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字理论的两个中心问题围绕：首先，椭圆曲线上的全球要点的构建，目的是更好地了解桦木和Swinnerton-Dyer猜想，其次，希尔伯特的十二个问题和明确的类领域理论，实际上是Quardratic Fields的分析构建领域的分析构建，是真正的Quadratic Fields的分析构建。 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在过去的一年中，我与Jan Vonk的最新作品被真实二次领域的环类字符扭曲，该研究揭示了以前出乎意料的奇异模量理论，用于真实的二次领域，与复杂乘法的经典理论相似。 这两件作品都提供了互补和有前途的途径，以更好地了解我在2000年左右引入的Stark-Heegner点的构建，从那时起，他们的支撑一直是我的主要研究重点。 我与维克多（Victor）的工作围绕着我们称为“广义KATO类”的对象：Selmer椭圆曲线组的全球课程（在适当的数字字段上，实际二次二次领域的类字段是一个特别诱人的特殊情况），这是由chow型或更高chimura valieties of Shimura chimura of Shimura of Shimura of Shimura chow groun的P-Adic aadic变形引起的。我们正在进行的努力旨在将这些阶级与库默理论的同态性同态性下的史塔克·希纳（Stark-Heegner）点的图像进行比较。虽然不足以建立Stark-Heegner点的全球性质，而该点在分析上被定义为纯粹的本地对象，但将它们与全球Selmer类联系起来是朝这个方向迈出的决定性的一步。从表面上讲完全不同的角度来看，我对Jan的发现是在更广泛的（仍然是猜想的）复杂乘法理论的更广泛的框架中重塑了真正的二次乘法理论，在这种二次二次领域中，在这种近二次领域中，突出型模块化功能的作用是由我们称为“刚性杂色循环的僵化”，似乎是对未来的Prols for Prols的Prols nor Prols for Prols for nurcimid of。的确，我们现在处理令人信服的策略来使这张图片的某些部分无条件地解决，这有可能为希尔伯特的第十二个问题提供令人满意的解决方案，以扩展Gross-Zagier和Kudla-Rapoport-Yang的基本工作，以实现P-Zagier的基本工作。库德拉（Kudla）的计划最终扩展到P-ADIC，可以为希尔伯特（Hilbert）第十二个问题提供关键，这也许是我最近与Jan Vonk的工作中最重要的见解。