Mathematical Sciences: Theory of P-Adic Modular Forms and Hecke Algebras

本基金的研究涉及p进赫克代数的理论。该理论试图给出一种系统的方法来构造与p进或复尖形相关的p进l函数及其在p进整数中系数的幂级数环上的伽罗瓦表示。此外，主要研究者希望找到这些伽罗瓦表示与所构造的p进L-函数之间的算术关系。这个理论已经发展为有理数，但在一般的数字领域还有很多工作要做。将会应用到Iwasawa的虚二次场理论。模形式的p进理论是本研究的主题。模形式是一种以特定方式变换的函数，是当今数论研究的中心。p进数理论用数论p进数代替了通常的实数或复数。利用通常的理论获得类似物是本研究的一个主要焦点。

The research in this grant concerns the theory of p-adic Hecke algebras. This theory attempts to give a systematic way to construct p-adic L-functions associated with p-adic or complex cusp forms and their Galois representations over power series rings with coefficients in the p-adic integers. Also the principal investigator hopes to find arithmetic relations between these Galois representations and the constructed p-adic L- functions. This theory has been developed for the rational numbers but much remains to be done over general number fields. There will be applications to the Iwasawa theory of imaginary quadratic fields. The p-adic theory of modular forms is the subject of this research. Modular forms are functions that transform in certain ways and are at the center of much research in number theory today. The p-adic theory replaces the usual real or complex numbers with the number theoretic p-adic numbers. Obtaining analogs with the usual theory is a major focus of this research.