Arithmetic of Automorphic Forms and Special L-Values

l函数在现代数论中扮演着重要的角色。也许最著名的例子是它们在著名的伯奇和斯温纳顿-戴尔猜想中所扮演的角色。给定一条椭圆曲线，为了在有理数上定义的简单性，该猜想以一种惊人的精确方式将椭圆曲线的各种算术不变量（如莫德尔-韦尔群的秩，猜想有限的state - shafarevich群的大小）与附在椭圆曲线上的l函数在s=1处的值联系起来。最后一个定义是把椭圆曲线的所有质数的局部信息放在一起，这个定义只有在s的实部足够大（实际上是3/2）时才有意义，为了在s=1时有意义，我们需要建立它的解析延图。这要感谢安德鲁·怀尔斯关于椭圆曲线的模性的著名工作，它是通过识别椭圆曲线的l函数与模形式的l函数来实现的。实际上，刚才所描述的推测性的情况可以扩展到更普遍的情况。也就是说，对于一个算术对象，通常称为动机，人们将一个l -函数联系起来，人们相信l -函数在其特殊的l值中编码了关于潜在动机的重要信息（Bloch-Kato猜想）。然而，这些l函数，即使是在算术几何领域中定义的，也只能通过将它们与自选形式的l函数相识别，才能用现有的知识来研究它们。动机l函数和自同构l函数之间的这种联系表明，自同构l函数的特殊值可能具有有趣的算术性质。实际上，本研究项目的主要目的是研究各种自同构形式，即hermite模形式，Siegel模形式和Siegel- jacobi模形式的特殊l值的代数和p进性质。对于hermite和Siegel模形式，我们的目标是构造阿贝尔和非阿贝尔p进测度，它们构成了Iwasawa理论主要猜想（交换或非交换）的表述中不可或缺的组成部分，而这又是解决上述算术不变量和特殊l值之间的bloh - kato猜想所隐含的关系的唯一工具。西格尔-雅可比形式的情况就不同了。它们的l -函数目前还不能被识别为由伽罗瓦表示得到的l -函数，实际上它们与志村变种没有关系。然而，它们确实具有算术结构，在本研究中，目的是解决与Shimura变体相关的自同构形式的各种众所周知的现象（特殊l值的代数性，klingen型爱森斯坦级数的Garrett猜想等）是否仍然适用于Siegel-Jacobi模形式的问题。

L-functions are known to play a central role in modern number theory. Perhaps the most well known example is the role they play in the famous Birch and Swinnerton-Dyer conjecture. Given an elliptic curve, for simplicity defined over the rational numbers, the conjecture relates various arithmetic invariants of the elliptic curve (such as the rank of the Mordell-Weil group, the size of the conjecturally finite Tate-Shafarevich group) in an astonishingly precise way to the value at s=1 of the L-function attached to the elliptic curve. This last one is defined by putting together local information, over all prime numbers, of the elliptic curve, and as defined makes sense only for the real part of s large enough (actually 3/2) and in order to make sense at s=1 one needs to establish its analytic continuation. This is now known thanks to the celebrated work of Andrew Wiles on the modularity of elliptic curves, and it is achieved by identifying the L-function of the elliptic curve with an L-function of a modular form. Actually the picture just described conjecturally extends to a much general situation. Namely to an arithmetic object, usually called a motive, one associates an L-function which it is believed to encode in its special L-values important information about the underlying motive (Bloch-Kato conjectures). However these L-functions, even though they are defined in the realm of arithmetic geometry, they can be studied with the current status of knowledge only by identifying them with the L-function of an automoprhic form. This connection between motivic and automorphic L-functions suggests that special values of automoprhic L-functions may enjoy interesting arithmetic properties. Indeed the main aim of this research project is to investigate algebraic and p-adic properties of special L-values of automorphic forms of various kinds, namely Hermitian, Siegel and Siegel-Jacobi modular forms. For Hermitian and Siegel modular forms we aim to the construction of abelian and non-abelian p-adic measures, which constitute an indispensable ingredient in the formulation of the Main Conjectures of Iwasawa Theory (commutative or not), which in turn is the only tool available to tackle the aforementioned relation implied by the Bloch-Kato conjecture between arithmetic invariants and special L-values. The situation is different with respect to the Siegel-Jacobi forms. Their L-function is not known at present to be identified to an L-function obtained from Galois representations, and actually they are not related to Shimura varieties. However they do enjoy arithmetic structure and in this research grant the aim is to address the question whether the various well-understood phenomena for automorphic forms associated to Shimura varieties (algebraicity of special L-values, Garrett's conjecture on Klingen-type Eisenstein series etc) are still valid for Siegel-Jacobi modular forms.