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Local theta correspondence: a new study through the theories of types and l-modular representations

局部 theta 对应：通过类型和 l 模表示理论进行的新研究

基本信息

批准号：
EP/V061739/1
负责人：
Shaun Ainsley Ross Stevens
金额：
$ 53.14万
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：
2022
资助国家：
英国
起止时间：
2022 至无数据
项目状态：
未结题

来源：
https://gtr.ukri.org/projects?ref=EP%2FV061739%2F1
关键词：
Local theta correspondence new study

项目摘要

You are wandering through the British Museum. Along the way, you end up in Room 4, which is dedicated to Egyptian sculpture. Why would we send you here to understand mathematical concepts? Not because of some sarcophagus or statue, but rather a large black slab inscribed with three versions of the same decree from ancient Egypt: the Rosetta Stone. These three texts are in Greek, Demotic and Hieroglyphic.Even though the inscriptions themselves do not deal with mathematics, the complex history of its decipherment illustrates the concept, and difficulty, of "correspondences". Before 1799, when the stone was found, occidental Egyptologists were faced with the major difficulty that they could not read Hieroglyphic (or Demotic, to some extent). But, hoping that the three texts contain only minor differences, the Rosetta Stone allowed these texts to be used to understand each other. Unfortunately, there was the added complication that some parts of each text are missing, though the three versions glued together give the full decree.The Rosetta Stone played a key and singular role in the struggle to decipher hieroglyphs, as it allowed one to build bridges or partial dictionaries - a mathematician would say "establish correspondences" - between the various languages (with missing parts!) at stake.In the setting of this project, the role of languages is taken by the "irreducible smooth representations of G and of H, where (G,H) is a dual pair in a symplectic group". The so-called "theta correspondence" associates to certain irreducible smooth representations of G, an irreducible smooth representation of H: so, if we think of G as Greek and H as Hieroglyphic, then the theta correspondence is a Rosetta Stone, giving a translation from certain words in Greek to words in Hieroglyphic. To complete the picture, there is also an analogue of Demotic ("galois representations into the Dual group of G") and a correspondence which translates Greek to Demotic: the "Langlands correspondence", which has been a focus of effort for a wide range of mathematicians over the last 50 years. Moreover, unlike the Rosetta Stone, there are infinitely many pairs (G,H) that one can consider (so infinitely many "stones"), which means a lot of information and cases involved.In this project, we will study the theta correspondence in a more refined way. If we take an imprint of the Rosetta Stone to a certain small depth, then we see only a partial contour of each word - and different words may give the same partial contour. Nonetheless, we may still be able to find a correspondence between the Greek partial contours and those in Hieroglyphic - and one which matches the original theta correspondence so that, if a Greek and Hieroglyphic word match, then their partial contours also match. We can even allow the depth to vary and look for a correspondence which matches all of these together. In our project, these partial contours are "l-modular representations", where l is a prime number representing the depth. While there are reasons that a simple correspondence cannot happen for certain primes l, we expect to find both a correspondence for the remaining l and partial results (for example, a weaker correspondence) for the difficult primes l, as well as a pathway towards a correspondence "in families" which would help explain all of these simultaneously.The theta correspondence, and the Langlands correspondence, are of interest because they encode a lot of arithmetic meaning - ultimately, this means information about properties encoded in the integers ... -1, 0, 1, 2, ... Indeed, even knowing which "words of Greek" we are able to translate tells us a lot, and the correspondence has numerous applications across Mathematics, developed over a century or so, ranging from representation theory to analytic number theory.

你正在大英博物馆里闲逛。沿着，你最终在4号房间，这是专门为埃及雕塑。为什么我们要送你来这里理解数学概念？不是因为一些石棺或雕像，而是一块巨大的黑色石板，上面刻着古埃及同一法令的三个版本：罗塞塔石碑。这三个文本分别是希腊文、世俗文和象形文字。尽管铭文本身并不涉及数学，但其复杂的破译历史说明了“对应”的概念和难度。在1799年发现这块石头之前，西方考古学家面临的主要困难是他们无法阅读象形文字（或在某种程度上，通俗文字）。但是，希望这三个文本只包含微小的差异，罗塞塔石碑允许这些文本被用来理解对方。不幸的是，每一个文本的某些部分都缺失了，这增加了复杂性，尽管三个版本粘在一起给出了完整的法令。罗塞塔石碑在破译象形文字的斗争中发挥了关键和独特的作用，因为它允许人们在各种语言之间建立桥梁或部分词典-数学家会说"建立对应关系"（缺少部分!）在这个项目的背景下，语言的作用是由"G和H的不可约光滑表示，其中（G，H）是辛群中的对偶对"。所谓的"theta对应"与G的某些不可约的光滑表示相关联，即与H的不可约的光滑表示相关联：因此，如果我们认为G是希腊语，H是象形文字，那么theta对应就是罗塞塔石碑，将某些希腊语单词翻译成象形文字单词。为了完成这幅图，还有一个类似于Demotic（“Galois表示到G的对偶群”）的对应关系，以及一个将希腊语翻译成Demotic的对应关系：“Langlands对应关系”，这是过去50年来广泛数学家努力的焦点。此外，与罗塞塔石碑不同，人们可以考虑的对（G，H）有无限多个（所以有无限多个"石头"），这意味着涉及大量的信息和案例。在这个项目中，我们将以更精细的方式研究θ对应。如果我们把罗塞塔石碑的印记带到某个很小的深度，那么我们只能看到每个单词的部分轮廓-不同的单词可能会给出相同的部分轮廓。尽管如此，我们仍然能够找到希腊语部分轮廓与象形文字部分轮廓之间的对应关系，并且与原始的theta对应关系相匹配，因此，如果希腊语和象形文字的单词匹配，那么它们的部分轮廓也匹配。我们甚至可以让深度变化，并寻找一个对应匹配所有这些在一起。在我们的项目中，这些部分轮廓是“l-模表示”，其中l是表示深度的素数。虽然有一些原因，一个简单的对应不能发生某些素数l，我们希望找到一个对应的其余l和部分结果（例如，一个较弱的对应）为困难的素数l，以及一条通往对应"在家庭"，这将有助于解释所有这些同时。θ对应，和朗兰兹对应，是有趣的，因为它们编码了大量的算术意义-最终，这意味着有关整数中编码的属性的信息... -1，0，1，2，...事实上，即使知道我们能够翻译哪些"希腊语单词"，也能告诉我们很多东西，而且这种对应关系在整个数学领域有着无数的应用，从表示论到解析数论，它已经发展了大约世纪。