Average 2-Selmer Sizes in Function Fields: Poonen–Rains with Power-Law Corrections — E8 Intelligence Research

Q: What does this research mean for the field?

The average size of 2-Selmer groups over function fields aligns with the Poonen-Rains heuristics, incorporating a power-law correction of O(q^{-g}). Novelty: ClaimNovelty.CONFIRMATORY. Consensus alignment: ConsensusAlignment.NEUTRAL.

Andrew Stewart Caldin

doi:10.5281/zenodo.22245324

September 2, 2026Open Access

Average 2-Selmer Sizes in Function Fields: Poonen–Rains with Power-Law Corrections — E8 Intelligence Research

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Key Points

To determine the average size of 2-Selmer groups over function fields and assess the validity of Poonen–Rains heuristics incorporating power-law corrections.
Analyzed the exact sequence relating elliptic curve quotients, 2-Selmer groups, and 2-torsion Shafarevich–Tate groups over global function fields.
Computed asymptotic limits using theta-series formulations at q = 1/2 and lattice volume heuristics associated with the Gaussian unitary ensemble across genus g and field constant q.
Identified the average 2-Selmer group size over function fields as σ₂ + O(q⁻ᵍ), matching the conjectured baseline σ₂ = ∑_{n≥0} 2⁻ⁿ² ≈ 2.1289.
Demonstrated that the power-law correction term q⁻ᵍ reflects root lattice symmetries and the partition function of a Gaussian unitary ensemble.

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Implication

Mathematical analysis uncovers average 2-Selmer group sizes in function fields, indicating power-law corrections consistent with Poonen–Rains heuristics.

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Andrew Stewart Caldin (2026) studied this question.

synapsesocial.com/papers/6a9ab0dc5d9e33f25c631f3d https://doi.org/10.5281/zenodo.22245324

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Average 2-Selmer Sizes in Function Fields: Poonen–Rains with Power-Law Corrections — E8 Intelligence Research

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Discussion

Cite This Study

Average 2-Selmer Sizes in Function Fields: Poonen–Rains with Power-Law Corrections — E8 Intelligence Research

Key Points

No abstract yet

Implication

Discussion

Cite This Study

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