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

Garrett-Rankin-Selberg 型和 Stark-Heegner 点的欧拉系统

• 批准号: 155499-2013
• 负责人: Darmon, Henri
• 金额: $ 4.08万
• 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: 2017
• 资助国家: 加拿大
• 起止时间: 2017-01-01 至 2018-12-31
• 项目状态: 已结题
• 来源: https://www.nserc-crsng.gc.ca/ase-oro/Details-Detailles_eng.asp?id=635544
• 关键词: Euler; Systems; Garrett; Rankin; Selberg

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".

大约十二年前，我发现了一种完全明确但目前完全是推测的椭圆曲线上全局点的构造，我将其称为“Stark-Heegner 点”。在最简单的非平凡设置中，这些点预计将在实二次域的阿贝尔扩展上定义，因此有助于我们理解此类域的“显式类域论”。这项发现资助提案的主要目标是无条件构造 Selmer 椭圆曲线群中的全局上同调类，这些椭圆曲线应该通过 Kummer 理论的连接同态从 Stark-Heegner 点产生。这种构造的关键工具是两种新型欧拉系统，它们源自 Gross-Kudla-Schoen 对角线循环的 p-adic 变形和 Beilinson-Flach 元素，我一直在与我的合作者（最重要的是 Victor Rotger 和 Massimo Bertolini）一起探索这些元素。自 2010 年以来。这两个欧拉系统是加藤贝林森元素欧拉系统的自然推广，我将这三个系统称为“欧拉系统” Garrett-Rankin-Selberg 类型”，因为 Garrett 和 Rankin-Selberg 公式在将它们与 L 函数的特殊值联系起来方面发挥了关键作用。对这些欧拉系统的研究已经使我和我的合作者本着科茨和怀尔斯早期基本结果的精神得出了伯奇和斯温纳顿-戴尔猜想的新案例。与当前项目最相关的是当相关的 L 函数在中心点非零时，Q 上的模椭圆曲线的 Mordell-Weil 群的分量的有限性附加到实二次域的特征。这种对“解析零阶”实二次域阿贝尔特征的 Birch 和 Swinnerton-Dyer 的新进展提出了希望，即该方法的扩展将导致有关神秘 Stark-Heegner 点的所需信息，对应于以下情况伯奇和斯温纳顿-戴尔猜想“处于分析第一级”。