Quantitative strong approximation for ternary quadratic forms III

We prove asymptotic formulas for counting (primitive) integral points with local conditions on the (punctured) affine cone defined by a non-singular integral ternary quadratic form, and we relate our results to the Brauer–Manin obstruction. Our approach is based on the $δ$-variant of the Hardy–Littlewood circle method developed by Heath-Brown.

💡 Research Summary

This paper studies the quantitative strong approximation problem for the punctured affine cone defined by a non‑singular integral ternary quadratic form F(x₁,x₂,x₃). Let W = {F = 0} ⊂ ℝ³ be the affine cone and W⁰ = W \ {0} its punctured version. For a fixed modulus L ∈ ℕ and a congruence class Γ ∈ W⁰(ℤ/Lℤ), the author counts integral points satisfying the equation F(x)=0, the congruence x ≡ Γ (mod L), and optionally the primitivity condition (gcd of coordinates equal to 1). A smooth, compactly supported weight function w : ℝ³ → ℝ is used to smooth the counting.

The main analytic tool is Heath‑Brown’s δ‑method, a variant of the Hardy–Littlewood circle method. The method introduces a δ‑symbol to detect the equation F(x)=0 and leads to a double sum over a “Poisson variable” c ∈ ℤ³ and a modulus q. By splitting q = q₁q₂ with (q₁,Ω)=1 and q₂|Ω (where Ω = 8LΔ, Δ the discriminant of F), the exponential sums factor into a part coprime to Ω, which can be evaluated using quadratic Gauss sums, and a part supported on the bad primes dividing Ω, which is bounded trivially.

Two types of local data are defined:

• The p‑adic local density σₚ(L,Γ) = lim_{k→∞} #{x ∈ (ℤ/pᵏℤ)³ : F(x)≡0 (mod pᵏ), x≡Γ (mod p^{ordₚ(L)})}/p^{2k}.
• The modified singular series 𝔖_{L,Γ} = ∏ₚ (1−1/p) σₚ(L,Γ), which converges absolutely.

The weighted singular integral I(w) = ∫_{ℝ³} w(t) e(θF(t)) dt also appears. The paper proves two central asymptotic formulas.

Theorem 1.2 (non‑primitive case).
For any weight w, there exists a constant H_{L,Γ}(w) such that \