Louis Mordell

Louis Mordell in Nice , 1970

Louis Joel Mordell (born January 28, 1888 in Philadelphia , USA ; † March 12, 1972 in Cambridge , England ) was an American-British mathematician who mainly worked in number theory , especially the theory of Diophantine equations .

Youth and Studies

Mordell was born in Philadelphia, the third of eight children of Lithuanian Jewish parents who immigrated to the United States a few years before he was born. His mathematical talent was already noticed at school, which he trained in self-study, for example from Leonhard Euler . He earned the money to travel to England to take exams for a math scholarship at Cambridge University. In retrospect, surprisingly, he came first and won the scholarship. While studying under Thomas John l'Anson Bromwich and Henry Frederick Baker , he was third ("Third Wrangler") in the Tripos in 1909 (the two first places went to the now less well-known Daniell and Neville), the last such exams before the overdue reform by Godfrey Harold Hardy . His interest in solving equations in whole numbers (Diophantine equations) arose during his studies. He won the 2nd Smith Prize for an essay on the solutions to the "Mordell equation" , which Pierre de Fermat had already considered, and which he used methods from elementary congruence arithmetic in the sense of Carl Friedrich Gauß , ideal theory and the connection to cubic shapes. At that time he did not know Axel Thue's work on this equation and gave explicit solutions for many values ​​of . With Thue's work it was proven that this equation only has a finite number of solutions. At first Mordell had considered the existence of an infinite number of solutions to be possible. Mordell continued his work on third and fourth degree Diophantine equations (elliptic curves) but was very disappointed when one of his major papers was rejected by the prestigious Journal of the London Mathematical Society in 1913 - he attributed this to the neglect of number theory in England. ${\displaystyle y^{2}=x^{3}+k}$ ${\displaystyle k}$

Career

In 1913, after an application for a fellowship at Cambridge was unsuccessful, he became a teacher at Birkbeck College at the University of London . In 1917 he applied the theory of modular functions in number theory and proved S. Ramanujan's conjecture about the tau function . The theory was later rediscovered and expanded by the German number theorist Erich Hecke (including the theory of Hecke operators ). In contrast to algebraic methods of the German school around Emil Artin and Helmut Hasse, for example, Mordell used mostly "elementary" methods in his work , or in other words, mathematics in the style of Euler and Gauss up to the 19th century. ${\displaystyle \tau (n)}$

In 1920 he took a position at Manchester College of Technology before becoming a professor at the university there in 1922. In 1922 he published a paper in which he proved that the group of rational points on elliptic curves is finitely generated (he called the "finite basis theorem"), which was extended to all algebraic curves by André Weil in 1929 ( Mordell's theorem) Because ). Mordell used a version of Fermat's method of infinite descent. At the same time he suspected ( Mordell's conjecture ) that there are only finitely many rational points for curves defined by gender , which was proven by Gerd Faltings in 1983 (therefore also called "Faltings' theorem"). With Harold Davenport and Kurt Mahler , who also teach in Manchester , he turned the university there into a center for number theory in England, visited by foreign guests such as Hans Heilbronn , Derrick Henry Lehmer , Claude Chabauty . His fields of work included Gauss sums, exponential sums (important for solving Diophantine equations mod , such as in André Weil's work, in which he proves the Riemann hypothesis for irreducible algebraic curves and makes his assumptions ), Minkowski's geometry of numbers, cubic surfaces (best known here is the theorem of his student Beniamino Segre that they only have 0, 1 or an infinite number of rational points). ${\displaystyle \mathbb {Q} }$ ${\displaystyle g>1}$ ${\displaystyle p}$

In 1924 Mordell was accepted as a member (" Fellow ") in the Royal Society , which in 1949 awarded him the New Year's Medal . In 1945 he succeeded Godfrey Harold Hardy as a mathematics professor at Cambridge University. From 1943 to 1945 he was head of the London Mathematical Society , whose De Morgan Medal he received in 1941. After his retirement in 1953, he held numerous visiting professorships around the world and continued to write numerous papers on number theory. In 1936 he gave a plenary lecture at the International Congress of Mathematicians in Oslo ( Minkowski's Theorems and Hypotheses on Linear Forms ) and in 1962 he was an Invited Speaker at the ICM in Stockholm (On Lerch's class number formulas for binary quadratic forms).

Private life

Mordell was a passionate mountaineer (also a rock climber in his youth), swimmer and bridge player. He had been married since 1916 and had a son Donald (professor of engineering in Canada) and a daughter. His students include Paul Erdős , John Arthur Todd , Patrick du Val , Beniamino Segre and Kurt Mahler . After Harold Davenport, he had a simple, helpful character and a natural talent for friendship. He supported his students and colleagues without ulterior motives and also contributed a lot to the explanation of his subject with many overview articles and book reviews in essay form.

literature

• Louis Mordell: On Mr. Ramanujan's empirical expansion of modular functions . Proc. Cambridge Philosophical Society, Vol. 19, 1917, pp. 117-124.
• Louis Mordell: Note on certain modular relations considered by Messrs. Ramanujan, Darling and Rogers . Proc. London Math. Society, Vol. 20, 1921, pp. 408-416.
• Louis Mordell: Three lectures on Fermat's Last Theorem . Cambridge University Press 1921.
• Louis Mordell: On the rational solutions of the indeterminate equation of the 3rd and 4th degrees . Proc. Cambridge Philosophical Society, Vol. 21, 1922, pp. 179-192 (Mordell-Weil Theorem, Mordell Assumption).
• Louis Mordell: Diophantine Equations . Academic Press, London 1969, ISBN 0125062508 .
• Louis Mordell: Reminiscences of an octogenarian mathematician . American Mathematical Monthly, Vol. 78, 1971, pp. 952-961.
• Louis Mordell (with foreword by Olaf Neumann): Two papers on number theory . VEB Deutscher Verlag der Wissenschaften, Berlin, 1972.

about Mordell and his work:

• Cassels: Biography in Biographical Memoirs of the Fellows of the Royal Society . In: Dictionary of scientific biography . Bulletin London Mathematical Society, Vol. 6, 1974, p. 69.
• Ireland, Rosen: A classical introduction to modern number theory . Springer-Verlag 1990.
• Nagell: Introduction to Number Theory . 1951 (proof of Mordell-Weil's theorem).
• Jean-Pierre Serre : Lectures on the Mordell-Weil Theorem . Vieweg 1997.

See also

• Erdös-Mordell's inequality

Individual evidence

1. ^ Entry on Mordell; Louis Joel (1888-1972) in the Archives of the Royal Society , London

Web links

• John J. O'Connor, Edmund F. Robertson :  Louis Mordell. In: MacTutor History of Mathematics archive .
• Biography of Davenport, Acta Arithmetica (PDF; 5.2 MB)