Poonen–Rains Heuristics and Bhargava–Shankar Bounds on Elliptic Curve Ranks — E8 Intelligence Research

FINDING: Poonen's heuristics predict a uniform upper bound on elliptic curve ranks over ℚ, with Selmer group distributions governed by random matrix/adelic sieving; Bhargava–Shankar rigorously prove average rank ≤ 7/6 via 2-Selmer averages. MATH: - **Poonen–Rains heuristic**: For a prime p, the p-Selmer group distribution matches the cokernel of a random alternating matrix over ℤ_p — probability that dim Sel_p(E) = d is proportional to p^{-d²} · ∏_{i=1}^d (1 - p^{-2i})^{-1} · (normalization). - **Uniform rank conjecture**: ∃ absolute constant B such that rank E(ℚ) ≤ B for all elliptic curves E/ℚ. Poonen's sieve predicts B = 21 (from p=2 Selmer constraints) — not proven, but consistent with all data. - **Bhargava–Shankar exact average**: lim_{X→∞} (1/N(X)) Σ_{ht(E)≤X} rank(E(ℚ)) ≤ 7/6 ≈ 1.1667. This comes from exact average size of 2-Selmer = 3 (i.e., dim Sel_2 = 1 on average), plus parity and root number arguments. - **Key constant**: 3 — the average size of the 2-Selmer grou Author: Andrew Stewart Caldin, Independent Researcher, UK. Part of the E8 Intelligence Research series. Platform: e8intelligence.com