Symplectic Symmetries in Selmer Ranks of Quadratic Twists — E8 Intelligence Research

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Selmer group ranks in quadratic twist families of elliptic curves are governed by symplectic root system symmetries, with mod-2 torsion modules exhibiting congruence structures tied to Weyl group actions. Novelty: ClaimNovelty.INCREMENTAL. Consensus alignment: ConsensusAlignment.NEUTRAL.

Andrew Stewart Caldin

doi:10.5281/zenodo.22245111

September 2, 2026Open Access

Symplectic Symmetries in Selmer Ranks of Quadratic Twists — E8 Intelligence Research

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Key Points

Investigate the algebraic and symmetry structures governing 2-Selmer group ranks and parity across quadratic twist families of elliptic curves.
Analyzed 2-Selmer rank parity in quadratic twists using the parity conjecture relating 2-Selmer rank to the analytic rank of L-functions mod 2.
Evaluated the symplectic structure of mod-p torsion modules endowed with the Weil pairing.
Formulated a graph-theoretic algorithm over the Gaussian integers computing the phi-Selmer group using weighted graphs based on odd Gaussian primes and Legendre symbols.
Demonstrated that Selmer group ranks in quadratic twist families are governed by symplectic root system symmetries, specifically connecting actions of Sp_{2g}(F_2) to C_g root systems.
Established that mod-2 torsion modules exhibit congruence structures directly constrained by Weyl group operations.
Showed that phi-Selmer groups for specific elliptic curves over Q(i) map directly to weighted graph invariants determined by Legendre symbols.

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Theoretical analysis reveals symplectic root system symmetries in Selmer group ranks of quadratic twists, indicating algebraic structures governed by Weyl group actions.

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Andrew Stewart Caldin (2026) studied this question.

synapsesocial.com/papers/6a9a861f5d9e33f25c63199b https://doi.org/10.5281/zenodo.22245111

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Symplectic Symmetries in Selmer Ranks of Quadratic Twists — E8 Intelligence Research

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Symplectic Symmetries in Selmer Ranks of Quadratic Twists — E8 Intelligence Research

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