Smith's Proof of Goldfeld's Conjecture via Selmer Groups and Cassels-Tate — E8 Intelligence Research
Year: 2026
Type: preprint
Abstract: FINDING: Alex Smith's proof of Goldfeld's conjecture via 2∞-Selmer groups and Cassels-Tate pairing, plus Poonen's random maximal isotropic subspace model for Selmer groups. | MATH: Goldfeld's conjecture: average rank of elliptic curves over ℚ is 1/2. Smith's result: for 2∞-Selmer groups of quadratic twists, the distribution matches a random maximal isotropic subspace of a symplectic ℤ₂-module. Cassels-Tate pairing: a nondegenerate alternating bilinear form on the 2-primary Selmer group, inducing a self-duality. Poonen's model: Selmer groups behave like random maximal isotropic subspaces in a finite symplectic vector space over ℤ/2ⁿℤ, with probability mass ∝ q^{-k²} (q=2). | CONNECTION: The symplectic structure of the Cassels-Tate pairing is the arithmetic shadow of the intersection form on a genus-g surface (root system D_g, Weyl group). The 2-adic filtration of Selmer groups mirrors the dyadic lattice ℤ₂ — a base-2 analogue of base-60's place-value structure. The isotropic condition ( Author: Andrew Stewart Caldin, Independent Researcher, UK. Part of the E8 Intelligence Research series. Platform: e8intelligence.com
Author: Caldin, Andrew Stewart
Language: English
Subfield: Algebra and Number Theory
Field: Mathematics
Domain: Physical Sciences
Open Access status: green
OpenAlex ID: W7210273514
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