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

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

Andrew Stewart Caldin

doi:10.5281/zenodo.22245323

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 evaluate the average size of 2-Selmer groups over function fields and test the validity of Poonen–Rains heuristics under power-law corrections.
Analyzed the exact sequence relating the Mordell–Weil group, the 2-Selmer group, and the 2-torsion of the Tate–Shafarevich group.
Modeled asymptotic bounds across function fields parameterized by field size constant q and curve genus g.
Determined that the average 2-Selmer group size scales asymptotically as σ₂ + O(q⁻ᵍ), where σ₂ = ∑ 2⁻ⁿ² ≈ 2.1289.
Identified that the power-law correction term O(q⁻ᵍ) reflects root lattice symmetries and partition functions analogous to the Gaussian unitary ensemble.

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Implication

Theoretical analysis demonstrates power-law corrections to Poonen–Rains heuristics for 2-Selmer groups in function fields, indicating deep links to modular forms and root lattices.

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Andrew Stewart Caldin (2026) studied this question.

synapsesocial.com/papers/6a9a862b5d9e33f25c631c04 https://doi.org/10.5281/zenodo.22245323

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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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