Average n-Selmer Size Equals Divisor Sum σ(n) over F_q(t) — E8 Intelligence Research
Year: 2026
Type: preprint
Abstract: FINDING: The average size of n-Selmer groups of elliptic curves over F_q(t) in the large-q limit equals the sum of divisors function σ(n) — a direct arithmetic-statistical law with deep lattice-theoretic content. | MATH: Let Sel_n(E/F_q(t)) denote the n-Selmer group. The geometric average (over a suitable moduli stack of elliptic curves, q→∞) satisfies: ⟨#Sel_n(E)⟩ = σ(n) = Σ_{d|n} d. For n=2: σ(2)=3 (classical Poonen–Rains result). For n=3: σ(3)=4. For n=4: σ(4)=7. This is not a heuristic — it is proven in the function-field large-q limit (Landesman, et al.). The proof uses the cohomology of the moduli space of elliptic curves, specifically the stable cohomology of the symplectic group Sp_{2g}(Z) and the Grothendieck–Lefschetz trace formula over finite fields. | CONNECTION: **Root system A_{n-1}**: The sum of divisors σ(n) is precisely the number of positive roots in the root system A_{n-1} counted with multiplicity by height? More precisely: σ(n) = Σ_{d|n} d = number of elements 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: W7210294258
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