Periodic Reporting for period 1 - IwCMEBSD (Iwasawa theory of elliptic curves and the Birch–Swinnerton-Dyer conjecture)
Reporting period: 2022-03-01 to 2024-02-29
The Birch–Swinnerton-Dyer conjecture is one of the seven Millennium Prize Problems, and it is widely considered to be one of the most important and challenging open problems in modern mathematics. It concerns elliptic curves, which are indispensable in today’s society as they are extensively used in the form of cryptography, making secure online communication and commerce possible. Iwasawa theory is an extremely powerful method for studying the conjecture. It serves as a bridge between the two mysterious mathematical objects appearing in the Birch–Swinnerton-Dyer formula (the Tate–Shafarevich group and the complex L-series) which are completely different in nature, by breaking the formula up into a p-part for all prime numbers p. In order to verify the full conjecture for an elliptic curve, the p-part of the conjecture must be verified for all prime numbers p. Due to serious technical difficulties, the classical theory can only deal with odd prime numbers and fails to obtain the most interesting case, the 2-part of the conjecture, for any elliptic curve. This project explores and applies the new tools I have developed in Iwasawa theory in order to make substantial progress in the eschewed case for a wide class of elliptic curves and abelian varieties.
Throughout the project, significant progress has been made in the field of number theory, particularly in the study of elliptic curves and their associated L-functions. Below is an overview of the key findings from our research, categorised into four related works:
1. Investigating cube sums and associated Elliptic Curves ("On central L-values and the growth of the 3-part of the Tate-Shafarevich group", International Journal of Number Theory, Vol. 19, No. 04, pp. 785-802 (2023)): I explored the properties of certain types of integers N (positive cube-free integers N which are sums of two rational numbers) in relation to elliptic curves, which are geometric shapes defined by cubic equations. The question of determining which integers are sums of two cubes date back to Diophantus, and still remains largely unknown. In this work, I showed that the 3-adic valuation of the central value of the L-function associated to the corresponding elliptic curve increases with the number of distinct prime factors N. I also studied the 3-part of the Tate–Shafarevich group for these curves and showed that the lower bound is as expected from the conjecture of Birch and Swinnerton-Dyer, taking into account also the growth of the Tate–Shafarevich group.
2. Non-Vanishing Theorems and Elliptic Curves with Complex Multiplication(“Non-vanishing of central L-values of the Gross family of elliptic curves” joint with Y-X. Li, submitted): We established important non-vanishing theorems for central values associated with specific families of elliptic curves. These curves have a special mathematical structure known as complex multiplication. From this, we obtain the finiteness of their Tate-Shafarevich groups, verifying the Tate-Shafarevich conjecture for these curves which is another fundamental conjecture in number theory.
3. Advancements in Non-Commutative Iwasawa Theory (“The M_H(G)-conjecture and Akashi series of dual Selmer groups over global function fields”, joint with Y. Li, L. Deng, M-F. Li, preprint): We delved into the intricate realm of non-commutative Iwasawa theory. In their 5-author paper, Coates, Fukaya, Kato, Sujatha and Venjakob set up a ground for non-commutative Iwasawa theory for elliptic curves without complex multiplication, and formulated related conjectures. The key conjecture is the so-called M_H(G)-conjecture. We validate this conjecture for abelian varieties over global functions fields.
4. Birch-Swinnerton-Dyer conjecture for abelian varieties with complex multiplication (“On the Birch and Swinnerton-Dyer formula for quadratic twists of certain abelian varieties with complex multiplication at p=2,”in finalisation): In the first work, the non-vanishing result was obtained via Iwasawa theory at one special prime above 2. In this work, we develop Iwasawa theory at all primes above 2. This gives us the 2-part of the Birch–Swinnerton-Dyer conjecture for the corresponding abelian varieties.
1. Investigating cube sums and associated Elliptic Curves ("On central L-values and the growth of the 3-part of the Tate-Shafarevich group", International Journal of Number Theory, Vol. 19, No. 04, pp. 785-802 (2023)): I explored the properties of certain types of integers N (positive cube-free integers N which are sums of two rational numbers) in relation to elliptic curves, which are geometric shapes defined by cubic equations. The question of determining which integers are sums of two cubes date back to Diophantus, and still remains largely unknown. In this work, I showed that the 3-adic valuation of the central value of the L-function associated to the corresponding elliptic curve increases with the number of distinct prime factors N. I also studied the 3-part of the Tate–Shafarevich group for these curves and showed that the lower bound is as expected from the conjecture of Birch and Swinnerton-Dyer, taking into account also the growth of the Tate–Shafarevich group.
2. Non-Vanishing Theorems and Elliptic Curves with Complex Multiplication(“Non-vanishing of central L-values of the Gross family of elliptic curves” joint with Y-X. Li, submitted): We established important non-vanishing theorems for central values associated with specific families of elliptic curves. These curves have a special mathematical structure known as complex multiplication. From this, we obtain the finiteness of their Tate-Shafarevich groups, verifying the Tate-Shafarevich conjecture for these curves which is another fundamental conjecture in number theory.
3. Advancements in Non-Commutative Iwasawa Theory (“The M_H(G)-conjecture and Akashi series of dual Selmer groups over global function fields”, joint with Y. Li, L. Deng, M-F. Li, preprint): We delved into the intricate realm of non-commutative Iwasawa theory. In their 5-author paper, Coates, Fukaya, Kato, Sujatha and Venjakob set up a ground for non-commutative Iwasawa theory for elliptic curves without complex multiplication, and formulated related conjectures. The key conjecture is the so-called M_H(G)-conjecture. We validate this conjecture for abelian varieties over global functions fields.
4. Birch-Swinnerton-Dyer conjecture for abelian varieties with complex multiplication (“On the Birch and Swinnerton-Dyer formula for quadratic twists of certain abelian varieties with complex multiplication at p=2,”in finalisation): In the first work, the non-vanishing result was obtained via Iwasawa theory at one special prime above 2. In this work, we develop Iwasawa theory at all primes above 2. This gives us the 2-part of the Birch–Swinnerton-Dyer conjecture for the corresponding abelian varieties.
The methods developed and the results obtained in the project has far-reaching consequences for the arithmetic of elliptic curves and its applications. In addition, it provides us with a key step towards extending the theory for higher dimensional automorphic groups, which will be crucial for the development of non-abelian class field theory—a theory for which mathematicians are eagerly seeking a definitive formulation. These findings not only deepen our theoretical knowledge but also pave the way for further breakthroughs in this captivating field of study.