Random Isotropic Subspaces and Average Selmer Rank 1/2 in Function Fields — E8 Intelligence Research

Abstract

FINDING: Poonen constructs a probability measure on maximal isotropic subspaces of a locally compact quadratic space over F_p, proving such a random subspace is discrete almost surely and its intersection with a fixed compact open subspace has a computable distribution — this models average 2-Selmer ranks in function fields, yielding average rank 1/2. MATH: - Quadratic space \( V \) over \( \mathbb{F}_p \), locally compact. - Maximal isotropic subspace \( L \subset V \), with \( L \cap K \) for fixed compact open \( K \subset V \). - Probability measure \( \mu \) on the set of closed maximal isotropic subspaces. - Key result: \( \dim(L \cap K) \) follows a distribution with probability generating function involving \( p^{-n(n-1)/2} \) (Gaussian binomial coefficients). - Average 2-Selmer rank in function fields: \( \mathbb{E}[\dim \mathrm{Sel}_2(E/K)] = 1/2 \) for elliptic curves over \( \mathbb{F}_q(t) \) (Poonen–Rains heuristic, later proven by Landesman et al. for large \( Author: Andrew Stewart Caldin, Independent Researcher, UK. Part of the E8 Intelligence Research series. Platform: e8intelligence.com