Arithmetic of L-Values

9701782史蒂文斯是首席研究员，他打算继续研究L函数的特殊值及其与算术几何的关系。拟议的研究将通过为研究自同构形式提供新的工具并通过提供后者的具体例子来提高我们对自同构形式和p-进上同调的理解。PI最近证明了Mazur，Tate，Teitelbaum和Coleman的一个猜想，提出了关于自同构型的解析族，它们的p-进L函数，伽罗瓦表示族和超收敛F-晶族的新问题。拟议的研究将阐明这些家族如何在特殊的非晶点退化，并将根据重量方向的变形来描述这些点的单晶性，从而以潜在有用的方式扩大单晶性的标准图景。这一研究也是对加藤最近关于模曲线的L函数和K_2的取值工作的补充，为在p-进解析族中变形加藤理论提供了工具。在相关的工作中，PI希望建立一个p-进Eichler-Shimura对应，将他的超收敛模符号理论与Katz的超收敛模形式理论联系起来，并通过推广PI早期工作中发展的p-进theta提升来构造非普通半整权模形式的解析族。最后，PI打算将这些思想推广到其他约化代数群上的自同构形式。这一研究为构造非普通自同构表示的p进解析族、伽罗瓦表示的自然变形空间和多元p进L函数提供了很有前途的工具。这与一些新的研究有关，包括p-进单行，Jochnowitz关于半整数权形式上的theta算子平方根的猜想，以及伽罗瓦表示的p-进变形理论。这个项目属于算术几何的一般领域--一个融合了数论和几何这两个最古老的数学领域的学科。事实证明，这种结合非常有成效--最近解决了几代人经受住的问题。在其众多后果中，有一种是新的纠错码。这种代码对于现代计算机(硬盘)和光盘都是必不可少的。

9701782 Stevens The principal investigator intends to continue his work on special values of L-functions and their relation to arithmetic geometry. The proposed research would improve our understanding of both automorphic forms and p-adic cohomology by providing new tools for studying the former and by providing concrete examples of the latter. The PI's recent proof of a conjecture of Mazur, Tate, Teitelbaum, and Coleman raises new questions about analytic families of automorphic forms, their p-adic L-functions, and families of galois representations and overconvergent F-crystals. The proposed research would clarify how such families degenerate at special non-crystalline points and would describe monodromy at such points in terms of a deformation in the weight direction, thus enlarging the standard picture of monodromy in potentially useful ways. This research would also complement Kato's recent work on values of L-functions and K_2 of modular curves by providing tools for deforming Kato's theory in p-adic analytic families. In related work, the PI hopes to develop a p-adic Eichler-Shimura correspondence that would relate his theory of overconvergent modular symbols to Katz's theory of overconvergent modular forms and to construct analytic families of non-ordinary half-integral weight modular forms by generalizing a p-adic theta lifting developed in earlier work of the PI. Finally, the PI intends to generalize these ideas to automorphic forms on other reductive algebraic groups. This research offers promising tools for the construction of p-adic analytic families of non-ordinary automorphic representations together with natural deformation spaces of Galois representations, and multivariable p-adic L-functions. This is connected with a number of new investigations, including p- adic monodromy, Jochnowitz's conjectures on the square root of theta operator on half-integral weight forms, and the p-adic deformation theory of Galois representations. This project falls into the general area of arithmetic geometry -a subject that blends two of the oldest areas of mathematics: number theory and geometry. This combination has proved extraordinarily fruitful - having recently solved problems that withstood generations. Among its many consequences are new error correcting codes. Such codes are essential for both modern computers (hard disks) and compact disks.