Workshop on function field arithmetic

这项提议的目的是于2011年6月在帝国理工学院举办为期两天的函数域算术研讨会。这将是第一次在英国举行的关于函数域算术的会议，其主要目的之一是将这一高度发达和享有盛誉的现代数论领域普及到英国数学界。函数域的算法围绕着Drinfeld模和t-Motives的概念。它们被证明是解决纯数学中几个核心猜想的强大工具，即全局和局部朗兰兹猜想、Ramanujan-Petersson猜想、Jaquet-朗兰兹对应和Carayol-Deligne猜想。另一方面，尽管他们的研究最初是为了推进上述猜想的发展，但现在它本身就是一门学科，一种极其丰富的理论，它对经典数论研究的每个对象都有一个对应物，包括模簇和模形式、L函数和伽马函数、伽罗瓦表示和霍奇理论、伯努利数和超越结果，它们经常紧密地反映，但有时诱人的背离数域的算术。此外，函数域的算术往往比数域的算术先进得多，因此该领域仍然是数论主要猜想的重要试验场。这个领域还与表示论、代数和算术几何、群论甚至组合学有很深的联系。预计这一领域的许多重大成果仍有待发现，这是非常自然的。一些顶尖的专家已经接受了演讲，参加博士课程的学生将接触到涵盖大量主题的最新方法。讲习班的另一个目的是促进与会者之间的讨论和对这一令人兴奋的领域的进一步研究。

The purpose of this proposal is to organise a two day workshop on the arithmetic of function fields at Imperial College in June 2011. This will be the first meeting on the arithmetic of function fields held in the UK and one of its main aims is to popularise this highly developed and prestigious area of modern number theory, which is still rapidly evolving, to the UK mathematical community. The arithmetic of function fields centres around the concepts of Drinfeld modules and t-motives. They proved to be an immensely powerful tool in resolving several central conjectures in Pure Mathematics, namely the global and local Langlands conjectures, the Ramanujan-Petersson conjecture, the Jaquet-Langlands correspondence, and the Carayol-Deligne conjecture. On the other hand although their study were originally undertaken in order to make advance on the above-mentioned conjectures, now it is a discipline on its own, an extremely rich theory which have a counterpart for every object which is studied by classical number theory, including modular varieties and modular forms, L- and Gamma-functions, Galois representations and Hodge theory, Bernoulli numbers and transcendence results and which often closely mirrors, but sometimes tantalising diverges from the arithmetic of number fields. Moreover very often the state of art in the arithmetic of function fields is far more advanced than in the case of number fields, hence the area still serves its role as an important testing ground for the major conjectures of number theory. The area also has deep connections to representation theory, algebraic and arithmetic geometry, group theory and even combinatorics. It is very natural to expect that many major results in this area are still to be discovered. Some of the leading experts have already accepted to speak and attending Ph. D. students will be exposed to the latest methods spanning a large selection of topics. The other aim of the workshop is to stimulate discussion among its participants and further research in this exciting area.