Arithmetical Algebraic Geometry

Ulmer博士提出了算术代数几何的三个项目，涉及Galois表示，模形式和椭圆曲线，都在数域和函数域上。 第一个项目是利用Colmez和方丹关于p-adic Hodge理论的最新工作，研究p-adic数的Galois群的某些表示的模素数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一个局部点的子群;这个子群包含全局点。他提出利用这些局部点来研究函数域上Birch和Swinnerton-Dyerforelliptic曲线的猜想。 Ulmer博士提出的第三个项目是研究一类新的问题，这些问题是受L-级数经典的非消失结果的启发而产生的。 新的问题来自于使用Grothendieck的L-函数分析重新解释消失结果，并导致纯粹的几何问题。 在某些情况下，这些问题可以用Katz和Sarnak的monodromy结果来处理。这个提议福尔斯属于算术代数几何的一般领域--一个融合了两个最古老的数学领域：数论和几何的学科。 这种结合已经证明是非常富有成效的，最近解决了几代人努力解决的问题。 其众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，以及用于互联网上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.