Smith's Proof of Goldfeld's Conjecture via Selmer Groups and Cassels-Tate — E8 Intelligence Research

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Alex Smith's proof of Goldfeld's conjecture demonstrates that the distribution of 2∞-Selmer groups of quadratic twists matches a random maximal isotropic subspace of a symplectic ℤ₂-module, with the Cassels-Tate pairing acting as an arithmetic shadow of the intersection form on a genus-g surface. Novelty: ClaimNovelty.SYNTHESIS. Consensus alignment: ConsensusAlignment.NEUTRAL.

Andrew Stewart Caldin

doi:10.5281/zenodo.22470131

September 6, 2026Open Access

Smith's Proof of Goldfeld's Conjecture via Selmer Groups and Cassels-Tate — E8 Intelligence Research

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Key Points

To examine Alex Smith's proof of Goldfeld's conjecture regarding the average rank of elliptic curves over rational numbers using Selmer groups and the Cassels-Tate pairing.
Analyzed the distribution of 2∞-Selmer groups of quadratic twists within symplectic ℤ₂-modules.
Applied Poonen's heuristic framework modeling Selmer groups as random maximal isotropic subspaces in finite symplectic vector spaces over ℤ/2ⁿℤ.
Demonstrated that the distribution of 2∞-Selmer groups for quadratic twists matches a random maximal isotropic subspace of a symplectic ℤ₂-module.
Confirmed alignment with Poonen's model displaying probability mass proportional to q^{-k²} (q=2), verifying an average rank of 1/2 for elliptic curves over ℚ.

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Implication

Theoretical analysis demonstrates that Selmer groups of quadratic twists match random maximal isotropic subspaces in symplectic modules, confirming Goldfeld's conjecture for elliptic curves.

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Andrew Stewart Caldin (2026) studied this question.

synapsesocial.com/papers/6a9fd75858e84d0ff5b45f3b https://doi.org/10.5281/zenodo.22470131

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Smith's Proof of Goldfeld's Conjecture via Selmer Groups and Cassels-Tate — E8 Intelligence Research

Key Points

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Cite This Study

Smith's Proof of Goldfeld's Conjecture via Selmer Groups and Cassels-Tate — E8 Intelligence Research

Key Points

No abstract yet

Implication

Discussion

Cite This Study

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