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Explicit and l-modular theta correspondence

显式和 l-模 theta 对应

基本信息

批准号：
EP/G001480/1
负责人：
Shaun Ainsley Ross Stevens
金额：
$ 33.77万
依托单位：
University of East Anglia
依托单位国家：
英国
项目类别：
Research Grant
财政年份：
2008
资助国家：
英国
起止时间：
2008 至无数据
项目状态：
已结题

来源：
https://gtr.ukri.org/projects?ref=EP%2FG001480%2F1
关键词：
Explicit modular theta correspondence

项目摘要

Suppose there is a big wedding: let us say that President Sarkozy of France and Carla Bruni are getting married. Before this most important day in their lives, the betrothed are preparing two parties: the hen party and the stag party. We are working for Hello magazine and two of our reporters, Howe and Waldspurger, have discovered that, at these two parties, there will be exactly the same number of people and, moreover, each man in the stag party is going out with one woman in the hen party (and vice versa).The editors of Hello magazine (Henniart, Harris,...) have asked for more information about this wedding. If possible we would like to know which people are in each party and who is going out with whom. Why are we interested in such a thing? The answer is very simple: it will be impossible to enter Sarkozy's party (due to the security) but it may be easier for one of our photographers to get into the hen party. If we see that Kate is at the hen party and I have proved before that Kate is going out with William then we will deduce that William is at Sarkozy's party!But, as you might see, this is a very difficult problem. How can one say who is going out with whom (Kate with William) without knowing who is in each party (Kate and William)? The strategy is the following: first we simplify the problem with arguments like blond-haired men are with black-haired women or men who study mathematics are with women whostudy philosophy ... Then we have to put names to people in each category and finally we have to prove who is with whom.In the analogy, each party is a p-adic group and the people in the party are ``representations'' of these groups -- these are abstract mathematical objects which have deep connections with Number Theory (which is essentially the study of the most basic mathematical objects: the natural numbers 1,2,3,...). The matching between the two parties is called the theta correspondence and the problem is to make it explicit: which representations appear and with whom are they paired? The first step is to put names to the representations of each p-adic group -- this is a classification problem. For some groups this is easier than for others, and this is the importance of the theta correspondence: having understood about the representations one group, we can use the theta correspondence to deduce information about therepresentations of the other.The theta correspondence and its predecessors have had major (mathematical) applications through the last 150 years; an explicit understanding of it will lead to many more.

假设有一个盛大的婚礼：让我们说，法国总统萨科齐和卡拉布吕尼要结婚。在这一生中最重要的一天之前，订婚的人要准备两个聚会：母鸡聚会和雄鹿聚会。我们为《你好》杂志工作，我们的两名记者豪和瓦尔德斯伯格发现，在这两个聚会上，人数完全相同，而且，在雄鹿聚会上，每个男人都和一个女人在母鸡聚会上约会（反之亦然）。想了解更多关于这场婚礼的信息如果可能的话，我们想知道每一方都有哪些人，谁和谁一起出去。为什么我们对这样的事情感兴趣？答案很简单：这将是不可能进入萨科齐的党（由于安全），但它可能是更容易为我们的摄影师之一进入母鸡党。如果我们看到凯特在母鸡聚会上，而我之前已经证明了凯特和威廉出去了，那么我们就可以推断出威廉在萨科齐的聚会上！但是，正如你所看到的，这是一个非常困难的问题。在不知道每个派对（凯特和威廉）都有谁的情况下，怎么能说谁要和谁出去（凯特和威廉）呢？战略如下：首先，我们把问题简单化，比如金发男人和黑发女人，或者学数学的男人和学哲学的女人……然后我们必须给每个类别中的人命名，最后我们必须证明谁和谁在一起。在类比中，每个政党都是一个p-adic群体，而党内的人是这些群体的“代表”-这些都是与数论有着深刻联系的抽象数学对象（本质上是研究最基本的数学对象：自然数1，2，3，.）。双方之间的匹配被称为theta对应，问题是要让它明确：哪些表征出现，它们与谁配对？第一步是给每个p-adic组的表示命名--这是一个分类问题。对于某些群体来说，这比其他群体更容易，这就是θ对应的重要性：在理解了一个群体的表征之后，我们可以使用θ对应来推断另一个群体的表征信息。θ对应及其前身在过去的150年里有着重大的（数学）应用;对它的明确理解将导致更多的应用。