Arithmetic of L-Values

9701782 Stevens首席研究员打算继续研究l函数的特殊值及其与算术几何的关系。本研究为研究自同构形式和p进上同提供了新的工具，并提供了p进上同的具体例子，从而提高了我们对自同构形式和p进上同的理解。PI最近对Mazur、Tate、Teitelbaum和Coleman的猜想的证明提出了关于自同构形式的解析族、它们的p进l函数、伽罗瓦表示族和过收敛f晶体的新问题。拟议的研究将阐明这些家族是如何在特殊的非结晶点上退化的，并将根据重量方向的变形来描述这些点上的单态，从而以潜在有用的方式扩大单态的标准图景。本研究也为加藤在p进解析族中的理论变形提供了工具，补充了加藤最近关于l函数值和模曲线的K_2的工作。在相关工作中，PI希望发展一个p进的Eichler-Shimura对应关系，将他的过收敛模符号理论与Katz的过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进提升来构建非普通半积分权模形式的解析族。最后，将这些思想推广到其他约化代数群上的自同构形式。本研究为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变量p进l函数的构造提供了有希望的工具。这与一些新的研究有关，包括p进单性，Jochnowitz关于半积分权形式上的算子的平方根的猜想，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，这门学科融合了数学中两个最古老的领域：数论和几何。事实证明，这一组合非常富有成效——最近解决了几代人经受住了考验的问题。其诸多后果之一是新的纠错码。这些代码对现代计算机（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.