Explicit class field theory and the Birch and Swinnerton-Dyer conjecture

显式类场论以及伯奇和斯温纳顿-戴尔猜想

• 批准号: RGPIN-2018-04062
• 负责人: Darmon, Henri
• 金额: $ 8.3万
• 依托单位: McGill University
• 依托单位国家: 加拿大
• 项目类别: Discovery Grants Program - Individual
• 财政年份: 2022
• 资助国家: 加拿大
• 起止时间: 2022-01-01 至 2023-12-31
• 项目状态: 已结题
• 来源: https://www.nserc-crsng.gc.ca/ase-oro/Details-Detailles_eng.asp?id=758943
• 关键词: Explicit; class; field; theory; Birch

My research program will revolve around two central open questions in number theory: firstly, the construction of global points on elliptic curves with the goal of better understanding the Birch and Swinnerton-Dyer conjecture, and secondly, Hilbert's twelfth problem and explicit class field theory, of which the analytic construction of class fields of real quadratic fields is the simplest prototypical special case. I intend to build on the significant progress I have achieved towards these questions in the last five years, most notably, (1) my work with Victor Rotger on p-adic deformations of diagonal cycles in the Chow groups of triple products of modular curves and Kuga-Sato varieties, which has led in particular to the proof of new cases of the Birch and Swinnerton-Dyer conjecture in analytic rank zero, for elliptic curves over Q twisted by ring class characters of real quadratic fields, and (2) my more recent work with Jan Vonk in the past year, which has revealed a previously unexpected theory of singular moduli for real quadratic fields enjoying striking parallels with the classical theory of complex multiplication. Both works offer complementary and promising avenues for better understanding the construction of Stark-Heegner points that I introduced around 2000, whose shoring up has been my primary research focus since that time. My work with Victor revolves around objects which we call "generalised Kato classes": global classes in the Selmer groups of elliptic curves (over appropriate number fields, class fields of real quadratic fields being a particularly tantalising special case) arising from p-adic deformations of special geometric objects in Chow groups or higher Chow groups of Shimura varieties. Our ongoing efforts aim to compare these classes with the images of Stark-Heegner points under the connecting homomorphism of Kummer theory. While not sufficient to establish the global nature of Stark-Heegner points, which are defined analytically as purely local objects, relating them to global Selmer classes is a decisive step in that direction. From an ostensibly quite different angle, my discovery with Jan that Stark-Heegner points can be recast in the broader framework of a (still conjectural) theory of complex multiplication for real quadratic fields in which the role of meromorphic modular functions is played by what we call "rigid meromorphic cocycles", seems to be full of promise for future progress. Indeed, we now dispose of convincing strategies for making some parts of this picture unconditional, potentially leading to a satisfying solution to Hilbert's twelfth problem for real quadratic fields based on extending fundamental work of Gross-Zagier and of Kudla-Rapoport-Yang to a p-adic setting. That an eventual extension of Kudla's program to the p-adics could offer a key to Hilbert's twelfth problem is perhaps the most significant insight to emerge from my recent work with Jan Vonk.

我的研究计划将围绕数论中的两个中心开放问题：第一，为了更好地理解Birch和Swinnerton-Dyer猜想，椭圆曲线上整体点的构造，第二，Hilbert第十二问题和显式类域理论，其中实二次域的类域的解析构造是最简单的典型特例。我打算在过去五年我在这些问题上取得的重大进展的基础上再接再厉，最值得注意的是，(1)我与Victor Rotger关于模曲线和Kuga-Sato簇的三重积的Chow群中对角圈的p-进变形的工作，这尤其导致证明了解析秩为零的Birch和Swinnerton-Dyer猜想的新情况，对于由实二次域的环类特征扭曲的Q上的椭圆曲线，以及(2)我最近与Jan Vonk在过去一年的工作，它揭示了一个先前出人意料的实二次域的奇异模理论，它与经典的复数乘法理论有着惊人的相似之处。这两部作品为更好地理解我在2000年左右介绍的斯塔克-海格纳点的构建提供了互补和有前途的途径，自那时以来，斯塔克-海格纳点的支撑一直是我的主要研究重点。我与Victor的工作围绕着我们称为“广义加藤类”的对象展开：椭圆曲线的Selmer群(在适当数域上，实二次域的类域是一个特别诱人的特例)中的全局类，由Shimura变种的Chow群或更高的Chow群中的特殊几何对象的p-adi变形产生。我们正在进行的工作旨在将这些类与Kummer理论的连接同态下的Stark-Heegner点的像进行比较。虽然不足以确定Stark-Heegner点的全局性质，但将它们与全局Selmer类联系起来是朝着这个方向迈出的决定性一步。从一个表面上完全不同的角度来看，我和Jan的发现，Stark-Heegner点可以在实二次域的复乘(仍然是猜测的)理论的更广泛框架中重塑，在该理论中，亚纯模函数的作用由我们所称的“刚性亚纯上循环”来扮演，似乎对未来的发展充满希望。事实上，我们现在处理令人信服的策略，使这幅图的某些部分是无条件的，潜在地导致基于将Gross-Zagier和Kudla-Rapoport-Yang的基本工作扩展到p-进设置的实二次场的Hilbert第十二问题的满意解决方案。库德拉的项目最终扩展到P-ADICS可能会为希尔伯特的第十二个问题提供一把钥匙，这可能是我最近与简·冯克合作得出的最重要的见解。