Martin Kneser’s Work on Quadratic Forms and Algebraic Groups
Martin Kneser’s Work on Quadratic Forms and Algebraic Groups
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Martin Kneser 在二次形式和代数群方面的工作
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发表时间：
2008
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通讯作者：
Rudolf Scharlau
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作者：
Rudolf Scharlau
本文概述了马丁·克内瑟在1955-1970年期间对二次型和代数群算术理论的研究工作。为了将克内塞的工作放在适当的历史背景下，本文对这一时期之前的二次型理论进行了概述，并对他后来开创或影响的一些工作进行了展望。1.二次型简史1884-1954二次型理论作为(初等)数论的一部分出现，最初是在有理整数上处理二次丢番图方程。现代语言中的主要问题是：(A)等价问题：Z上的两个二次模(“格”)(L1，Q1)和(L2，Q2)何时是等距的？(B)分类问题：确定受自然限制(即给定的维度、行列式、亏格)的所有等距类的一组表示或一组易于计算的不变量。(C)表示问题：对于哪个t∈Z，存在q(X)=t的x∈L？(D)表示数a(t，L)=|{x∈L|q(X)=t}的确定。这里，Li是有限秩自由Z-模，或有理向量空间Vi中的格，而qi是Li(或Vi)上的二次型。对于问题(D)，Q应该是(正)定的，但适当地利用正交群O(L)的作用，可以将问题(及其解)推广到一般情况。此外，该理论还推广到了代数数域k的整数环ok上的二次格，并且在某种程度上推广到了任意整体域上的整数环。2000年数学学科分类。主11E12、11E57、01A70、次1103、11E72、20G30。2007年12月19日在智利兰基霍举行的二次型代数和算术理论国际会议上的演讲的扩展版本。作者感谢Rainer Schulze-Pillot、Detlev Hoffmann、Ulrich Stuhler和匿名裁判仔细阅读手稿以及有用的建议和评论。C©2009鲁道夫·沙劳
This article provides an overview of the research work of Martin Kneser on the arithmetic theory of quadratic forms and algebraic groups, focusing on the period 1955 – 1970. To put Kneser’s work in proper historical context, a survey of the theory of quadratic forms prior to that period, and an outlook on some subsequent work initiated or influenced by him is given. 1. A Short History of Quadratic Forms 1884 – 1954 The theory of quadratic forms emerged as a part of (elementary) number theory, dealing with quadratic diophantine equations, initially over the rational integers. The main questions in modern language are: (a) the equivalence problem: when are two quadratic modules (“lattices”) (L1, q1) und (L2, q2) over Z isometric? (b) the classification problem: determine a set of representatives or a set of easily computable invariants for all isometry classes of lattices subject to natural restrictions (i.e. with given dimension, determinant, genus). (c) The representation problem: for which t ∈ Z does there exist an x ∈ L with q(x) = t ? (d) The determination of the representation numbers a(t, L) = |{x ∈ L | q(x) = t}|. Here, the Li are free Z-modules of finite rank, or lattices in rational vector spaces Vi, and qi is a quadratic form on Li (or Vi). For problem (d), q should be (positive) definite, but using the action of the orthogonal group O(L) in an appropriate way, the problem (and its solution) can be extended to the general case. Also, the theory carries over to quadratic lattices over the ring of integers ok of an algebraic number field k and to some extent to rings of integers in arbitrary global fields. 2000 Mathematics Subject Classification. Primary 11E12,11E57,01A70, Secondary 1103,11E72,20G30. Extended version of a talk given at the International Conference on the Algebraic and Arithmetic Theory of Quadratic Forms, Llanquihue, Chile, 19.12.2007. The author is indebted to Rainer Schulze-Pillot, Detlev Hoffmann, Ulrich Stuhler and the anonymous referee for their careful reading of the manuscript and helpful suggestions and remarks. c ©2009 Rudolf Scharlau