What does this research mean for the field?

The average rank of elliptic curves over ℚ is bounded by 7/6, and the average size of n-Selmer groups is exactly n+1 for n=2, 3, 4, and 5. Novelty: ClaimNovelty.SYNTHESIS. Consensus alignment: ConsensusAlignment.NEUTRAL.

What question did this study set out to answer?

To establish an explicit upper bound on the average algebraic rank of elliptic curves over the rational numbers using the distribution of Selmer group sizes.

September 3, 2026Open Access

Average Rank of Elliptic Curves Bounded by 7/6 via Selmer Group Sizes — E8 Intelligence Research

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To establish an explicit upper bound on the average algebraic rank of elliptic curves over the rational numbers using the distribution of Selmer group sizes.
Evaluated the asymptotic distribution of elliptic curves over rational numbers ordered by height.
Calculated the exact average sizes of n-Selmer groups for n = 2, 3, 4, and 5.
The average algebraic rank of elliptic curves over ℚ is bounded above by 7/6.
The average size of n-Selmer groups equals exactly n + 1 for n = 2, 3, 4, and 5.

Implication

Mathematical analysis reveals an upper bound of 7/6 on the average rank of elliptic curves over rational numbers, suggesting tight constraints on rational point generation.

Abstract

FINDING: Bhargava–Shankar proved the average rank of elliptic curves over ℚ is bounded (≤ 7/6), with average size of n-Selmer groups exactly n+1 for n=2,3,4,5. | MATH: limX→∞ (1/N(X)) Σ_{E: ht(E) Author: Andrew Stewart Caldin, Independent Researcher, UK. Part of the E8 Intelligence Research series. Platform: e8intelligence.com

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Cite This Study

Andrew Stewart Caldin (2026) studied this question.

synapsesocial.com/papers/6a9935d8636c6408cfa7e64a https://doi.org/10.5281/zenodo.22225595

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Implication

Mathematical analysis reveals an upper bound of 7/6 on the average rank of elliptic curves over rational numbers, suggesting tight constraints on rational point generation.

Average Rank of Elliptic Curves Bounded by 7/6 via Selmer Group Sizes — E8 Intelligence Research

Key Points

Implication

Abstract

Discussion

Cite This Study

Average Rank of Elliptic Curves Bounded by 7/6 via Selmer Group Sizes — E8 Intelligence Research

Key Points

Implication

Abstract

Discussion

Cite This Study

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