Eigenvalues of the Canonical Height Pairing and the Unbounded Rank Conjecture — E8 Intelligence Research
- 1Independent Researcher, United Kingdom
Abstract
FINDING: The canonical height pairing on elliptic curves defines a positive-definite quadratic form whose matrix eigenvalues encode the curve's rank and lattice structure — the rank conjecture asks whether this rank is unbounded over families. MATH: - Elliptic curve: E: y2 = x3 + ax + b, a,b ∈ ℚ - Mordell–Weil group: E(ℚ) ≅ ℤr ⊕ Etors - Canonical height: ĥ(P) = limn→∞ 4−n h(2n P) - Height pairing: ⟨ P,Q ⟩ = 1/2[ĥ(P+Q) − ĥ(P) − ĥ(Q)] - Matrix Mij = ⟨ Pi, Pj ⟩ for independent points → eigenvalues λ1 ≥ λ2 ≥ … ≥ λr > 0 - Rank r = number of positive eigenvalues (or dimension of kernel of M over ℝ) - Birch–Swinnerton-Dyer: r = ords=1 L(E,s) - Rank conjecture: sup{r : ∃ E/ℚ with rank r} = ∞ (open; current record \(r \ge 28
Author: Andrew Stewart Caldin, Independent Researcher, UK. Part of the E8 Intelligence Research series. Platform: e8intelligence.com
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BibTeX
@misc{caldin2026eigenvalues,
title = {{Eigenvalues of the Canonical Height Pairing and the Unbounded Rank Conjecture — E8 Intelligence Research}},
author = {Andrew Stewart Caldin},
howpublished = {Zenodo},
year = {2026},
month = oct,
doi = {10.5281/zenodo.23635613},
url = {https://zenodo.org/records/23635613},
note = {CC BY 4.0},
}
RIS
TY - UNPB
TI - Eigenvalues of the Canonical Height Pairing and the Unbounded Rank Conjecture — E8 Intelligence Research
AU - Andrew Stewart Caldin
PY - 2026
DA - 2026/10/25
PB - Zenodo
DO - 10.5281/zenodo.23635613
UR - https://zenodo.org/records/23635613
AB - FINDING: The canonical height pairing on elliptic curves defines a positive-definite quadratic form whose matrix eigenvalues encode the curve's rank and lattice structure — the rank conjecture asks whether this rank is unbounded over families. MATH: - Elliptic curve: E: y^2 = x^3 + ax + b, a,b ∈ ℚ - Mordell–Weil group: E(ℚ) ≅ ℤ^r ⊕ E_tors - Canonical height: ĥ(P) = lim_n→∞ 4^−n h(2^n P) - Height pairing: ⟨ P,Q ⟩ = 1/2[ĥ(P+Q) − ĥ(P) − ĥ(Q)] - Matrix M_ij = ⟨ P_i, P_j ⟩ for independent points → eigenvalues λ_1 ≥ λ_2 ≥ … ≥ λ_r > 0 - Rank r = number of positive eigenvalues (or dimension of kernel of M over ℝ) - Birch–Swinnerton-Dyer: r = ord_s=1 L(E,s) - Rank conjecture: sup{r : ∃ E/ℚ with rank r} = ∞ (open; current record \(r \ge 28 Author: Andrew Stewart Caldin, Independent Researcher, UK. Part of the E8 Intelligence Research series. Platform: e8intelligence.com
ER -