Average Selmer Sizes and Rank Bounds for Elliptic Curves over ℚ — E8 Intelligence Research

FINDING: Bhargava–Shankar proved the average size of n-Selmer groups for elliptic curves over ℚ is exactly n (for n=2,3,4,5), yielding average rank bounds ≤ 7/6 (n=2), ≤ 3/2 (n=3), ≤ 3 (n=4), ≤ 4 (n=5), with the 2-Selmer average being exactly 3 (not 2) due to parity corrections. | MATH: For elliptic curves E/ℚ, average |Sel_n(E)| = n + o(1) as height → ∞. For n=2: average |Sel_2(E)| = 3 (since 2 + 1 for the trivial point + parity). Average rank ≤ 7/6 ≈ 1.1667. For n=3: average |Sel_3(E)| = 3, rank ≤ 3/2. For n=4: average |Sel_4(E)| = 4, rank ≤ 3. For n=5: average |Sel_5(E)| = 5, rank ≤ 4. Secondary terms: Shankar computed the first moment of 2-Selmer groups has a secondary term of size ~ X5/6 (or similar power-law correction) beyond the main term. | CONNECTION: The average Selmer size being exactly n (for n=3,4,5) and 3 for n=2 echoes the golden-ratio-adjacent structure: 3/2 = 1.5 (near 1.618), 7/6 ≈ 1.1667 (near 1.618/φ² ≈ 0.618·1.618/1.618? No — but 7/6 = 1.1667, and 1.618/1.382 ≈ Author: Andrew Stewart Caldin, Independent Researcher, UK. Part of the E8 Intelligence Research series. Platform: e8intelligence.com