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核心开放问题：第一，椭圆曲线上全局点的构造，目的是更好地理解伯奇和斯温纳顿-戴尔猜想；第二，希尔伯特的第十二个问题和显式类域论，其中实二次域的类域的解析构造是最简单的典型特例。我打算在过去五年中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 合作研究了模曲线和 Kuga-Sato 簇的 Chow 群中对角线循环的 p-adic 变形，这特别导致了解析等级为零的 Birch 和 Swinnerton-Dyer 猜想的新案例的证明，对于 Q 上的椭圆曲线 被实二次域的环类特征扭曲，以及（2）去年我与 Jan Vonk 的最新工作，它揭示了一种先前意想不到的实二次域奇异模理论，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 年左右引入的 Stark-Heegner 点的构造提供了互补且有前景的途径，从那时起，对 Stark-Heegner 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之为“广义加藤类”的对象：椭圆曲线 Selmer 群中的全局类（在适当的数域上，实二次域的类域是一个特别诱人的特殊情况），由 Chow 群或 Shimura 变种的更高 Chow 群中特殊几何对象的 p 进变形产生。我们正在进行的努力旨在将这些类与库默理论的连接同态下的 Stark-Heegner 点的图像进行比较。虽然不足以建立 Stark-Heegner 点的全局性质（这些点在分析上被定义为纯粹的局部对象），但将它们与全局 Selmer 类联系起来是朝这个方向迈出的决定性一步。从表面上完全不同的角度来看，我与 Jan 发现 Stark-Heegner 点可以在实二次域的复乘法（仍然是猜想的）理论的更广泛的框架中重铸，其中亚纯模函数的作用由我们所谓的“刚性亚纯余循环”发挥，这似乎对未来的进展充满了希望。事实上，我们现在处理了令人信服的策略，使这幅图的某些部分成为无条件的，基于将 Gross-Zagier 和 Kudla-Rapoport-Yang 的基础工作扩展到 p-adic 设置，有可能为真正二次域的希尔伯特第十二个问题提供令人满意的解决方案。 Kudla 的程序最终扩展到 p-adics 可能为希尔伯特第十二个问题提供关键，这也许是我最近与 Jan Vonk 合作中得出的最重要的见解。