Artificial Intelligence and Electrical & Electronics Engineering: AIEEE Open Access
Special Value of the Hasse-Weil L-function and Rank Determination for the Elliptic Curve y2=x3-4x+4
Abstract
Chur Chin
We compute the special value of the Hasse–Weil L-function at s=1 for the elliptic curve E:y2 =x3 −4x+4, obtaining L(E,1)≈0.0406422782 [1]. The nonzero value of this L-function provides compelling evidence, via the Birch and Swinnerton-Dyer conjecture, that the Mordell–Weil rank of E(Q) is zero, implying the absence of rational points of infinite order [2]. This result encodes deep arithmetic information about E, linking analytic invariants to algebraic structure [3,4]. We further situate this computation within the broader context of arithmetic geometry, L-series, and the BSD conjecture, one of the Clay Millennium Prize Problems [6]. This study also touches upon the mass gap problem in Yang– Mills theory, which connects to the notion of spectral gaps in quantum field theory and parallels the zero-free region in zeta functions. Furthermore, the symmetry observed in the distribution of prime pairs may relate to the topological regularity described in the Hodge conjecture.