卡尔·西格尔Carl Ludwig Siegel

卡尔·西格尔 的父亲在邮局工作。西格尔 于 1915 年进入柏林大学，当时正值第一次世界大战期间，并听了 费迪南德·格奥尔格·弗罗贝尼乌斯 和 马克斯·普朗克 的讲座。西格尔 写道 [15]：-

通过亲自教授[初学者课程]，教授们可以在仅仅几节课后，从学生提交的作业中看出哪些学生更有天赋，并据此指导他们的工作。这就是我自己第一次接触我的老师 费迪南德·格奥尔格·弗罗贝尼乌斯 和 马克斯·普朗克 的方式……

起初他的意图是学习天文学，但 费迪南德·格奥尔格·弗罗贝尼乌斯 的影响使他转向 数论，这将成为他职业生涯的主要研究课题。然而在 1917 年，他不得不中断学业，被征召服兵役。军旅生活肯定不适合 西格尔，他最终作为军队的失败者之一被退伍，因为尽管他们尽了最大努力，仍未能让他适应军队生活。人们不得不相信 西格尔 会把这视为成功而非失败。

战争结束后，西格尔 于 1919 年开始在哥廷根继续他的学业。他在哥廷根的博士学位论文由 埃德蒙·朗道 指导，西格尔 随后继续攻读他的 教授资格论文（Habilitation）。他的学位论文写于 1920 年，[1]：-

……是丢番图逼近史上的一个里程碑。

它扩展了约瑟夫·刘维尔首先注意到的一个想法，然后由阿克塞尔·图厄推进，他证明了给定一个有理数$r$和任意$e > 0$，只有有限多个有理数$\large\frac{p}{q}\normalsize$（以其最简形式）使得

$|r - \large\frac{p}{q}\normalsize |$ ≤ $\large\frac {1}{q^2+1+e}$。

西格尔改进了这一点，他证明了只有有限多个有理数$\large\frac{p}{q}\normalsize$使得如果$r$是一个次数为$n$的代数数

$|r - \large\frac{p}{q}\normalsize | ≤ \large\frac{1}{q^{m}\normalsize}$，其中$m = 2√n$。

阿图尔·舍恩弗利斯于1914年被任命为法兰克福约翰-沃尔夫冈-歌德大学的教授，那一年这所新大学成立。他被任命时61岁，1922年退休时，西格尔被任命为教授以接替他在法兰克福的职位。尽管阿图尔·舍恩弗利斯在退休后的六年间住在法兰克福，但当西格尔接任教授职位时，他作为活跃数学家的日子已经结束。然而，法兰克福的教职员中有几位年轻数学家，他们与西格尔一起创建了一个优秀的数学中心。

恩斯特·黑林格和阿图尔·舍恩弗利斯一样，在1914年新成立的法兰克福大学开办时被聘为教授，而Szász在同一年被聘为Privatdozent。Szász于1921年晋升为教授，保罗·爱泼斯坦于1919年被聘用，马克斯·登于1921年被聘用。这是一个实力雄厚、令人兴奋的系，西格尔于1922年加入。

有若干活动是这四位数学家西格尔、恩斯特·黑林格、保罗·爱泼斯坦和马克斯·登合作进行的。其中之一是马克斯·登于1922年发起的数学史讨论班。西格尔在[15]中写道：-

如今回想起来，研讨班中那些共处的时光是我一生中最快乐的记忆之一。即便在当时，我也很享受这项活动，它让我们每周四下午四点到六点聚在一起。后来，当我们分散到世界各地，我在别处的种种幻灭经历中体会到，能拥有不存个人野心、无私合作的学术同事，而不是只从高位发号施令，是多么难得的幸运。

数学史研讨班持续了十三年。他们定下一条规则：要研读所有数学著作的原文，尽管这减少了参与研讨班的学生人数，但从未少于六人。他们研读了包括欧几里得、阿基米德、斐波那契、吉罗拉莫·卡尔达诺、西蒙·斯蒂文、弗朗索瓦·韦达、约翰内斯·开普勒、吉拉尔·笛沙格、勒内·笛卡儿、皮埃尔·德·费马、克里斯蒂安·惠更斯、伊萨克·巴罗和詹姆斯·格雷果里在内的数学家的著作。研讨班的宗旨是[5]：-

……增进参与学生对于讲座中所呈现结果的理解，并为教师提供细致考察往昔杰出著作的审美满足。

数学史研讨班并不是西格尔在法兰克福参与的唯一个研讨班，因为教授们还组织了一个初级研讨班和一个研讨班。在西格尔被任命后，学生人数迅速增加。起初他只教几个学生，并且[15]：-

……我记得在一门高级课程中只有两个学生。有一天他们俩都迟到了，因为被大学财务主管耽搁了。当他们到达时，他们震惊地发现我没等他们就开始讲课，而且已经写满了整整一黑板。

到1928年，西格尔在微分与积分微积分课程中教143名学生，并且不得不投入许多时间批改学生习题。正是在这个时候，学生人数达到最高点，随后又开始下降。

1933年1月30日希特勒上台，1933年4月7日的《公务员法》为将犹太教师从大学中清除提供了手段。这并未影响西格尔，他是雅利安人（用西格尔所憎恶的当时术语来说），而且在这个阶段，这并未影响保罗·爱泼斯坦、恩斯特·黑林格或马克斯·登，他们虽然是犹太人，但符合一项豁免条款，该条款豁免了在第一次世界大战中为德国作战的非雅利安人。然而，Szász被解除了职务。尽管西格尔未受《公务员法》影响，但他憎恨纳粹政权，这对他而言是一段非常不快乐时光的开始。

1935年，西格尔在美国普林斯顿的高等爱德华·斯图迪研究所度过了一年。他回到法兰克福后发现，他的犹太同事们的处境已变得糟糕得多。1935年秋纽伦堡党代表大会做出决定后，保罗·爱泼斯坦、恩斯特·黑林格和马克斯·登被迫离职。他们留在法兰克福，无法授课。1937年末，西格尔接受了哥廷根的教授职位，并于1938年初迁往那里。在哥廷根，他[15]：-

……过着一种颇为退隐的生活。

哥廷根的生活仍受纳粹政策的影响，数学家们对政治压力作出了不同的反应。例如，哥廷根的哈塞想接受其助手的任教资格论文，但西格尔和古斯塔夫·赫格洛茨认为这是哈塞的政治决定而非数学决定，并阻止了该任教资格的通过。

1939年，纳粹政权将德国拖入战争，西格尔感到自己再也无法留在故土。1940年初，他离开德国，先后在丹麦和挪威讲学。1940年3月，他在挪威与马克斯·登会面。马克斯·登因担心性命安危而逃离德国，当西格尔拜访他时，他正在特隆赫姆教书。西格尔在港口看到了德国商船，直到后来离开挪威前往美国后，他才发现所见到的那些船只正是德国入侵部队的先遣队。

西格尔将他旅居美国的时光描述为[15]：-

……自我放逐于美国。

他于1940年至1951年在普林斯顿的高等斯图迪研究所工作，并于1946年被任命为那里的终身教授。然而，1951年他回到德国，并在哥廷根度过了余下的职业生涯。

论文[8]在七个标题下列出了西格尔对数学的杰出贡献。它们是：

1. 用有理数逼近代数数及其在丢番图方程中的应用。
2. 超越性问题，特别是某些函数在代数点处的值。
3. ζ函数，包括在类数上的应用。
4. 数的几何及其在代数数论中的应用。
5. 戈弗雷·哈罗德·哈代-利特尔伍德方法，包括代数数的爱德华·华林型问题。
6. 二次型：分析理论与模形式。
7. 天体力学。

西格尔尤以在数论方面的工作而闻名，他在这一领域占有显著地位。西奥多·施奈德是西格尔的学生，他于1982年在德国数学联合会作了三次讲座，介绍西格尔对数论的贡献。这些讲座收录于[13]中，描述了西格尔在数论中最重要的成果。这些成果包括他对上文所述阿克塞尔·图厄定理的改进，该改进见于他1920年的学位论文，以及该定理对某些二元多项式丢番图方程的应用，证明了数域上亏格至少为1的仿射曲线只有有限个整点，这发表于1929年。也许他1929年发表的两部分论文就是[1]：-

……他最深刻、最具原创性的。

在1929年的论文中，西格尔对超越理论做出了重大贡献，特别是提出了一种新方法，用于证明某些$E$-函数值的代数独立性。他证明了，如果$J^{0}$是下标为0的弗里德里希·威廉·贝塞尔函数，那么对于任何非零代数整数$r$，他证明了$J^{0}(r)$是超越数。

在此之前，他于1922年撰写了关于理查德·戴德金的代数数域的zeta函数的函数方程的论文，并在1921/23年对加性问题做出了贡献，例如代数数域的爱德华·华林型问题。他在1944年对后一主题做出了进一步贡献。西格尔在1935/37年关于二次型解析理论的研究具有根本重要性，他在考虑系数来自代数数域的二次型方面开辟了新天地。

Klingen在[9]中讨论了西格尔对复分析的贡献。他特别研究了多复变自守函数、西格尔的模函数，这些导致了更深刻的理解。在这一广泛领域中，西格尔考虑了不连续群及其基本域的理论、模函数之间以及模形式之间的代数关系，以及模形式的Fourier series。

西格尔在天体力学方面的工作，在他最喜欢的话题列表中仅次于数论，Rüssmann在[12]中对此进行了讨论。该论文列出了西格尔对该学科做出的八项主要贡献。他研究了：

1. $n$体问题以及海因里希·布伦斯关于代数积分的定理。
2. 限制性三体问题及其积分，其中使用了西格尔在(i)中证明的结果。
3. 月球的轨道，这同样本质上是一个三体问题。西格尔对乔治·威廉·希尔所发展的月球理论给出了大为改进的版本。
4. 三体问题的拉格朗日解。西格尔发展了确定平衡点附近周期轨道的一般方法。
5. 小除数问题，其中西格尔首次得到了收敛性结果。
6. 乔治·戴维·伯克霍夫正规形式。他考察了乔治·戴维·伯克霍夫关于利用形式幂级数在平衡点附近求解解析哈密顿微分方程的微扰理论解的工作。西格尔给出了不具有收敛到正规形式的变换的系统的例子。
7. 对稳定性理论的贡献。

一个有趣的插曲发生在20世纪60年代，它让我们对西格尔的数学研究方法有了很多了解。塞尔日·兰在1962年发表了Diophantine geometry，两年后路易斯·乔尔·莫德尔写了一篇批评性的评论。随后西格尔写信给路易斯·乔尔·莫德尔[11]：-

大约一年前，当我第一次看到[Lang的丢番图几何]时，我厌恶地发现我自己对该主题的贡献被歪曲得面目全非，变得难以理解。当你提到瑞普·凡·温克尔时，我的感受得到了很好的表达！

作者的整体风格与我们在数论大师——约瑟夫·拉格朗日、卡尔·弗里德里希·高斯，或在较小规模上，戈弗雷·哈罗德·哈代、埃德蒙·朗道——的作品中所钦佩的简洁和诚实感相矛盾。刚才Lang又出版了一本关于代数数的书，在我看来，比前一本更糟。我看到一头猪闯进美丽的花园，把所有的花和树都拱了出来。

不幸的是，有许多“同路人”已经败坏了代数和函数论的很大一部分；然而，直到现在，数论尚未被触及。这些人让我想起国家社会主义者的无耻行为，他们唱道：“Wir werden weiter marschieren, bis alles in Scherben zerfällt!'' Ⓣ（我们将继续前进，直到一切化为碎片！）

我担心，如果当前这种无意义抽象的趋势——我称之为空集理论——不能被阻止，数学将在本世纪末之前消亡。...

让·迪厄多内在[1]中写道：-

西格尔终身未婚，将一生献给了研究。

但让·迪厄多内解释了他为什么认为西格尔几乎没有博士生：-

……他论文的完美和彻底性没有给使用相同技术进行改进留下多少空间，[这]使许多研究生望而却步，因为要做得比他更好需要新方法。然而，西格尔喜欢教学，甚至基础课程，他出版了关于数论、天体力学和多复变函数论的教科书。

他获得了许多荣誉，其中或许最负盛名的是1978年的沃尔夫奖。

Carl Siegel's father worked for the post office. Siegel entered the University of Berlin in 1915, in the midst of World War I, and attended lectures by Frobenius and Planck. Siegel wrote [15]:-

By conducting [beginners' classes] personally the professors could see, after only a few lectures, which of the students were the more gifted by the work they handed in, and the professors could direct their work accordingly. This was the way I myself first came into contact with my teachers Frobenius and Planck ...

Initially his intention had been to study astronomy, but Frobenius's influence took him towards number theory which would became the main research topic of his career. In 1917, however, he had to interrupt his studies when he was called for military service. Most certainly military life did not suit Siegel and he was eventually discharged from the army as one of their failures, for despite their best efforts they had failed to have him adapt to army life. One would have to believe that Siegel would have classed this as a success rather than a failure.

After the war had ended, Siegel continued his studies at Göttingen, beginning in 1919. His doctoral dissertation at Göttingen was supervised by Edmund Landau and Siegel then continued to study for his habilitation. His dissertation, written in 1920, [1]:-

... was a landmark in the history of Diophantine approximations.

It extended an idea first noted by Liouville, then pushed forward by Thue who proved that, given a rational number $r$ and any $e > 0$ there are only finitely many rational numbers $\large\frac{p}{q}\normalsize$ (in their lowest terms) such that

$|r - \large\frac{p}{q}\normalsize |$ ≤ $\large\frac {1}{q^2+1+e}$.

Siegel improved this by showing that there are only finitely many rational numbers $\large\frac{p}{q}\normalsize$ such that if $r$ is an algebraic number of degree $n$

$|r - \large\frac{p}{q}\normalsize | ≤ \large\frac{1}{q^{m}\normalsize}$, where $m = 2√n$.

Schönflies had been appointed as professor at the Johann-Wolfgang-Goethe-University of Frankfurt in 1914, the year in which the new university opened. He was aged 61 when he was appointed and when he retired in 1922 Siegel was appointed as professor to succeed him at Frankfurt. Although Schönflies spent the six years of his retirement in Frankfurt, his days as an active mathematician were over by the time Siegel took up the professorship. There were, however, several young mathematicians on the staff at Frankfurt who would with Siegel create an excellent centre for mathematics.

Hellinger, like Schönflies, had been appointed as a professor to the new university of Frankfurt when it opened in 1914, and Szász had been appointed as a Privatdozent in the same year. Szász was promoted to professor in 1921, Epstein was appointed in 1919, and Dehn in 1921. It was a strong and exciting department which Siegel joined in 1922.

There were a number of activities on which the four mathematicians Siegel, Hellinger, Epstein, and Dehn collaborated. One was the history of mathematics seminar instigated by Dehn in 1922. Siegel wrote in [15]:-

As I look back now, those communal hours in the seminar are some of the happiest memories of my life. Even then I enjoyed the activity which brought us together each Thursday afternoon from four to six. And later, when we had been scattered over the globe, I learned through disillusioning experiences elsewhere what rare good fortune it is to have academic colleagues working unselfishly together without thought to personal ambition, instead of just issuing directives from their lofty positions.

The history of mathematics seminar was to last for thirteen years. They made a rule that they would study all the mathematical works in their original languages and although this reduced the number of students who participated in the seminar, there was never less than six. They studied the works of mathematicians including Euclid, Archimedes, Fibonacci, Cardan, Stevin, Viète, Kepler, Desargues, Descartes, Fermat, Huygens, Barrow, and Gregory. The aim of the seminar was [5]:-

... to increase the understanding of the participating students for the results presented in lectures and to provide the teachers with aesthetic satisfaction of examining the outstanding works of past times in close detail.

The history of mathematics seminar was not the only one which Siegel participated in at Frankfurt, for the professors organised also a proseminar and a seminar. Student numbers rapidly built up after Siegel was appointed. At first he taught only a few students and [15]:-

... I remember having only two in one of the advanced courses. One day they were both late for class, having been delayed at the university bursar. When they arrived, they were shocked to find I had begun without them and had already filled a whole section of the blackboard.

By 1928 Siegel was teaching 143 students in the differential and integral calculus course, and had to put in many hours work correcting students exercises. It was at this time that the student numbers reached a maximum, then they began to drop again.

On 30 January 1933 Hitler came to power and on 7 April 1933 the Civil Service Law provided the means of removing Jewish teachers from the universities. This did not affect Siegel who was an Aryan (to use the terminology of the time which Siegel hated) and, at this stage it did not affect Epstein, Hellinger or Dehn who, although Jewish, fell under a clause which exempted non-Aryans who had fought for Germany in World War I. Szász, however, was dismissed from his post. Although Siegel was not affected by the Civil Service Law, he hated the Nazi regime and this was the beginning of a very unhappy time for him.

In 1935 Siegel spent a year at the Institute for Advanced Study at Princeton in the United States. He returned to Frankfurt to find that the problems of his Jewish colleagues had become much worse. After decisions at the Nuremberg party congress in the autumn of 1935, Epstein, Hellinger and Dehn were forced from their posts. They remained in Frankfurt, unable to teach. In late 1937 Siegel accepted a professorship at Göttingen and he moved there in early 1938. At Göttingen he [15]:-

... led a somewhat retiring life.

Life in Göttingen was still influenced by the Nazi policies and mathematicians reacted in different ways to the political pressures. For example Hasse in Göttingen wanted to accept the habilitation thesis of his assistant, but Siegel and Herglotz felt that this was a political rather than mathematical decision by Hasse and stopped the habilitation being accepted.

The Nazi regime had taken Germany to war in 1939 and Siegel felt that he could no longer remain in his native land. In early 1940 he left Germany, lecturing first in Denmark and then in Norway. In March 1940 he met up with Dehn in Norway. Dehn had fled from Germany in fear of his life and was teaching in Trondheim when Siegel visited him. Siegel saw German merchant ships in the harbour and only later, having left Norway for the United States, did he discover that the ships he had seen were the advanced party of the German invasion force.

Siegel described his time in the United States as [15]:-

... self imposed exile in America.

He worked at the Institute for Advanced Study at Princeton from 1940 until 1951, being appointed to a permanent professorship there in 1946. However, in 1951 he returned to Germany and again worked at Göttingen for the rest of his career.

The paper [8] lists Siegel's impressive contributions to mathematics under seven headings. These are:

1. Approximation of algebraic numbers by rationals and applications thereof to Diophantine equations.
2. Transcendence questions, in particular values of certain functions at algebraic points.
3. Zeta functions including applications to class numbers.
4. Geometry of numbers and its applications to algebraic number theory.
5. Hardy-Littlewood method, including Waring-type problems for algebraic numbers.
6. Quadratic forms: analytic theory and modular forms.
7. Celestial mechanics.

Siegel is especially famed for his work on the theory of numbers where he held an eminent role. Schneider, who was a student of Siegel's, gave three lectures on Siegel's contributions to number theory to the German Mathematical Union in 1982. These are reproduced in [13] and describe Siegel's most important results in number theory. These include his improvement of Thue's theorem, described above, given in his 1920 dissertation, and its application to certain polynomial Diophantine equations in two unknowns, proving an affine curve of genus at least 1 over a number field has only a finite number of integral points in 1929. Perhaps his two part paper which appeared in 1929 is [1]:-

.. his deepest and most original.

In the 1929 paper Siegel made a substantial contribution to transcendence theory, especially a new method for the algebraic independence of values of certain $E$-functions. He proved that if $J^{0}$ is the Bessel function of index 0, then for any non-zero algebraic integer $r$ he showed that $J^{0}(r)$ is transcendental.

He had earlier than this in 1922, written papers on the functional equation of Dedekind's zeta functions of algebraic number fields and in 1921/23 made contributions to additive questions such as Waring type problems for algebraic number fields. He made further contributions to this latter topic in 1944. Siegel's research on the analytic theory of quadratic forms in 1935/37 was of fundamental importance and he broke new ground in considering quadratic forms in which the coefficients were from an algebraic number field.

Klingen, in [9], discusses Siegel's contributions to complex analysis. In particular he studied automorphic functions in several complex variables, Siegel's modular functions, which have led to a much deeper understanding. In this general area Siegel considered the theory of discontinuous groups and their fundamental domains, algebraic relations between modular functions and between modular forms, and Fourier series of modular forms.

Siegel's work in celestial mechanics, which came next to number theory in his list of favourite topics, is discussed by Rüssmann in [12]. The paper lists eight major contributions which Siegel made to the subject. He studied:

1. the $n$-body problem and the theorem of Bruns on algebraic integrals.
2. the restricted problem of three bodies and their integrals, which used the results Siegel had proved in (i).
3. the orbit of the moon, again essentially a three-body problem. Siegel gave a much improved version of lunar theory as developed by Hill.
4. the Lagrangian solutions for the three-body problem. Siegel developed general methods to determine periodic orbits near the equilibrium points.
5. the problem of small divisors, where Siegel first obtained convergence results.
6. Birkhoff normal forms. He examined Birkhoff's work on perturbation theory solutions for analytical Hamiltonian differential equations near an equilibrium point using formal power series. Siegel gave examples of systems which did not possess convergent transformations into a normal form.
7. contributions to stability theory.

An interesting episode, which tells us a lot about Siegel's approach to mathematics, occurred in the 1960s. Serge Lang published Diophantine geometry in 1962 and Mordell wrote a critical review of it two years later. Siegel then wrote to Mordell [11]:-

When I first saw [Lang's Diophantine geometry], about a year ago, I was disgusted with the way in which my own contributions to the subject had been disfigured and made unintelligible. My feeling is very well expressed when you mention Rip van Winkle!

The whole style of the author contradicts the sense for simplicity and honesty which we admire in the works of the masters in number theory - Lagrange, Gauss, or on a smaller scale, Hardy, Landau. Just now Lang has published another book on algebraic numbers which, in my opinion, is still worse than the former one. I see a pig broken into a beautiful garden and rooting up all flowers and trees.

Unfortunately there are many "fellow-travellers" who have already disgraced a large part of algebra and function theory; however, until now, number theory had not been touched. These people remind me of the impudent behaviour of the national socialists who sang: "Wir werden weiter marschieren, bis alles in Scherben zerfällt!'' Ⓣ

I am afraid that mathematics will perish before the end of this century if the present trend for senseless abstraction - as I call it: theory of the empty set - cannot be blocked up. ...

Dieudonné, writes in [1]:-

Siegel, who never married, devoted his life to research.

But Dieudonné explains why he believes that Siegel had few doctoral students:-

... the perfection and thoroughness of his papers did not leave much room for improvement with the same technique, [and this] discouraged many research students because to do better than he required new methods. Siegel enjoyed teaching, however, even elementary courses, and he published textbooks on the theory of numbers, celestial mechanics, and the theory of functions of several complex variables.

He was awarded many honours, perhaps the most prestigious of which was the Wolf Prize in 1978.