We give a survey of a variety of recent results about the distribution and some geometric properties of points (x, y) on modular hyperbolas \({xy \equiv a\;(\mod m)}\). We also outline a very diverse range of applications of such results, discuss multivariate generalisations and suggest a number of open problems of different levels of difficulty.

中文翻译：

---

模块化双曲线

我们对有关模块化双曲线\（{xy \ equiv a \;（\ mod m）} \）上点（x，y）的分布和某些几何性质的最新结果进行了调查。我们还概述了此类结果的多种应用，讨论了多元归纳法，并提出了许多难度不同的开放问题。