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

设K是数域，G_K是K的绝对Galois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时精确地以一阶消失。最后一个假设意味着K要么是一个全真实的数域，要么是一个几乎全真实的域（简称ATR）。七十年代初，哈罗德·斯塔克（Harold Stark）做了深入的研究，将这种L-函数在s=0时的一阶导数与K的阿贝尔扩张中的单位联系起来。这个猜想是惊人的，因为它涉及到一个特殊的L-函数值（解析数据）与某个单位在一个阿贝尔扩展的K（代数数据）。哈罗德·斯塔克可以在K是有理数域或K是虚二次数域的特殊情况下证明这个猜想。他可以证明这一点有关的一阶导数的L-功能的一个特殊价值的一个很好的全纯函数，满足许多对称性，一个所谓的模块功能。我研究的主要目标是找到“正确的数学对象”，在K是一个完全真实的域或ATR域的一般情况下，它应该取代模函数所扮演的角色。到目前为止，这个数学对象似乎是非常难以捉摸的，预计在这个问题上取得的任何进展将汇集不同的领域，如分析，代数和微分几何。 所有这些提到的主题在数论中以不同的形式出现，我们数论家希望澄清它们之间的关系，以便有一个更统一的理解。

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.