New Rank Records For Elliptic Curves Having Rational Torsion (2003.00077v1)

Abstract: We present rank-record breaking elliptic curves having torsion subgroups Z/2Z, Z/3Z, Z/4Z, Z/6Z, and Z/7Z.

• The paper sets new rank records for elliptic curves with rational torsion subgroups including Z/2Z, Z/3Z, Z/4Z, Z/6Z, and Z/7Z.
• The research utilizes and improves computational methods, such as the Mestre-Nagao technique and efficient sieving, to search for high-rank curves.
• Key results, such as a rank 20 curve for Z/2Z torsion and rank 15 for Z/3Z, provide evidence relevant to the conjecture on the boundedness of elliptic curve ranks.

Advances in High-Rank Elliptic Curves with Rational Torsion Subgroups

The paper "New Rank Records for Elliptic Curves Having Rational Torsion" by Noam D. Elkies and Zev Klagsbrun presents substantial advances in discovering high-rank elliptic curves over rational torsion subgroups. This research contributes to the ongoing debate regarding the boundedness of ranks in elliptic curves as posited by the Mordell-Weil theorem. The authors successfully break existing rank records for several torsion groups, namely $\mathbb{Z}/2\mathbb{Z}$, $\mathbb{Z}/3\mathbb{Z}$, $\mathbb{Z}/4\mathbb{Z}$, $\mathbb{Z}/6\mathbb{Z}$, and $\mathbb{Z}/7\mathbb{Z}$, while making strides in computational methodologies.

Methodological Advances

The research leverages the established method of Mestre and Nagao to search for elliptic curves of high rank. This involves the exploitation of elliptic fibrations over $\mathbb{Q}(t)$ to identify instances where specializations achieve notably high rank. By employing computations involving the Birch and Swinnerton-Dyer conjecture and modular curve constructions, they improve the constraints on their search, increasing both the scope and accuracy of their findings.

A critical innovation in this paper is the utilization of computational techniques based on Nagao’s sieving method. This approach accelerates the search process by allowing large-scale computations for rank evaluations efficiently. The authors take advantage of modular arithmetic properties and algorithmic sieving to reduce computational load, especially for estimating rank scores $S(t, B)$.

Numerical Results and Records

Elkies and Klagsbrun successfully establish new rank records for the torsion subgroups studied. Notably, a curve of rank 20 was identified for the $\mathbb{Z}/2\mathbb{Z}$ subgroup, marking the highest known rank for this group. For $\mathbb{Z}/3\mathbb{Z}$ and $\mathbb{Z}/4\mathbb{Z}$, the paper identifies curves reaching ranks of 15 and 13 respectively. Additionally, they accomplish a rank of 9 for curves with $\mathbb{Z}/6\mathbb{Z}$ torsion. These results not only set new records but also exceed previously held heuristic bounds which were considered optimistic.

Theoretical and Practical Implications

The results of this paper bear significant theoretical implications, particularly in the context of the conjectured boundedness of elliptic curve ranks. The minimal breakage of prior records despite extensive searches provides evidence that curve ranks might not increase indefinitely, suggesting potential upper bounds that are smaller than conjectured by certain heuristic models. This insight is crucial in the paper of rational points on elliptic curves, which has profound implications in number theory and cryptography.

On the practical side, the advances in computational techniques introduced here can be employed in other upper-bound explorations in arithmetic geometry and number theory. The paper explicitly posits open questions, such as the precise scaling of prime bounds in relation to search regions and the intricate relationship between Tamagawa factors and rank estimation. Addressing these questions could refine the methodological approaches to elliptic curve research further.

Future Directions

Future developments could involve exploring these open questions and their impact on heuristic approaches to bounding elliptic curve ranks. Additionally, incorporating Bayesian methods for estimating probabilities of high-rank findings more precisely would be a worthwhile exploration that would build on the framework presented here.

Overall, Elkies and Klagsbrun provide a robust groundwork and remarkable results which push the limitations of computational number theory, and they invite further exploration into the capabilities and limitations of existing methods around elliptic curves and their ranks.