Arithmetic of L-values

L值PI的算术：Glenn Stevens Proposal 0071065项目摘要主要研究者建议继续他关于L-函数的特殊值及其与算术几何的关系的工作。拟议的研究将通过提供研究前者的新工具和提供后者的具体例子来提高我们对自守形式和p-adic上同调的理解。具体地说，该研究将扩展显式互易定律的适用性，以包括伽罗瓦表示的p进分析族。 这将为研究二元p-自适应L-函数的算术性质提供一个新的工具。 它还将阐明伽罗瓦表示的解析族如何在半稳定的非晶态点退化，并将在这些点上用权方向的变形来描述单值性，从而以潜在有用的方式扩大了单值性的标准图像。 本研究也将补充加藤最近关于L-函数的值和模曲线的K-理论的工作，为加藤理论在p-adic解析族中的变形提供工具。 在相关的工作中，PI希望开发一个p-adic Eichler-Shimura对应，将他的理论与Katz的理论联系起来，并通过推广PI早期工作中开发的ap-adic theta提升来构建非普通半整权模形式的分析族。最后，PI打算将这些想法推广到其他约化代数群上的自守形式。 这一研究为构造非普通自守表示的p-adic解析族、Galois表示的自然变形空间和多元p-adic L-函数提供了有希望的工具。 这是与一些调查，包括p-adic monodromy，Jochnowitz'astructures的平方根的theta运营商，和p-adic变形理论的伽罗瓦representations.The调查这一建议属于generalmathematical领域的算术几何。 这个超现代的研究领域结合了数学的两个最古老的分支：数论和几何。 这种结合产生的新见解正在产生越来越强大的工具来解决像费马大定理这样的长期存在的问题，这些问题抵制了三个多世纪数学家的最强努力。 此外，虽然算术几何有时被认为是最纯粹的纯数学，它也一直在开发有见地的新技术，导致在诸如纠错码和密码学等应用领域的巨大进步。

Arithmetic of L-valuesPI: Glenn Stevens Proposal 0071065Project AbstractThe principal investigator proposes to continue his work on specialvalues of L-functions and their relation to arithmetic geometry.The proposed research would improve our understanding of bothautomorphic forms and p-adic cohomology by providing new tools forstudying the former and by providing concrete examples of the latter.Specifically, the research would extend the applicability of ExplicitReciprocity Laws to include p-adic analytic families of galoisrepresentations. This, in turn, would provide a new tool forstudying the arithmetic properties of two-variable p-adicL-functions. It would also clarify how analytic families of galoisrepresentations degenerate at semistable non-crystalline points andwould describe monodromy at such points in terms of a deformation inthe weight direction, thus enlarging the standard picture ofmonodromy in potentially useful ways. This research would alsocomplement Kato's recent work on values of L-functions and K-theory ofmodular curves by providing tools for deforming Kato's theory inp-adic analytic families. In related work, the PI hopes to developa p-adic Eichler-Shimura correspondence that would relate histheory of overconvergent modular symbols to Katz's theory ofoverconvergent modular forms and to construct analytic families ofnon-ordinary half-integral weight modular forms by generalizing ap-adic theta lifting developed in earlier work of the PI. Finally,the PI intends to generalize these ideas to automorphic forms onother reductive algebraic groups. This research offers promisingtools for the construction of p-adic analytic families ofnon-ordinary automorphic representations together with naturaldeformation spaces of Galois representations, andmultivariable p-adic L-functions. This is connected with anumber of investigations, including p-adic monodromy, Jochnowitz'sconjectures on the square root of the theta operator, and thep-adic deformation theory of Galois representations.The investigations of this proposal belong to the generalmathematical area of Arithmetic Geometry. This ultramodern researcharea combines two of the oldest branches of mathematics: numbertheory and geometry. New insights arising out of thiscombination are producing increasingly powerful tools to solvelongstanding problems like Fermat's Last Theorem, which haveresisted the strongest efforts of over three centuries ofmathematicians. In addition, though Arithmetic Geometry is sometimesregarded as the purest of pure mathematics, it has also beendeveloping insightful new techniques leading to dramatic progress insuch applied areas as error-correcting codes and cryptography.