Selmer Averages over Function Fields Equal the Divisor Sum σ(n) — E8 Intelligence Research
Year: 2026
Type: preprint
Abstract: FINDING: Average size of n-Selmer groups over function fields F_q(t) in the large-q limit equals sum-of-divisors function σ(n), connecting arithmetic statistics to classical multiplicative number theory. | MATH: lim_{q→∞} Avg |Sel_n(E/F_q(t))| = σ(n) = Σ_{d|n} d; for n=2, average = 3; for n=3, average = 4; for n=4, average = 7; for n=5, average = 6; for n=6, average = 12. This is the geometric analogue of the Poonen–Rains/Bhargava–Shankar heuristics for number fields, where average |Sel_n| is conjecturally σ(n) for elliptic curves over Q. | CONNECTION: σ(n) is multiplicative and its growth is sublinear — but the *ratios* σ(n)/n relate to highly composite numbers; for n=2,3,4,5,6 the averages are 3,4,7,6,12 — note 3/2=1.5, 4/3≈1.333, 7/4=1.75, 6/5=1.2, 12/6=2. No direct 0.618/1.618 ratio appears, but the *Weyl chamber* in the title hints at root systems (A_n, D_n) whose Coxeter numbers (n+1, 2n-2) appear in Selmer group statistics via the geometry of moduli of curves — the sum-of-diviso Author: Andrew Stewart Caldin, Independent Researcher, UK. Part of the E8 Intelligence Research series. Platform: e8intelligence.com
Author: Caldin, Andrew Stewart
Language: English
Subfield: Geometry and Topology
Field: Mathematics
Domain: Physical Sciences
Open Access status: green
OpenAlex ID: W7210291431
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