**Max Deuring**'s parents were liberally minded people as is evident from the fact that they were subscribers to the liberal Göttingen newspaper until the Nazi party came to power in 1933 and closed the paper down. Max was brought up in Göttingen completing his school education at the Oberrealschule there. In 1926 he entered the University of Göttingen where he studied mathematics and physics. Apart from the winter semester of 1928/29 which he spent in Rome, his university studies were completely undertaken at Göttingen. His interests were in number theory, and at the this time Hilbert's ideas dominated the area but new ideas were flourishing, particularly driven by Emmy Noether's new algebraic approach.

Deuring was captivated by the new approach to algebra which Emmy Noether was developing at Göttingen and, from 1927 on, she advised him on the research for his doctorate. He submitted his doctoral dissertation *Arithmetische Theorie der algebraischen Funktionen* in 1930, the degree being conferred in the following year.One of the great strengths shown by Deuring at this stage, and throughout his career, was an ability to make simplifications and generalisations to existing work. For example in his first paper *Verzweigungstheorie bewerteter Körper* (1931) he generalised Hilbert's theory of prime divisors in Galois extensions to more general abstract fields. Again his ability to simplify proofs and gain better understanding with his knowledge of class field theory is shown in his second paper published in 1931 *Zur Theorie der Normen Relativzyklischer Körper.*

Bartel van der Waerden had also undertaken research at Göttingen. He was four years older than Deuring but the two had become friends, both being part of the circle of young mathematicians around Emmy Noether. After short spells at the University of Rostock and the University of Groningen, van der Waerden had been appointed professor of mathematics at the University of Leipzig in 1931 and Deuring was appointed as his assistant. By 1935, with an excellent publication record behind him, including his famous book *Algebren* published in that year, Deuring wanted to qualify as a lecturer and submit his habilitation thesis. He would have liked to become a lecturer at Leipzig but he was aware that this route was blocked for political motives. Helmut Hasse was by this time director of the Mathematical Institute at Göttingen and he encouraged Deuring to habilitate there. However, Erhard Tornier, a staunch Nazi supporter, although accepting the quality of Deuring's work, nevertheless found him politically unacceptable. Tornier was supported by militant Nazi sympathisers among the students, and Deuring was refused the right to lecture by the Ministry. He remained at Leipzig but was not qualified to lecture.

Perhaps Deuring's greatest mathematical idea came in 1936. We will look below at these impressive contributions. However before that we will follow the rest of Deuring's career. In 1937 he was allowed to proceed with his career at the University of Jena. After six years in Jena, he was appointed as an extraordinary professor at the University of Posen in 1943. This was not the most attractive of appointments for Deuring since the University of Posen had been set up by the Nazis in Poznán, Poland. The university had opened in February 1940 in the territory annexed from Poland at the start of World War II, only offering economics and political science when it first opened. Deuring remained in Poznán until the end of the war when he returned to his home town of Göttingen. In 1947 he was appointed to Marburg then, in the following year, to the University of Hamburg. Herglotz had held a chair at Göttingen until he was forced to resign due to ill health in 1946. The chair remained unfilled until 1950 when Deuring was appointed. He continued to hold this chair until 1976 when he retired and was made professor emeritus.

Hasse proved in 1932 an analogue of the Riemann hypothesis for the zeta function associated with an elliptic curve over a finite field. Deuring's 'greatest mathematical idea' which we referred to above was an idea on how to generalise Hasse's result but to do this he had to introduce a radically new approach. Deuring was in Leipzig in 1936, having failed to be accepted at Göttingen in the previous year as we explained above. He wrote to Hasse on 9 May (see for example [6]):-

In the last few weeks, I have tried to generalize your results for elliptic function fields to fields of higher genus. I have succeeded in doing so, all the way to the construction of the ring of multipliers and the proof that it is algebraic. Since you may already be further ahead in these questions, I enclose the introduction to a projected paper. There the algebraic results are only stated. I have complete proofs of them; but they are still monstrous.

What Deuring had found was a way to approach extending Hasse's results from curves of genus 1, the elliptic curves, to curves of higher genus. It involved his algebraic theory of correspondences. Hasse was impressed and replied to Deuring [6]:-

... At any rate, I am sure that you have laid the ground for coming to terms with the Riemann hypothesis in arbitrary function fields. I am convinced that I will be able to give a proof of the Riemann hypothesis by linking my own approaches, which I have thought about these past weeks, with your results. I will think this over as soon as possible, also with a view to smoothing out your proofs.

In fact it was neither Hasse nor Deuring who finally proved the result which both were trying to obtain. However Deuring presented his new ideas on the algebraic theory of correspondences in two papers, both of which were published by Crelle's journal. The first part of *Arithmetische Theorie der Korrespondenzen algebraischer Funktionenkörper* appeared in 1937, the second part in 1940. The generalisation of the Riemann hypothesis to function fields of arbitrary genus was first achieved by André Weil in 1948, but he did so using Deuring's theory of correspondences. The result was first published in Weil's book *Sur les courbes algébriques et les variétés qui s'en déduisent.*

In 1937 Deuring wrote the following in the introduction to *On Epstein's zeta function* which shows how deeply he was attacking problems of zeta functions:-

It is the purpose of this note to present some formulae which connect the zeta functions of quadratic forms or Epstein's zeta-functions to the Riemann zeta-function and functions related to it. Although these formulae did not yield any new results concerning deeper questions on zeta-functions, I nevertheless believe them of sufficient interest and hope that they will be of use in further investigations.

Deuring continued to produce important papers. In 1941 in *Die Typen der Multiplikatorenringe elliptischer Funktionenkörper* he classified endomorphism rings of elliptic function fields having laid some of the groundwork in his paper *Invarianten und Normalformen elliptischer Funktionenkörper.* A paper of 1942 is described by Roquette (who was a student of Deuring during his time in Hamburg) in [4]:-

On several occasions in the work of Hasse and Deuring, there appeared a situation which today we would call "good reduction" of algebraic function fields or curves. In his paper[Reduktion algebraischer Funktionenkörper nach Primdivisoren des Konstantenkörpers]Deuring developed a coherent general theory of good reduction which covered all special cases which were encountered so far. Although this paper appeared one year later than the1941paper on endomorphism rings, it was completed earlier, and Deuring used it in an essential way in his study on endomorphism rings.

During his years at Göttingen, Deuring made many trips abroad but had two longer visits, one to the Institute for Advanced Study at Princeton, the other at the Tata Institute of Fundamental Research in Bombay, India, in 1958. On this second visit he gave a series of lectures which were published as *Lectures on the theory of algebraic functions of one variable* in 1973. As a reviewer states:-

One expects the best from this author and he gives no less in these lectures.

We should note that the paper [2] contains a list of 25 doctoral students advised by Deuring.

He was honoured with election to a number of academies such as the Academy of Sciences and Literature in Mainz, the Göttingen Academy of Sciences, and the German Academy of Natural Scientists Leopoldina.

**Article by:** *J J O'Connor* and *E F Robertson*