Arithmetical Algebraic Geometry

Ulmer博士提出了算术代数几何的三个项目，涉及伽罗瓦表示、模形式和椭圆曲线，包括数域和函数域。提出的第一个项目是利用Colmez和Fontaine最近在p进霍奇理论上的工作，研究p进伽罗瓦群的某些表示的约简模a素数p。在第二个项目中，Ulmer博士在函数场上的椭圆曲线上的合适位置构造了局域点的子群；此子组包含全局点。他提出利用这些局域点来研究函数域上的Birch曲线和Swinnerton-Dyerforelliptic曲线的猜想。ulmerdr .提出的第三个项目是研究一类新的从经典的l -级数的非消失结果中得到启发的品种的生态同调问题。新的问题来自于利用格罗滕迪克对l函数的分析对消失结果的重新解释，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的单一性结果来处理。这一建议属于算术、代数、几何的一般领域——这一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，这种结合取得了非凡的成果，最近解决的问题经受住了几代人的努力。它的诸多后果包括用于计算机存储设备（如光盘和硬盘驱动器）的新型纠错码，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.