Gross-stark conjectures, abelian L-functions and algebraic cycles

Gross-stark 猜想、阿贝尔 L 函数和代数圈

• 批准号: 371968-2009
• 负责人: Chapdelaine, Hugo
• 金额: $ 1.53万
• 依托单位: Université Laval
• 依托单位国家: 加拿大
• 项目类别: Discovery Grants Program - Individual
• 财政年份: 2011
• 资助国家: 加拿大
• 起止时间: 2011-01-01 至 2012-12-31
• 项目状态: 已结题
• 来源: https://www.nserc-crsng.gc.ca/ase-oro/Details-Detailles_eng.asp?id=483814
• 关键词: Gross; stark; conjectures; abelian; functions

Let K be a number field and let G_K be the absolute Galois group of K. In the case where one deals with one dimensional representations of G_K, Class field theory provides a correspondence between these L-functions and finite abelian extensions of K. The L-functions coming from one dimensional representations are called abelian L-functions. In some cases, these abelian L-functions vanish precisely with order one at s=0. The last assumption implies that K is either a totally real number field or an almost totally real field (ATR for short). In the early seventies, Harold Stark made deep conjectures which relate the first derivative of such L-functions at s=0 with units lying in abelian extensions of K. This conjecture is striking since it relates a special value of an L-function (analytic data) with a certain unit in an abelian extension of K (algebraic data). Harold Stark could prove this conjecture in the special case where K is the field of rational numbers or K is an imaginary quadratic number field. He could prove it by relating the first derivative of the L-function to a special value of a nice holomorphic function which satisfies many symmetries, a so-called modular function. The main goal of my research is to find the "right mathematical object" that should replace the role played by the modular functions in the general case where K is either a totally real field or an ATR field. So far, this mathematical object seems to be very elusive and it is expected that any progress made on this problem will bring together different areas like analysis, algebra and differential geometry. All these mentioned themes appear in various guises in number theory and we, number theorists, would like to clarify their relationships in order to have a more unified understanding.

设K是数域，G_K是K的绝对伽罗瓦群。在处理G_K的一维表示的情况下，类域理论提供了这些L函数与K的有限阿贝尔扩张之间的对应。来自一维表示的L函数称为阿贝尔L函数。在某些情况下，这些阿贝尔L函数在S=0时以一阶精确地消失。最后一个假设意味着K要么是全实数域，要么是几乎全实数域(简称ATR)。二十世纪七十年代初，Harold Stark提出了一个深层次的猜想，它把这种L函数在S=0的一阶导数与K的阿贝尔扩张中的单位联系起来。这个猜想之所以引人注目，是因为它把一个L函数(分析数据)的特定值与K(代数数据)的阿贝尔扩张中的某个单位联系起来。在K是有理数域或K是虚二次数域的特殊情况下，Harold Stark可以证明这一猜想。他可以通过将L函数的一阶导数与满足多种对称性的好的全纯函数的特定值联系起来来证明这一点，这就是所谓的模函数。我研究的主要目的是在K是全实域或ATR域的一般情况下，找到应该取代模函数所起作用的“正确的数学对象”。到目前为止，这个数学对象似乎非常难以捉摸，预计在这个问题上取得的任何进展都将汇集不同的领域，如分析、代数和微分几何。所有这些主题都以不同的形式出现在数论中，我们作为数论家，希望澄清它们之间的关系，以便有更统一的理解。