论文标题
更高的rédei互惠和锥体上不可或缺的积分
Higher Rédei reciprocity and integral points on conics
论文作者
论文摘要
修复整数$ l $，以便$ | l | $是Prime $ 3 $ modulo $ 4 $。令$ d> 0 $为平方英尺，让$ n_d（x，y）$为$ \ mathbb {q}（\ sqrt {d}）$的主要二进制二进制形式。在亚历山大·史密斯（Alexander Smith）的突破性基础上，我们为$ n_d（x，y）= l $ in Integers $ x $和$ y $ as $ d $ as $ d $的溶解度提供了渐近公式，在$ l $中可分解的方形$ d $。 作为推论，我们给出了$ l> 0 $的情况，这是一个事件的渐近公式，即“ hasse单位索引” $ \ mathbb {q}（\ sqrt {-l}，\ sqrt {d}）$是$ 2 $ as $ d $ as $ d $，而不是所有正面的SquareFree integers $ 2 $。我们还改善了Fouvry和Klüners的结果，以及Chan，Milovic和作者对负Pell方程的溶解度的最新结果。我们的主要新工具是由于Rédei而对经典互惠法的概括。
Fix an integer $l$ such that $|l|$ is a prime $3$ modulo $4$. Let $d > 0$ be a squarefree integer and let $N_d(x, y)$ be the principal binary quadratic form of $\mathbb{Q}(\sqrt{d})$. Building on a breakthrough of Alexander Smith, we give an asymptotic formula for the solubility of $N_d(x, y) = l$ in integers $x$ and $y$ as $d$ varies among squarefree integers divisible by $l$. As a corollary we give, in case $l > 0$, an asymptotic formula for the event that the Hasse Unit Index of the field $\mathbb{Q}(\sqrt{-l}, \sqrt{d})$ is $2$ as $d$ varies over all positive squarefree integers. We also improve the results of Fouvry and Klüners and recent results of Chan, Milovic and the authors on the solubility of the negative Pell equation. Our main new tool is a generalization of a classical reciprocity law due to Rédei.
