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

大约12年前，我发现了一个完全明确的，但现在完全是抽象的，椭圆曲线上的全局点的构造，我称之为“斯塔克-希格纳点”。在最简单的非平凡的设置，这些点预计将被定义在交换扩展的真实的二次字段，因此将有助于我们的理解“显式类场论”等领域。这个发现补助金提案的主要目标是给一个无条件的建设全球上同调类的塞尔默组的椭圆曲线，应该产生从斯塔克-希格纳点通过连接同态的库默理论。这个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对角环和Beilinson-Flach元素的p-adic变形产生的，我从2010年起就与我的合作者（最重要的是维克托·罗杰和马西莫·贝托里尼）一起探索。这两个欧拉系统是一个自然的推广加藤的欧拉系统的贝林森元素，我把所有三个作为“欧拉系统的加勒特-兰金-塞尔伯格型”，因为发挥关键作用的公式加勒特和兰金-塞尔伯格在有关他们的特殊价值的L-功能。对这些欧拉系统的研究，已经使我和我的合作者们，本着科茨和怀尔斯早期基本结果的精神，找到了伯奇和斯温纳顿-戴尔猜想的新例子。最相关的项目在手是有限的组件的模椭圆曲线在Q附加到字符的真实的二次域时，相关的L-函数是非零的中心点。这一新的入侵到伯奇和斯温纳顿-戴尔的交换字符的真实的二次字段在“解析秩零”提出了希望，该方法的扩展将导致所需的信息神秘的斯塔克-海格纳点，相应的情况下，伯奇和斯温纳顿-戴尔猜想“在解析秩一”。

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".