We make effective $l^2 L^p$ decoupling for the parabola in the range $4 < p < 6$. In an appendix joint with Jean Bourgain, we apply the main theorem to prove the conjectural bound for the sixth-order correlation of the integer solutions of the equation $x^2 + y^2 = m$ in an extremal case. This proves unconditionally a result that was proven by Bombieri and Bourgain under the hypotheses of the Birch and Swinnerton-Dyer conjecture and the Riemann Hypothesis for $L$-functions of elliptic curves over $\mathbb{Q}$.

中文翻译：

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有效 l 2 抛物线解耦

我们对 $4 < p < 6$ 范围内的抛物线进行有效的 $l^2 L^p$ 解耦。在与 Jean Bourgain 的附录中，我们应用主定理来证明方程 $x^2 + y^2 = m$ 在极值情况下的整数解的六阶相关性的猜想界。这无条件地证明了 Bombieri 和 Bourgain 在 Birch 和 Swinnerton-Dyer 猜想的假设以及 $\mathbb{Q}$ 上椭圆曲线的 $L$ 函数的黎曼假设下证明的结果。