Refined BSD Conjecture: Linking Elliptic Curve Rank to L-Function Vanishing — E8 Intelligence Research
- 1Independent Researcher, United Kingdom
Abstract
FINDING: The Birch and Swinnerton-Dyer (BSD) conjecture links the algebraic rank of an elliptic curve (number of independent rational points) to the order of vanishing of its L-function at s=1, with recent progress on the 2-part for quadratic twists. | MATH: BSD conjecture: ords=1 L(E,s) = rank(E(ℚ)). The refined BSD formula: L(r)(E,1)/(r! ΩE) = (|Ш(E)|· ∏p cp · Tamagawa)/#E(ℚ)tors2. Key constants: ΩE (real period), cp (Tamagawa numbers), Ш(E) (Tate-Shafarevich group). Recent result (arXiv:1712.01271): for a large class of elliptic curves, explicit infinite families of quadratic twists with analytic rank 0 exist, and the 2-part of BSD holds for many — i.e., ords=1 L(Ed,s)=0 implies Ш(Ed)[2] is finite and its size matches the predicted value. | CONNECTION: Elliptic curves are 1-dimensional abelian varieties — their rat
Author: Andrew Stewart Caldin, Independent Researcher, UK. Part of the E8 Intelligence Research series. Platform: e8intelligence.com
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BibTeX
@misc{caldin2026refined,
title = {{Refined BSD Conjecture: Linking Elliptic Curve Rank to L-Function Vanishing — E8 Intelligence Research}},
author = {Andrew Stewart Caldin},
howpublished = {Zenodo},
year = {2026},
month = oct,
doi = {10.5281/zenodo.23446956},
url = {https://zenodo.org/records/23446956},
note = {CC BY 4.0},
}
RIS
TY - UNPB
TI - Refined BSD Conjecture: Linking Elliptic Curve Rank to L-Function Vanishing — E8 Intelligence Research
AU - Andrew Stewart Caldin
PY - 2026
DA - 2026/10/18
PB - Zenodo
DO - 10.5281/zenodo.23446956
UR - https://zenodo.org/records/23446956
AB - FINDING: The Birch and Swinnerton-Dyer (BSD) conjecture links the algebraic rank of an elliptic curve (number of independent rational points) to the order of vanishing of its L-function at s=1, with recent progress on the 2-part for quadratic twists. | MATH: BSD conjecture: ord_s=1 L(E,s) = rank(E(ℚ)). The refined BSD formula: L^(r)(E,1)/(r! Ω_E) = (|Ш(E)|· ∏_p c_p · Tamagawa)/#E(ℚ)_tors^2. Key constants: Ω_E (real period), c_p (Tamagawa numbers), Ш(E) (Tate-Shafarevich group). Recent result (arXiv:1712.01271): for a large class of elliptic curves, explicit infinite families of quadratic twists with analytic rank 0 exist, and the 2-part of BSD holds for many — i.e., ord_s=1 L(E^d,s)=0 implies Ш(E^d)[2] is finite and its size matches the predicted value. | CONNECTION: Elliptic curves are 1-dimensional abelian varieties — their rat Author: Andrew Stewart Caldin, Independent Researcher, UK. Part of the E8 Intelligence Research series. Platform: e8intelligence.com
ER -