# Carl Ludwig Siegel

Carl Ludwig Siegel in Göttingen , 1975

Carl Ludwig Siegel (born December 31, 1896 in Berlin , † April 4, 1981 in Göttingen ) was a German mathematician ; his specialty was number theory . He is considered one of the most important mathematicians of the 20th century.

## Life

Celebration of Siegel's doctorate , June 1920 in Göttingen: Siegel in the wagon, and from left to right Grandjot, Bessel-Hagen , Rogosinski , Ness, Windau, Walfisz , Krull , Emersleben , Kopfermann , Hedwig Wolff, Boskowits and Hellmuth Kneser .

Siegel was the son of a postman. From 1915 he studied astronomy , physics and mathematics in Berlin , among others with Ferdinand Georg Frobenius and Max Planck . Under the influence of Frobenius, he specialized in number theory . In 1917 he was called up. Because he refused to do military service, he was sent to a mental hospital. In his own words, he only survived the time because Edmund Landau , whose father had a clinic in the neighborhood, supported him. He continued his studies in Göttingen in 1919, this time sponsored by Richard Courant , and received his doctorate under Landau in 1920 with the thesis on the approximation of irrational numbers, which Thue's result had already found in Berlin as a fourth semester. As early as 1922 he became a professor in Frankfurt as the successor to Arthur Moritz Schoenflies . Siegel, who was deeply repugnant to National Socialism, made friends with the Jewish lecturers Ernst Hellinger and Max Dehn and stood up for them. This attitude made Siegel's appointment as successor to the chair of Constantin Carathéodory in Munich impossible.

In Frankfurt he took part with Dehn, Hellinger, Paul Epstein and others in a seminar on the history of mathematics, which was carried out at the highest level (the originals were always read). Siegel later saved this period from being forgotten in an essay. In the 1930s he tried in vain with the National Socialist government to keep his Jewish colleagues Landau, Dehn, Hellinger and Courant at the chairs. After spending a while at the Institute for Advanced Study in Princeton, New Jersey , in the mid-1930s , he decided, against the advice of his colleagues, to return to Germany. One motive was that he had difficulties adapting to American living conditions and found the atmosphere in Princeton to be prudish (he lived with a friend without being married). Another motive was that he wanted to help his Jewish colleagues Dehn and Hellinger in Frankfurt (he even wanted to reverse the replacement of Hellinger by the National Socialist Werner Weber) and also threatened to withdraw his pension there because of his absence.

Grave site in Göttingen

In 1938 Siegel returned to Göttingen as a professor, but decided in 1940 not to return to Germany after visiting Denmark and Norway . Shortly before the German occupation of Norway, he fled to the USA on a steamer . The emigration was made easier for him by the fact that he had no family, even if he left a close friend in Göttingen, the mathematician Hel Braun ; he remained unmarried throughout his life.

Siegel taught and worked from 1940 to 1951 at the Institute for Advanced Study in Princeton, where he was already in 1935. He received a permanent professorship there in 1946 and became a US citizen. In 1951 he returned to Göttingen, where he retired in 1959 (after which he held lectures for several years) and stayed until the end of his life. He gave four lectures at the Tata Institute of Fundamental Research in Bombay . He had been a corresponding member of the Göttingen Academy of Sciences since 1949 and a full member since 1951 . In 1958 he was elected a member of the Leopoldina and a corresponding member of the Bavarian Academy of Sciences .

His doctoral students include Helmut Klingen , Theodor Schneider , Kurt Mahler (as co-referee), Hel Braun , Helmut Rüßmann , Günter Meinardus , Christian Pommerenke , Jürgen Moser , Erhard Scheibe (in the last two cases also as a co-referee).

## plant

### Number theory

In his dissertation in 1920 , Siegel significantly improved Thue's estimate of the approximation of algebraic numbers by rational numbers, a result that he had already found as a third-semester student. It was tightened again (as best as possible) in 1955 by Klaus Friedrich Roth , who received the Fields Medal for it ( theorem of Thue-Siegel-Roth ). Siegel then applied his result in 1929 to achieve his most famous result, the proof that algebraic equations in integers only have a finite number of solutions as soon as the gender is g ≥ 1. Quadratic equations (gender zero, corresponding to sphere) naturally have an infinite number of solutions, e.g. B. Pythagorean triples. The sentence corresponding to Siegel's theorem for rational numbers is called Mordell's conjecture or, according to Faltings 'proof, “ Faltings ' theorem”.

Siegel considerably expanded the theory about transcendent numbers , which had been very weak up to that point, and developed appropriate decision criteria for when a number is transcendent, i.e. not the solution of an algebraic equation. Siegel introduced new methods, first for the proof of special values ​​of the solutions of differential equations of the second order, such as the Bessel functions . Gelfond and Schneider (who did his doctorate at Siegel and was his assistant) led a. a. with these methods later evidence of transcendence that solved one of Hilbert's problems (see Gelfond-Schneider theorem ).

He also researched the geometry of numbers (in Minkowski's sense), the theory of the zeta function (he found new results from Bernhard Riemann in his estate and expanded them), proved the functional equation for the Dedekind zeta function in algebraic number fields, worked on quadratic forms and found further rules for estimating solutions of Diophantine equations. In additive number theory, he investigated problems of the Waring type (maximum number of kth powers, which are necessary to represent any natural numbers as the sum of these kth powers) using analytical methods.

In his analytical theory of quadratic forms in several variables he proved his famous analytical class number formula for the number of representations of one form by another: on one side there is a kind of theta function , with the trace of the matrices in the exponent and summation over class representatives; on the other side of the equation there is an Eisenstein series , i.e. a modular form , again adding up over class representatives. These analytical structures simultaneously provide two ways of introducing Siegel's modular functions, which were sensational at the time around 1935, since little was known about function theory in several variables.

Siegel also found a result with Richard Brauer about the asymptotic behavior of the class numbers of algebraic number fields. Together with Hans Heilbronn , he proved that the class numbers of imaginary square number fields (defined by the adjunction of the root of (-n) to the rational numbers) diverge for large n , which was already suggested by Carl Friedrich Gauß . Together with Harold Stark and Max Deuring , he also saved the proof of the private scholar Kurt Heegner (1952) for the "class number 1" problem of imaginary quadratic number fields from Gauss (that is, that there were no other such number fields besides the then already known nine) for he used properties of modular functions. The reason for this was the new evidence from Harold Stark in the 1960s, which led to the reconsideration of the difficult-to-understand evidence from Heegner that was doubted at the time.

For him and Arnold Walfisz is set of Seal Walfisz named.

### Function theory

Siegel examined automorphic functions of several variables first as an aid for number theoretic problems, his analytical theory of quadratic forms 1935/7 in several variables. This led to the development of the theory of Siegel modular forms (analogues of the modular forms in the Siegel half-space ), which soon became the subject of research. He also examined the underlying discontinuous groups and their fundamental domains that generalize the theory of modular function and its modular group by Robert Fricke and Felix Klein . He also found new relationships between these functions and studied their Fourier coefficients (e.g. from Eisenstein series). In connection with the theory of his modular forms, Siegel speaks in some works of "symplectic geometry", a term that is used differently today.

### Differential equations and celestial mechanics

Here Siegel was primarily interested in questions related to celestial mechanics, in particular the three-body problem or more generally the n-body problem, questions of the regularization of the singular equations of motion (collisions), the existence of algebraic integrals of the equations of motion (continuing work by Ernst Heinrich Bruns ), the lunar theory (based on George William Hill ), the existence of quasi-regular orbits and their stability (in simpler analytical dynamic systems, Siegel discs ), questions of convergence of the perturbation function ("problem of the small denominators"), as well as the normal forms of Hamiltonian equations of motion near equilibrium points (on George David Birkhoff building). His book on celestial mechanics, written with Jürgen Moser , is also considered a classic and helped prepare the KAM theorem (named after Kolmogorow , Arnold and Moser ) , which is famous in this discipline .

### Siegel's position on the development of mathematics

Like hardly any other mathematician of the 20th century, Siegel was critical of the increasing abstraction and axiomatization of mathematics. In his opinion, the Bourbaki project was the culmination of a "catastrophic development". The model for him was the clarity of Gauss and Lagrange , as well as the exploration of concrete mathematical objects.

## Quotes, anecdotes

"I'm afraid that math will perish before the end of the century if the trend towards pointless abstraction - the theory of the empty set, as I call it - is not stopped."

- Carl Ludwig Siegel

He once gave the following remarkable assessment of Roger Apéry's proof of irrationality : ${\ displaystyle \ zeta (3)}$

"You can only hold the proof like a crystal in front of you"

- orally by Wilhelm Maak handed

"(A mathematician) von Hilbert format"

- Richard Courant , quoted from Constance Reid

"The collection (meaning the edition of his collected writings) stands as a monument to the genius of the author"

- Review in the mathematical annals

Siegel had a difficult character at times. For example, he literally "sank" the post-doctoral thesis of a well-known mathematician he was friends with ( Erich Bessel-Hagen ), which he was supposed to examine, on the ocean crossing to America because he was tired of reading it. Of course, he later regretted this and invited Bessel-Hagen to go on a trip to Greece as compensation.

Siegel also played the piano. At an evening entertainment he once challenged the audience in vain to identify the piece he was playing - he had played a Mozart composition backwards.

Siegel gave a lecture on celestial mechanics in Frankfurt in 1928, which he had put early in the morning to scare off listeners. He then only had four listeners, including Cornelius Lanczos , Willy Hartner and André Weil . When all four were late one day, they found that he had started the lecture without them and had already filled a blackboard.

## literature

from seal:

• Collected works , 3 volumes, Springer 1966, Volume 4, 1979
• with Jürgen Moser Lectures on Celestial mechanics , Springer 1971, or the older edition (still without Moser as a co-author) lectures on celestial mechanics , Springer, Basic Teachings of Mathematical Sciences 1956
• About some applications of Diophantine approximations , session reports of the Prussian Academy of Sciences, Math.-Phys. Class, 1929, No. 1, (his theorem on finitude solutions of integer equations)
• Lectures on quadratic forms , Tata Institute 1957
• On the reduction theory of quadratic forms , Tokyo: Publ. Math. Soc. Japan, 1959
• Advanced Analytic Number Theory , Tata Institute 1961
• Lectures on Riemann Matrices , Tata Institute 1963
• On the history of the Frankfurt Mathematical Seminar: Lecture by Carl Ludwig Siegel on June 13, 1964 in the Mathematical Seminar of the University of Frankfurt on the occasion of the fiftieth anniversary of the Johann Wolfgang Goethe University in Frankfurt , Frankfurter Universitätsreden NF 36, Frankfurt: Klostermann 1965
• English translation: On the history of the Frankfurt Mathematics Seminar , Mathematical Intelligencer Vol. 1, 1978/9, Issue 4
• Lectures on the analytical theory of quadratic forms , 3rd edition, Göttingen, Peppmüller 1963 (Lectures Institute for Advanced Study 1934/35)
• Transcendental Numbers , BI University Pocket Book 1967 (Original: Transcendental Numbers , Princeton UP 1949)
• Lectures on Function Theory , 3 volumes, Göttingen, Mathematical Institute (held from 1953 to 1955, in Volume 3 also on his Siegel module functions)
• English edition "Topics in complex function theory", 3 volumes, Wiley (Interscience Tracts in Pure and Applied Mathematics), Volume 1, 1969 (Elliptic functions and uniformization theory), Volume 2, 1971 (Automorphic Functions and Abelian Integrals), Volume 3, 1973 (Abelian Functions and Modular Functions of Several Variables)
• Lectures on the geometry of numbers , Springer 1989 (first New York University 1946)
• Lectures on the singularities of the three-body problem , Tata Institute 1967
• Letter to Louis J. Mordell, March 3, 1964.

Many of Siegel's lectures in Göttingen (e.g. on analytical number theory, quadratic forms and function theory) can be obtained from the mathematical institute there (see here ).

Commons : Carl Ludwig Siegel  - Collection of images, videos and audio files

## Individual evidence

1. Karl Grandjot (1900-1979), see short biographies at the DMV , there are also short biographies of William Ness (* 1898), Willi Windau (1889-1928) and Hedwig Wolff (* 1900).
2. ^ Dieudonne, Dictionary of Scientific Biography
3. Constance Reid David Hilbert
4. Freddy Litten: The Carathéodory successor in Munich (1938–1944)
5. Harald Bohr called Siegel's return in a letter to Courant in 1935 as incredibly foolish , unbelievably foolish . Siegmund-Schultze Mathematicians fleeing from Nazi Germany , Princeton University Press 2009, p. 160
6. In a letter to Courant in 1935, he wrote that it would be meaningless to escape Göring's sadism just to come under the yoke of Mrs. Eisenhart's view of morality . What is meant is the wife of Luther P. Eisenhart , who had strict social regulations in Princeton. Reinhard Siegmund-Schultze Mathematicians fleeing from Nazi Germany , Princeton University Press, 2009, p. 247. Among other things, Hel Braun was later his girlfriend during his second emigration ; and in a letter to Oswald Veblen in 1946 he complained bitterly that she was denied her residence permit, which he compared to Gestapo methods - shortly afterwards he apologized for it.
7. ^ Letter from Siegel to Courant April 20, 1935, Siegmund-Schultze Mathematicians fleeing under the Nazis , p. 159
8. ^ Sanford Segal Mathematicians under the Nazis , p. 67, quoted from a letter from Siegel to Veblen in which he explains his motives.
9. ^ André Weil Science Francaise? , Nouvelle Revue Francaise, January 1955, p. 102, with Professor A is meant Siegel (with B Claude Chevalley).
10. The lectures have been published: On quadratic forms 1957, On Riemann Matrices 1963, On Singularities of the three body problem 1967, Advanced Analytic Number Theory 1961
11. Holger Krahnke: The members of the Academy of Sciences in Göttingen 1751-2001 (= Treatises of the Academy of Sciences in Göttingen, Philological-Historical Class. Volume 3, Vol. 246 = Treatises of the Academy of Sciences in Göttingen, Mathematical-Physical Class. Episode 3, vol. 50). Vandenhoeck & Ruprecht, Göttingen 2001, ISBN 3-525-82516-1 , p. 226.
12. ^ Member entry by Carl Siegel at the German Academy of Sciences Leopoldina , accessed on February 15, 2016.
13. ^ Carl Ludwig Siegel obituary in the 1982 yearbook of the Bavarian Academy of Sciences (PDF file).
14. ^ A proof can be found in Serre, Lectures on the Mordell-Weil theorem, Vieweg 1998. A proof with the subspace theorem by Wolfgang Schmidt after Umberto Zannier and P. Corvaja is in Bombieri, Gubler, Heights in Diophantine Geometry, Cambridge UP 2006
15. A corresponding letter from Siegel to Louis Mordell , who sympathized with Siegel in this regard, is given by Serge Lang , a representative of the abstract direction of mathematics against whom Siegel's anger was directed, in his essay Mordell's Review ... , Notices AMS 1995. Siegel compared this direction with marching National Socialists ( These people remind me of the impudent behavior of the national socialists who sang “We'll march on until everything falls apart” ) and with pigs in a beautiful garden ( I see a pig broken into a beautiful garden and rooting up all flowers and trees. ) In 1960, as in 1956 (Scharlau, Das Glück Mathematician, Springer 2016, p. 73, at that time because Hirzebruch wavered between Göttingen and Bonn), Siegel prevented Friedrich Hirzebruch's appointment to Göttingen ( his own successor was Hans Grauert ) and as head of a forerunner planned at the time of the later Max Planck Institute for Mathematics, as he also saw in him a representative of this abstract mathematics (excerpts from the correspondence of the appointment negotiations with Lang, loc.cit.) .
16. ↑ Narrated by Hel Braun. You can find it e.g. B. Benjamin Yandell The Honors Class. Hilberts Problems and their solvers , AKPeters, 2002, among other Siegel anecdotes.
17. Memoirs of Goro Shimura , PDF file
18. Based on the report by Willy Hartner, quoted by Wolfgang Schwarz : From the history of number theory. Lecture elaboration 2000/2001