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：第一，椭圆曲线上全局点的构造，目的是更好地理解Birch和Swinnerton-Dyer猜想；第二，希尔伯特第十二问题和显式类场论，其中实数二次场的类场的解析构造是最简单的典型特例。我打算建立在我对这些问题取得了重大进展在过去的五年里,最值得注意的是,(1)我的工作与维克多Rotger p进变形Chow组三对角周期的产品模块化的曲线和Kuga-Sato品种,导致在特定的证明新病例的桦树和Swinnerton-Dyer推测分析等级为零,对椭圆曲线在问扭环类字符的二次领域,(2)我最近与Jan Vonk在去年的合作，揭示了一个以前意想不到的实二次域的奇异模理论，它与经典的复乘法理论有着惊人的相似之处。这两部作品都为更好地理解我在2000年左右提出的Stark-Heegner点的构建提供了互补和有希望的途径，从那时起，我的主要研究重点就是对其进行支持。我和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例），由特殊几何对象在Chow群或Shimura变种的更高Chow群中的p进变形引起。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点的图像进行比较。虽然不足以建立Stark-Heegner点的全局性，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发现Stark-Heegner点可以在更广泛的（仍然是推测的）实二次域的复乘法理论框架中重新定义，其中亚纯模函数的作用是由我们所谓的“刚性亚纯环”发挥的，这似乎对未来的进展充满了希望。事实上，我们现在处理了一些令人信服的策略，使这幅图的某些部分成为无条件的，有可能导致希尔伯特第十二问题的令人满意的解决方案，这是基于将格罗斯-扎吉尔和Kudla-Rapoport-Yang的基本工作扩展到p进设置。库德拉的计划最终扩展到p-adics，可能为希尔伯特第十二个问题提供一把钥匙，这可能是我最近与扬·冯克（Jan Vonk）合作中最重要的见解

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.**