non-split Cartan subgroup of level 27 cannot be the 3-adic Galois image of a non-CM elliptic curve.
- The authors determine that the set of rational points on the modular curve $X_{\ns}^+(27)(\Q)$ consists of 8 points, each corresponding to non-CM discriminants, using the quadratic Chabauty method.
- They also identify all $X_{\ns}^+(27)(F)$ for the quadratic field $F=\Q(\zeta_3)$, further pinpointing CM points characterized by specific discriminants.
The research presented is rooted in the extension and application of the quadratic Chabauty method over number fields, implemented using the computer algebra system {\tt Magma}. This method extends prior approaches by incorporating reduction via scalars and applying Nekovárič heights, allowing for the study of rational points on modular curves with higher Mordell–Weil ranks than previously feasible.
Theoretical Implications
The theoretical landscape of arithmetic geometry is enriched by providing a comprehensive classification of 3-adic Galois representations. This result extends the work of Mazur on rational points on modular curves and Rouse, Sutherland, and Zureick-Brown on elliptic curve images, concluding decades of research in this area. Furthermore, the method expands the toolkit available for studying rational points on modular curves, offering potential applications beyond the current problem scope.
Practical and Computational Implications
Practically, this work highlights the growing utility of computational techniques in arithmetic geometry. The paper demonstrates that, even in scenarios where traditional methods are computationally prohibitive, methods like quadratic Chabauty, particularly over number fields, can yield solutions to complex problems. This suggests robustness and flexibility that may pave the way for tackling other unresolved problems with large rank and genus constraints.
Conclusion
This study address essential gaps in understanding $3$-adic images in elliptic curve Galois representations, particularly concerning the non-split Cartan subgroup. By innovatively extending computational methodologies like quadratic Chabauty, the authors chart a path forward for future research into related algebraic and arithmetic problems. The paper's results are significant, providing closure to an important classification problem and extending methodologies that hold promise across various domains within arithmetic geometry.