In 1920 - Hecke had gone to Hamburg - Hasse went to Marburg where he completed his studies under Kurt Hensel, whose work on padic numbers was to have a profound influence on him. It was at this time that he worked out the "localglobal principle" now known by his name, and applied it with great success to the study of quadratic forms over the rationals, where both the representability of a number by a given form and the equivalence of two forms can be decided by local information alone. These two questions became, respectively, the focus of his doctoral dissertation "Über die Darstellbarkeit von Zahlen durch quadratische Formen im Körper der rationalen Zahlen" and that of his Habilitationsschrift "Über die Äquivalenz quadratischer Formen im Körper der rationalen Zahlen". Both were published in the prestigious Crelle's Journal (J. reine u. angew. Math.), vol. 152 (1923), of which he was later (1929) to become a coeditor.
The winter semester 1922-23 saw Hasse as Privatdozent at Kiel, where he married Clara Ohle. He frequently visited Hamburg and maintained close relations with Artin, Hecke, Ostrowski, Petersson, Schreier and other mathematicians there. Taking up a suggestion by Hilbert, he began to work on his "Klassenkörperbericht", the first comprehensive textbook on class field theory incorporating not only the foundations laid by Kronecker, Weber and Hilbert, but also the recent work of Furtwängler and Takagi.
When Hensel retired in 1930, Hasse became his successor in Marburg. Picking up a question which had arisen from E. Artin's dissertation about the zeta function of an algebraic curve over a finite field - tantamount to the appropriate analogue of the famous Riemann Hypothesis - he achieved the first breakthrough and established the conjectured property for zeta functions of elliptic curves (genus one). The general case was later settled by A. Weil. The Hasse(-Weil) theorem implies that the number N(p) of rational points of an elliptic curve over the finite field Z/pZ, where p is a prime, can differ from the mean value p+1 by at most twice the square root of p.
In 1934, Hasse became director of the famous mathematical institute at Göttingen, but the unique concentration of brilliant minds on which this reputation had been based was already falling apart. Hilbert had retired in 1930, H. Weyl had gone to Princeton, E. Landau and E. Noether were driven from their chairs, and Hasse (and, from 1937 onwards, C. L. Siegel) fought an uphill battle against the Nazi beaurocracy in order to maintain at least some vestiges of the former scientific standard. Not without some success; M. Deuring and M. Eichler, R. Nevanlinna and E. Witt and the brilliant but fanatic O. Teichmüller all spent at least some of their early years there.
During the war, Hasse was again associated with the navy - now heading a research institute in Berlin, and studying problems in ballistics.
Together with many of his students, Hasse finally moved to Hamburg in 1950 as Deuring's successor (who had returned to Göttingen to take over Landau's former chair from Kaluza). His good overall constitution helped him recover from a heart attack in 1955. He stayed at Hamburg until his retirement in 1966. Artin returned from his U.S. exile in 1958, and the old friendship was renewed until Artin's untimely death in 1962. Hasse also maintained friendly relations with the Academy of Sciences in East Berlin, of which he had become a member in 1949. (He was a member of several other academies as well.) Apart from original research articles which he kept turning out at his usual rate, he prepared his lectures on number theory for publication in book form.
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