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，我发现了一个完全明确的，但是目前完全是猜想的，在椭圆曲线上的全球要点的构造，我称之为“ Stark-Heegner Points”。在最简单的非平底环境中，预期这些点将在实际二次领域的Abelian扩展上定义，因此将有助于我们对此类领域的“显式阶级字段理论”的理解。这项发现赠款提案的主要目的是在Selmer的椭圆曲线群中提供无条件的构造，这些椭圆形曲线应该通过Kummer Theory的同性形态来从Stark-Heegner点产生。 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这三个都是“ Garrett-Rankin-Selberg类型的Euler Systems”，因为Garrett和Rankin-Selberg的公式在将它们与L-功能的特殊值相关联时起着关键作用。对这些Euler系统的研究已经使我的合作者和我以Coates和Wiles的基本早期结果精神精神构成了桦木和Swinnerton-Dyer猜想的新案例。与手头的项目最相关的是当中心点上关联的L功能不是零时，在Q附着在Q上的Mordell-Weil组的成分是Q附着在Q上的Q的有限性。这是“分析等级零”中真正二次领域的Abelian角色的桦木和Swinnerton-Dyer的新侵害，这使人们希望该方法的扩展将导致有关神秘的Stark-Heegner点的理想信息，以对应于桦木和Swinnerton-Dyer-Dyer-Dyer-Dyer-Dyer-Dyer-Dyer-Dyer-Dyer-Dyer-dyer the Insparient candextict candextict cank in Analitaltic Cank on Analitaltic canke n in Analitaltial cans on。