Large-q Limit of Selmer Groups over Function Fields Converges to Divisor-Sum Function — E8 Intelligence Research

FINDING: Average size of n-Selmer groups over function fields converges to divisor-sum function in large-q limit; explicit Selmer groups for cyclic covers of P¹ admit tractable local-cohomology description. MATH: - For elliptic curves over 𝔽_q(t), large-q limit: avg |Sel_n(E)| → Σ_{d|n} d (divisor sum function). - Selmer group: Sel^φ(J,k) ⊆ H¹(Gal(k^sep/k), J[φ]) defined by local conditions (Tate–Shafarevich, local descent). - Cyclic cover of P¹ of prime degree p: Jacobian J; Selmer group structure tied to ramification data and p-torsion of J. - Iwasawa theory: fine Selmer group Sel₀(E/K_∞) — kernel of restriction to Σ-local cohomology; growth governed by λ, μ invariants (not directly in search results, but context). CONNECTION: - Divisor sum Σ_{d|n} d is multiplicative and relates to *Eisenstein integers* (ring ℤ[ω], ω = e^{2πi/3}) and *root system A₂* — the hexagonal lattice. The average size 1, 3, 4, 7, 6, 12, … for n=1,2,3,4,5,6,… mirrors the *norm form* of Eisenstein Author: Andrew Stewart Caldin, Independent Researcher, UK. Part of the E8 Intelligence Research series. Platform: e8intelligence.com