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, :-
... 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 q and any e > 0 there are only finitely many rational numbers p/q (in their lowest terms) such that
|q - p/q| ≤ 1/(q2 +1+e).Siegel improved this by showing that there are only finitely many rational numbers p/q such that if q is an algebraic number of degree n
|q - p/q| ≤ 1/qm, 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 :-
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 :-
... 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 :-
... 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 :-
... 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 :-
... 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  lists Siegel's impressive contributions to mathematics under seven headings. These are:
- Approximation of algebraic numbers by rationals and applications thereof to Diophantine equations.
- Transcendence questions, in particular values of certain functions at algebraic points.
- Zeta functions including applications to class numbers.
- Geometry of numbers and its applications to algebraic number theory.
- Hardy-Littlewood method, including Waring-type problems for algebraic numbers.
- Quadratic forms: analytic theory and modular forms.
- 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  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 :-
.. 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 J0 is the Bessel function of index 0, then for any non-zero algebraic integer r he showed that J0(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 , 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 . The paper lists eight major contributions which Siegel made to the subject. He studied:
- the n-body problem and the theorem of Bruns on algebraic integrals.
- the restricted problem of three bodies and their integrals, which used the results Siegel had proved in (i).
- the orbit of the moon, again essentially a three-body problem. Siegel gave a much improved version of lunar theory as developed by Hill.
- the Lagrangian solutions for the three-body problem. Siegel developed general methods to determine periodic orbits near the equilibrium points.
- the problem of small divisors, where Siegel first obtained convergence results.
- 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.
- 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 :-
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!Dieudonné, writes in :-
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. ...
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.
Article by: J J O'Connor and E F Robertson
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