This expository article is an introduction to the Langlands functoriality conjectures and their applications to the arithmetic of representations of Galois groups of number fields. Thanks to the work of a great many people, the stable trace formula is now largely established in a version adequate for proving Langlands functoriality in the setting of endoscopy. These developments are discussed in the first two sections of the article. The final section describes the compatible families of ℓ-adic Galois representations that can be attached to automorphic forms with the help of Shimura varieties. To illustrate the relevance of Langlands functoriality to number theory, the article concludes with a description of the Sato–Tate conjecture for elliptic modular forms, recently proved in joint work of Barnet-Lamb, Geraghty, Taylor, and the author.

中文翻译：

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Langlands程序的算术应用

这篇说明性文章介绍了Langlands功能猜想及其在数域Galois组表示的算术中的应用。得益于许多人的努力，稳定的痕量配方现已在很大程度上建立了足以证明兰德兰具有内窥镜检查功能的版本。本文的前两部分讨论了这些发展。最后一节介绍的兼容家庭ℓ-adic Galois表示形式，可以在Shimura品种的帮助下附加到自守形式。为了说明Langlands函数与数论的相关性，本文以Sato-Tate椭圆模形式的猜想作为结尾，最近在Barnet-Lamb，Geraghty，Taylor和作者的联合研究中证明了这一观点。