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

你正漫步在大英博物馆。沿着这条路，你最终来到了4号房间，这是一个专门展示埃及雕塑的房间。为什么我们要派你来这里理解数学概念？不是因为石棺或雕像，而是因为一块巨大的黑色石板，上面刻着来自古埃及的同一法令的三个版本：罗塞塔石碑。这三个文本是希腊文、德文文和象形文字。尽管铭文本身并不涉及数学，但其破译的复杂历史说明了“通信”的概念和难度。在1799年发现这块石头之前，西方埃及学家面临着一个主要的困难，他们不能读懂象形文字(或者说在某种程度上是通俗的)。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相互理解。不幸的是，尽管三个版本粘合在一起给出了完整的法令，但每一篇文章的某些部分都被遗漏了。罗塞塔石碑在破译象形文字的斗争中发挥了关键而独特的作用，因为它允许人们在不同语言之间建立桥梁或部分词典--数学家会说“建立对应关系”(有缺失的部分！)在这个项目的背景下，语言的作用是通过“G和H的不可约光滑表示，其中(G，H)是辛群中的对偶”来承担的。所谓的“theta对应”与G的某些不可约的流畅表示相关联，G是H的不可约的流畅表示：因此，如果我们认为G是希腊语，而H是象形文字，那么theta对应是一块罗塞塔石碑，给出了从希腊语中的某些单词到象形文字的翻译。为了完成这幅画，还有一种类似的德米德(“伽罗瓦表示为G的对偶群”)和一种将希腊语翻译成德米德的通信：“朗兰兹通信”，这是过去50年来范围广泛的数学家努力的焦点。此外，与Rosetta Stone不同的是，我们可以考虑无限多的对(G，H)(所以有无限多的“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theta对应。如果我们把罗塞塔石碑的印记取到一定的深度，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和象形文字的部分轮廓之间的对应关系--而且这种对应关系与原始的theta对应关系相匹配，因此，如果希腊语和象形文字匹配，那么它们的部分轮廓也匹配。我们甚至可以允许深度变化，并寻找与所有这些都匹配的对应关系。在我们的项目中，这些部分轮廓线是“L模表示”，其中L是代表深度的素数。虽然有一些原因使得某些素数L不能发生简单的对应，但我们希望找到剩余的L的对应和困难素数L的部分结果(例如，较弱的对应)，以及通向有助于同时解释所有这些的“族中”对应的途径。theta对应和朗兰兹对应之所以有趣，是因为它们编码了大量的算术意义--归根结底，这意味着关于整数中编码的性质的信息...-1，0，1，2，...事实上，即使知道我们能够翻译的希腊单词也能告诉我们很多，这种对应在数学中有许多应用，从表示论到解析数论，这些数学在一个世纪左右的时间里得到了发展。

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.