Euler Systems of Garrett-Rankin-Selberg type and Stark-Heegner points

大约12年前，我发现了椭圆曲线上全局点的一种完全显式的、但目前完全是猜测的构造，我称之为“Stark-Heegner点”。在最简单的非平凡环境中，这些点应该定义在实二次场的阿贝尔扩张上，因此有助于我们理解此类场的“显式类场理论”。本文的主要目的是利用Kummer理论的连通同态，无条件地构造由Stark-Heegner点产生的椭圆曲线Selmer群中的整体上同调类。这种构造中的关键工具是两种新类型的欧拉系统，它们是由Gross-Kudla-Schoen对角循环和Beilinson-Flach元素的p进变形产生的，自2010年以来，我一直在与我的合作者(最重要的是Victor Rotger和Massimo Bertolini)一起探索这两种类型的欧拉系统。这两个欧拉系是加藤贝林逊元素的欧拉系的自然推广，我称这三个欧拉系为“加勒特-兰金-塞尔贝格型欧拉系”，因为加勒特和朗金-塞尔伯格公式在将它们与L函数的特殊值联系起来时所起的关键作用。根据Coates和Wiles的基本早期结果，对这些欧拉系统的研究已经将我和我的合作者带到了Birch和Swinnerton-Dyer猜想的新案例中。与手头上的项目最相关的是Q上的模椭圆曲线的Mordell-Weil群的分支的有限性，当相关的L函数在中心点非零时，这些分支附着于实二次域的特征标上。在“解析秩零”中实二次域的阿贝尔特征的Birch和Swinnerton-Dyer的这一新的探索提高了人们的希望，该方法的扩展将导致关于神秘的Stark-Heegner点的所需信息，对应于Birch和Swinnerton-Dyer猜想“在解析秩1中”的情况。

Around twelve years ago I discovered a completely explicit, but for now entirely conjectural, construction of global points on elliptic curves, which I called "Stark-Heegner points". In the simplest non-trivial setting, these points are expected to be defined over abelian extensions of real quadratic fields and would therefore contribute to our understanding of "explicit class field theory" for such fields. The main objective of this Discovery Grant proposal is to give an unconditional construction of the global cohomology classes in Selmer groups of elliptic curves that ought to arise from Stark-Heegner points via the connecting homomorphism of Kummer theory. The key tools in this construction are two new types of Euler systems arising from p-adic deformations of Gross-Kudla-Schoen diagonal cycles and Beilinson-Flach elements which I have been exploring with my collaborators (most importantly, Victor Rotger and Massimo Bertolini) since 2010. These two Euler systems are a natural generalisation of Kato's Euler system of Beilinson elements, and I refer to all three as "Euler systems of Garrett-Rankin-Selberg type" because of the key role played by the formulae of Garrett and Rankin-Selberg in relating them to special values of L-functions. The study of these Euler systems has already led my collaborators and me to new cases of the Birch and Swinnerton-Dyer conjecture in the spirit of the fundamental early results of Coates and Wiles. Most relevant to the project at hand is the finiteness of components of Mordell-Weil groups of modular elliptic curves over Q attached to characters of real quadratic fields when the associated L-function is non-zero at the central point. This new inroad into the Birch and Swinnerton-Dyer for abelian characters of real quadratic fields in "analytic rank zero" raises the hope that extensions of the method will lead to the desired information about the mysterious Stark-Heegner points, corresponding to cases of the Birch and Swinnerton-Dyer conjecture "in analytic rank one".