Riemann surfaces and the theta function
Riemann surfaces and the theta function
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黎曼曲面和 theta 函数
DOI：
10.1007/bf02391007
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发表时间：
1964
期刊：
影响因子：
3.7
通讯作者：
J. Lewittes
中科院分区：
文献类型：
--
作者：
J. Lewittes
本文的目的是对 Riemarm 的某些定理进行清晰的阐述，特别是下面的定理 8，该定理是他通过研究 0 函数作为解决雅可比反演问题的方法而获得的。仔细阅读黎曼的文集[4]，就会发现这是他非常感兴趣的话题。对于本文，可以参考 [4]，第 133-142、212-224、487-504 页，以及补编，第 1-59 页。在黎曼提出后的半个世纪里，许多数学家试图阐明并证明他的结果。在这方面我们可以提到克里斯托菲尔、诺特、韦伯、罗斯特和庞卡德。对旧文献的引用可以在书籍[2]、[3]和[6]中找到。尽管做出了所有这些努力，但我很难说是否已经对所声称的一切提供了完整的证明。在本文中，我们希望对其中一些有趣的结果以及一些新定理给出正确的证明。我们的方法本质上是黎曼及其追随者的方法，尽管语言可能稍微现代一些。我们方法的关键是考虑基点的作用，即第一类积分的下限，及其对黎曼常数向量K的影响。基点和K的作用似乎被大家忽视了，可能是因为Riemann的说法，[4]，p. 133 和 p。 213，在适当的归一化下，向量K消失。最后，与被束缚于球体的特定分支覆盖物相比，抽象黎曼曲面的概念具有明显的优势。在第一节中，我们证明了关于某些“乘法函数”的零点的基本定理。总的来说，在本节中，我们试图符合
The purpose of this paper is to present a clear exposition of certain theorems of Riemarm, notably Theorem 8 below, which he obtained from his study of the 0 function as a means of solving the Jacobi inversion problem. A perusal of Riemann's collected works,[4], shows that this was a topic of great interest to him. For this paper, one may consult [4], pp. 133-142, 212-224, 487-504, and the Supplement, pp. 1-59. Many mathematicians, in the half century after Riemann, tried to elucidate and justify his results. In this connection we may mention Christoffel, Noether, Weber, Rost, and Poincard. Citations of the older literature may be found in the books,[2],[3], and [6].Despite all these efforts, it is difficult for me to say whether or not complete proofs have been given to everything that has been claimed. In this paper, we hope to give correct proofs of some of these interesting results, along with some new theorems. Our method is essentially that of Riemann and his followers, although the language may be slightly more modern. The key to our method is consideration of the role of the base point, ie, lower limit of the integrals of first kind, and its influence on the vector K of Riemann constants. The roles of the base point and K seem to have been overlooked by all, probably because of the statement of Riemann,[4], p. 133 and p. 213, that, under a suitable normalization, the vector K vanishes. Finally, having available the concept of an abstract Riemann surface gives one a distinct advantage over being tied down to a particular branched covering of the sphere. In the first section, we prove the basic theorem concerning the zeros of certain" multiplicative functions". On the whole, in this section, we try to conform with the