A two-dimensional delta symbol method and its application to pairs of quadratic forms

1. Problem Statement

The paper addresses the problem of counting integral solutions to a system of two homogeneous quadratic equations in $s$ variables:
$$F_1(\mathbf{x}) = 0, \quad F_2(\mathbf{x}) = 0$$
where $F_1, F_2 \in \mathbb{Z}[x_1, \dots, x_s]$ define a non-singular complete intersection of codimension 2 in projective space. The goal is to derive an asymptotic formula for the number of integral points $\mathbf{x}$ with $|\mathbf{x}| \le P$ as $P \to \infty$.

Specifically, the authors aim to determine the minimal number of variables $s$ required to establish this asymptotic formula using the Hardy-Littlewood circle method. Previous results required $s \ge 11$ (Munshi) or handled only "split" cases (Heath-Brown–Pierce). The authors aim to lower this threshold to $s \ge 10$ unconditionally and $s \ge 9$ under the Generalized Lindelöf Hypothesis (GLH).

2. Methodology: The Two-Dimensional Delta Symbol

The core innovation of the paper is the development of a two-dimensional delta symbol method. This is a refinement of the classical circle method designed to handle systems of equations ($R \ge 2$) where the standard "major/minor arc" decomposition is insufficient for obtaining sharp bounds.

2.1. Limitations of Previous Methods

• Classical Circle Method: Splits the unit square $[0,1]^2$ into major arcs (near rationals $\mathbf{a}/q$) and minor arcs. For $R=2$, the geometry of rational approximations is complex; the distance between neighbors depends on the heights of affine lines, making a literal Farey dissection impossible.
• Nested Delta Symbols (Munshi): Used a nested approach but resulted in larger moduli $q$ than optimal, leading to less efficient Poisson summation steps.
• Function Field Analogs: Vishe previously solved this over $\mathbb{F}_q(t)$ using a 2D Farey dissection, but the non-Archimedean nature of that setting prevented direct translation to number fields.

2.2. The New Delta Symbol Identity

The authors prove Theorem 1.2, which provides a smooth partition of unity for the delta function $\delta_{\mathbf{n}}$ (where $\delta_{\mathbf{n}} = 1$ if $\mathbf{n}=\mathbf{0}$ and $0$ otherwise) in two dimensions:
$$\delta_{\mathbf{n}} = \sum_{1 \le q \le Q} \sum_{\mathbf{a} \pmod q}^* \int_{\mathbb{R}^2} p_{\Lambda(\mathbf{a}, q)}(\mathbf{w}) e\left( \left(\frac{\mathbf{a}}{q} + \mathbf{w}\right) \cdot \mathbf{n} \right) d\mathbf{w} + O(Q^{-N})$$
Key features of this identity:

• Lattice Dependence: The kernel function $p_{\Lambda(\mathbf{a}, q)}$ depends on the lattice $\Lambda(\mathbf{a}, q) = \{ k\mathbf{a} + q\mathbf{y} : k \in \mathbb{Z}, \mathbf{y} \in \mathbb{Z}^2 \}$.
• Optimal Support: The function is supported on $\mathbf{w}$ of size roughly $q^{-1}Q^{-1/2}$. This is the optimal size for a 2D Dirichlet approximation, allowing the parameter $Q$ to be chosen as $Q \asymp P^{4/3}$ (where $P$ is the size of the variables), which is smaller than in previous methods ($Q \asymp P^{3/2}$).
• Double Kloosterman Refinement: The structure of the lattice allows the authors to group terms by the lattice rather than just the denominator $q$. This enables averaging over $\mathbf{a}$ (Kloosterman refinement) and, crucially, averaging over $q$ (double Kloosterman refinement) to exploit cancellations in exponential sums.

3. Key Contributions and Technical Results

3.1. The Main Asymptotic Formula (Theorem 1.1)

The authors establish the asymptotic formula for the number of solutions $N(P)$:
$$N(P) = S I P^{s-4} + O(P^{s-4-\Delta})$$
where:

• $S$ is the singular series (arithmetic factor).
• $I$ is the singular integral (geometric factor).
• The error term exponent $\Delta$ depends on $s$:
  • Unconditional:
    • If $s$ is even and $s \ge 10$, $\Delta < 1/6$.
    • If $s$ is odd and $s \ge 11$, $\Delta < 1/15$.
  • Conditional (under GLH):
    • If $s = 9$ (odd), $\Delta < 1/15$.

This result verifies the Manin–Peyre conjecture for non-singular intersections of two quadrics of dimension $\ge 7$ (or 6 under GLH).

3.2. Decomposition of the Counting Function

The proof involves decomposing the counting function into three parts based on the size of the modulus $q$ and the "weight" function in the delta symbol:

1. Major Arcs ($N_0$): The main term derived from $q$ small and $\mathbf{w}$ near 0.
2. Minor Arcs Type 1 ($N_1$): Contributions from small $q$ but larger $\mathbf{w}$, handled using standard exponential sum bounds.
3. Minor Arcs Type 2 ($N_2$): Contributions from large $q$ and the specific structure of the lattice. This is the most difficult part, requiring a detailed analysis of exponential sums $S_{q, d\mathbf{c}}(u)$ and exponential integrals.

3.3. Handling Exponential Sums and Geometry

The authors perform a deep analysis of the exponential sums $S_{q, d\mathbf{c}}(u)$:

• They classify primes and vectors $\mathbf{c}$ into "good" and "bad" types based on the singularity of the quadratic form $F_{\mathbf{c}} = c_1 F_1 + c_2 F_2$.
• They utilize Poisson summation and geometry of numbers (specifically the geometry of the lattice $\Lambda(\mathbf{a}, q)$) to bound the contributions from the minor arcs.
• For the case $s=9$, they rely on the Generalized Lindelöf Hypothesis to obtain sufficient cancellation in the average of exponential sums over $q$.

4. Significance and Comparison

4.1. Improvement over Existing Bounds

• Munshi [29]: Required $s \ge 11$. The new method reduces this to $s \ge 10$ unconditionally.
• Heath-Brown–Pierce [16]: Handled the "split" case ($F_i(x,y) = G_i(x) + H_i(y)$) for $s \ge 10$. This paper removes the "split" restriction, handling general non-singular intersections.
• Function Fields: The result matches the conditional bound ($s=9$) known in the function field setting (Vishe [37]), bridging a significant gap between number fields and function fields.

4.2. Heuristic Superiority

In Section 14, the authors provide a heuristic comparison (Heuristic 2) showing that their method is superior for large $s$.

• The efficacy of a delta symbol method is roughly captured by $Q^{s/2} A^{-1/2}$, where $Q$ is the modulus parameter and $A$ is the size of the average over $\mathbf{a}$.
• Their method achieves $Q \asymp P^{4/3}$ and $A \asymp Q$, leading to an error term of size $P^{2(s-1)/3}$.
• Previous methods (like Munshi's) had $Q \asymp P^{3/2}$ and $A \asymp Q^{5/3}$, leading to a larger error term $P^{(3s-5)/4}$.
• The new method becomes strictly better for $s \ge 8$ and the gap widens as $s$ increases.

5. Conclusion

The paper presents a breakthrough in the analytic theory of Diophantine equations by constructing a robust two-dimensional delta symbol. This tool facilitates a double Kloosterman refinement in the circle method, allowing for the treatment of pairs of quadratic forms with fewer variables than previously possible. The results significantly advance the understanding of the distribution of rational points on complete intersections of quadrics and provide a framework that the authors argue will outperform existing methods for systems with more variables or higher degrees.