Optimal strong approximation for quadrics over [formula omitted

Abstract

Suppose q is a fixed odd prime power, F(x) is a non-degenerate quadratic form over Fq[t] of discriminant Δ in d≥5 variables x, and f,g∈Fq[t], λ∈Fq[t]d. We show that whenever deg⁡f≥(4+ε)deg⁡g+Oε,F(1), gcd⁡(Δ∞,fg)=O(1), and the necessary local conditions are satisfied, we have a solution x∈Fq[t]d to F(x)=f such that x≡λmodg. For d=4, we show that the same conclusion holds if we instead have deg⁡f≥(6+ε)deg⁡g+Oε,F(1). This gives us a new proof (independent of the Ramanujan conjecture over function fields proved by Drinfeld) that the diameter of any k-regular Morgenstern Ramanujan graph G is at most (2+ε)logk−1⁡|G|+Oε(1). In contrast to the d=4 case, our result is optimal for d≥5. Our main new contributions are a stationary phase theorem over function fields for bounding oscillatory integrals, and a notion of anisotropic cones to circumvent isotropic phenomena in the function field setting.