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.
