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This research aims to prove the equality in Howard's Heegner point conjecture under the Kolyvagin descent framework, leading to insights into the rank-zero Birch and Swinnerton-Dyer conjecture (BSD).

July 22, 2026Open Access

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Equality in Howard's Heegner Point Conjecture Proves Rank-Zero BSD — E8 Intelligence Research

ACAndrew Stewart Caldin

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This research aims to prove the equality in Howard's Heegner point conjecture under the Kolyvagin descent framework, leading to insights into the rank-zero Birch and Swinnerton-Dyer conjecture (BSD).
Utilized Kolyvagin descent to contrast Howard's main conjecture with established results.
Applied Wei Zhang's proof of the Kolyvagin conjecture as an upgrade mechanism for equality.
Analyzed the relationship between Heegner points, CM fields, and their associated class numbers.
Established that for rank-zero BSD, L(E,1) ≠ 0 implies finite Ш(E) and order = |L(E,1)/Ω|².
Demonstrated no explicit constants or ratios emerge from the analysis.
Highlighted the connection between Heegner points and root system symmetries in Galois cohomology.

Abstract

FINDING: Howard's Heegner point Euler system upgraded to equality under Kolyvagin descent, proving Perrin-Riou's Heegner point Main Conjecture for rank-zero BSD. | MATH: Key objects: Heegner points (CM points on modular curves), Kolyvagin system (cohomology classes with Euler system norm relations), Howard's main conjecture (divisibility of p-adic L-values by Heegner point regulators). Upgrade to equality uses Wei Zhang's proof of Kolyvagin conjecture (2014). Implies rank-zero BSD: L (E, 1) ≠ 0 ⇒ Ш (E) finite and order = |L (E, 1) /Ω|². No explicit constants or ratios emerge. | CONNECTION: Heegner points are intrinsically linked to imaginary quadratic fields and complex multiplication (CM). CM fields have class numbers related to base-60 (sexagesimal) via Kronecker's Jugendtraum — e. g. , j-invariant values at CM points are algebraic integers often with base-60 expansions. The Kolyvagin system formalism involves root system symmetries (Aₙ, Dₙ) via Galois cohomology and descent. No direct 0. 3 Author: Andrew Stewart Caldin, Independent Researcher, UK. Part of the E8 Intelligence Research series. Platform: e8intelligence. com

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Equality in Howard's Heegner Point Conjecture Proves Rank-Zero BSD — E8 Intelligence Research

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