Arithmetic-First BSD Computations Across Ranks 1–5: Tensor-Cube Verification, Isogeny Invariance and Mordell–Weil Lattice Saturation
Year: 2026
Type: preprint
Abstract: The Birch–Swinnerton-Dyer (BSD) conjecture establishes one of the deepest proposed connections in modern number theory: a relationship between the analytic behavior of elliptic-curve L-functions and the arithmetic structure of rational points. This research develops and investigates an arithmetic-first computational framework for the BSD quotient across elliptic curves of ranks 1–5, combining Mordell–Weil regulators, canonical heights, torsion subgroups, Tamagawa factors, real periods, and numerical L-function evaluations. Rather than treating the BSD formula solely as an analytic prediction, the study emphasizes the construction of arithmetic quantities from rational-point data and their subsequent comparison with analytic leading coefficients. Principal Research Results 1. BSD quotient computations across ranks 1–5 The study presents arithmetic and numerical investigations of BSD quotients across multiple Mordell–Weil ranks, including higher-rank examples. The computations examine the relationship between independently constructed arithmetic factors and the corresponding analytic leading terms. 2. Tensor-cube leading-term identities Using the factorization L(E × E × E, s) = L(Sym³ E, s) × L(E, s − 1)² the research investigates the fourth- and fifth-derivative leading terms at the central point s=2 for rank-two elliptic curves. Both functional-equation sign branches are examined, with reported numerical agreement exceeding 15 significant digits in the principal examples. 3. Verification across isogeny classes The tensor-cube calculations are extended from individual curves to complete isogeny classes. Three isogeny classes comprising eight elliptic curves are investigated, demonstrating numerical consistency between common analytic leading terms and the changing arithmetic factors of individual isogenous curves. For the four-member isogeny class 2175c, the reported agreement reaches approximately 33 decimal digits. 4. Mordell–Weil lattice saturation and regulator corrections The computations identify cases in which initially obtained rational points generate a subgroup of nontrivial index. Explicit index-three corrections demonstrate the importance of lattice saturation when calculating canonical-height regulators and comparing BSD quotients. 5. Broader arithmetic investigations The study also documents computational investigations involving descent, Tate–Shafarevich groups, p-adic calculations, and higher-rank examples, including the numerical behavior of BSD-related quantities under isogenies. Significance The central contribution is a consolidated computational investigation connecting rational-point arithmetic, regulator calculations, BSD leading coefficients, tensor-cube L-functions, and isogeny-class consistency. The results provide numerical evidence and reproducible computational examples for studying the arithmetic–analytic correspondence predicted by BSD. The tensor-cube leading-term identities follow from the underlying L-function factorization; their role here is to provide an additional computational framework for testing arithmetic consistency. The numerical evidence does not constitute a proof of the general BSD conjecture. Research Materials The accompanying archive contains the research manuscript, computational scripts, numerical results, and reproducibility documentation. Author: Joona Matti Ensio EskelinenYear: 2026Research areas: Number Theory, Arithmetic Geometry, Elliptic Curves, L-functions, Computational Mathematics
Author: Eskelinen, Joona Matti Ensio
Language: English
Subfield: Algebra and Number Theory
Field: Mathematics
Domain: Physical Sciences
Open Access status: green
OpenAlex ID: W7221878571
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