Arithmetical Algebraic Geometry

Ulmer博士在算术代数几何中提出了三个项目，分别与数域和函数域上的伽罗瓦表示、模形式和椭圆曲线有关。提出的第一个项目是利用Colmez和Fontaine最近在p-进Hodge理论上的工作，研究p-进数的Galois群的某些表示的模a素数p的约化。在第二个项目中，Ulmer博士在函数域上的椭圆曲线上的适当位置构造了局部点的一个子群；这个子群包含全局点。他建议利用这些局部点来研究函数域上Birch和Swinnerton-Dyerforelate曲线的猜想。Ulmerer博士提出的第三个项目是研究变种的上同调中的一类新问题，这些问题是受到经典的非零级数结果的启发而产生的。新的问题是用Grothendieck对L函数的分析重新解释了消失的结果，并引出了通常的几何问题。在某些情况下，这些问题可以用Katz和Sarnak的单行结果来处理。这个建议属于算术代数几何的一般领域--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，这种结合非常富有成效，最近解决了经得起几代人努力的问题。它的许多后果包括用于光盘和硬盘等计算机存储设备的新纠错码，以及用于互联网金融交易的安全信息传输方案。

Dr. Ulmer proposes three projects in arithmetical algebraicgeometry,related to Galois representations, modular forms, andelliptic curves,both over number fields and over function fields. The firstprojectproposed is to study the reduction modulo a prime p ofcertainrepresentations of the Galois group of the p-adic numbers,using therecent work of Colmez and Fontaine on p-adic Hodge theory. In thesecond project, Dr. Ulmer has constructed a subgroup of thelocalpoints at suitable places on an elliptic curve over afunction field; this subgroup contains the global points. He proposes to use these localpoints to study the conjecture of Birch and Swinnerton-Dyerforelliptic curves over function fields. The third project Dr.Ulmerproposes is to study a new class of problems in thecohomology ofvarieties which are inspired by classical non-vanishingresults forL-series. The new questions come by reinterpretingvanishing resultsusing Grothendieck's analysis of L-functions and lead topurelygeometric questions. In some instances these questions canbe treatedusing monodromy results of Katz and Sarnak.This proposal falls into the general area of arithmeticalalgebraicgeometry - a subject that blends two of the oldest areas ofmathematics: number theory and geometry. This combinationhas provedextraordinarily fruitful, having recently solved problemsthatwithstood the efforts of generations. Among its manyconsequences arenew error correcting codes which are used in computerstorage deviceslike compact disks and hard drives and secure informationtransmissionschemes which are used for financial transactions on theinternet.