What does this research mean for the field?

The average size of the n-Selmer group of elliptic curves over ℚ is σ(n) for n=2,3,4,5, which bounds the average rank to ≤ 7/6 and exhibits numerical connections to the ranks of specific crystallographic root systems. Novelty: ClaimNovelty.INCREMENTAL. Consensus alignment: ConsensusAlignment.NEUTRAL.

What question did this study set out to answer?

Evaluate the theoretical implications of the average n-Selmer group size equaling the divisor sum function σ(n) on establishing rank bounds for rational elliptic curves.

September 8, 2026Open Access

Average Selmer Size σ(n) Proves Elliptic Curve Rank Bound — E8 Intelligence Research

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Evaluate the theoretical implications of the average n-Selmer group size equaling the divisor sum function σ(n) on establishing rank bounds for rational elliptic curves.
Analyzed the average size of the n-Selmer group over rational numbers for n = 2, 3, 4, and 5.
Evaluated root number equidistribution and Tamagawa product averages to assess curve parity and algebraic rank bounds.
Mapped calculated divisor sums to the dimensions and ranks of exceptional Lie algebras and crystallographic root systems.
Determined average n-Selmer group sizes of 3, 4, 7, and 6 for n = 2, 3, 4, and 5, respectively, establishing an upper bound on average rank of 7/6.
Demonstrated that combining Selmer bounds with a 50% positive and 50% negative root number distribution yields an average rank of 1/2 for 100% of curves.
Identified that the average Tamagawa product equals 1 and that divisor values correspond directly to ranks of crystallographic root systems A₂, A₃, D₄, and A₅.

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Implication

Mathematical analysis reveals average Selmer group sizes bound elliptic curve ranks in rational curves, indicating underlying symmetries with crystallographic root systems.

Cite This Study

Andrew Stewart Caldin (2026) studied this question.

synapsesocial.com/papers/6a9fd78a58e84d0ff5b46446 https://doi.org/10.5281/zenodo.22471400

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Implication

Mathematical analysis reveals average Selmer group sizes bound elliptic curve ranks in rational curves, indicating underlying symmetries with crystallographic root systems.

Average Selmer Size σ(n) Proves Elliptic Curve Rank Bound — E8 Intelligence Research

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Implication

Cite This Study

Average Selmer Size σ(n) Proves Elliptic Curve Rank Bound — E8 Intelligence Research

Key Points

Discussion

Implication

Cite This Study

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