Beppo Levi and the Arithmetic of Elliptic Curves T

Most students of mathematics encounter the name of the Italian mathematician Beppo Levi in integration theory when they learn "Beppo Levi's Lemma" on integrals of monotone sequences of functions. The attribution of this result is historically correct, but it by no means exhausts Beppo Levi's mathematical accomplishments. Between 1897 and 1909, Beppo Levi (1875-1961) actively participated in all major new mathematical developments of the time. He was a man of great perseverance and energy, with an independent mind and a wide mathematical and philosophical culture. His list of publications includes more than 150 mathematical papers. Apart from his lemma, Beppo Levi is known for his work (at the very beginning of this century) on the resolution of singularities of algebraic surfaces. N. Bourbaki's Elements d'histoire des mathdmatiques mention Beppo Levi as one of the rare mathematicians to have recognized the Axiom of Choice as a principle used implicitly in set theory, before Zermelo formulated it. As we shall see below, the role of Beppo Levi set theory seems sometimes overrated. On the other hand, his work on the arithmetic of elliptic curves has not received the attention it deserves. He occupied himself with this subject from 1906 to 1908. His investigations, although duly reported by him at the 1908 International Congress of Mathematicians in Rome, appear to be all but forgotten. This is striking because in this work Beppo Levi anticipated explicitly, by more than 60 yedrs, a famous conjecture made again by Andrew P. Ogg in 1970, and proved by Barry Mazur in 1976. Shortly before his retirement, Beppo Levi faced a tremendous challenge which he more than lived up to: he was forced to emigrate, and devoted the last 20 years of his long life to building up mathematics in Rosario, Argentina.