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

我的研究计划将围绕数论中的两个中心开放问题：首先，椭圆曲线上的全局点的构造，以更好地理解Birch和Swinnerton-Dyer猜想，其次，希尔伯特第十二问题和显式类域理论，其中真实的二次域的类域的解析构造是最简单的原型特例。我打算建立在我在过去五年中对这些问题取得的重大进展的基础上，最值得注意的是，（1）我与维克托罗特尔关于模曲线和Kuga-Sato变种的三重积的Chow群中对角循环的p-adic变形的工作，这特别导致了Birch和Swinnerton-Dyer猜想在解析秩为零的新情况下的证明，对于椭圆曲线在Q扭曲的环类字符的真实的二次领域，和（2）我最近的工作与扬Vonk在过去的一年中，这揭示了一个以前意想不到的理论奇异模的真实的二次领域享有惊人的平行与经典理论的复杂的乘法。 这两部作品都为更好地理解我在2000年左右介绍的斯塔克-希格纳点的构造提供了互补和有希望的途径，从那时起，斯塔克-希格纳点的支撑一直是我的主要研究重点。 我的工作与维克托围绕对象，我们称之为“广义加藤类”：全球类的塞尔默组的椭圆曲线（在适当的数字领域，类领域的真实的二次领域是一个特别诱人的特殊情况）所产生的p进变形的特殊几何对象在周组或更高周组志村品种。我们正在进行的努力的目的是比较这些类的图像下的连接同态的库默理论的斯塔克-希格纳点。虽然不足以建立斯塔克-希格纳点的整体性质，这是定义为纯粹的局部对象分析，将它们与全球塞尔默类是一个决定性的一步。从一个表面上完全不同的角度来看，我发现与扬斯塔克-希格纳点可以重铸在一个更广泛的框架（仍然是抽象的）理论复乘法的真实的二次领域，其中的作用，亚纯模函数发挥了我们所谓的“刚性亚纯上循环”，似乎是充满希望的未来进展。事实上，我们现在处理令人信服的战略，使这一图片的某些部分无条件的，可能导致一个令人满意的解决方案，希尔伯特的第十二个问题的真实的二次域的基础上扩展的基本工作的格罗斯-扎吉尔和Kudla-Rapoport-杨的p-adic设置。库德洛的程序最终扩展到p-adics可以为希尔伯特第十二问题提供一把钥匙，这也许是我最近与扬·冯克（Jan Vonk）的工作中出现的最重要的见解。

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.**