Lucien Szpiro

Lucien Szpiro 2005

Lucien Szpiro (born December 23, 1941 in Paris ; † April 18, 2020 there ) was a French mathematician who dealt with number theory , arithmetic algebraic geometry and commutative algebra .

Life

Szpiro was an assistant teacher at a grammar school from 1963 to 1965. In 1965 he became an assistant and then senior assistant at the Sorbonne . From 1969 he conducted research (first as Attaché de Recherches, then Chargé de Recherches) for the CNRS at the University of Paris VII. He received his doctorate in 1971 under Pierre Samuel at the University of Paris-Süd XI in Orsay (Doctorat d'État). His dissertation was also from the seminars of Claude Chevalley , Alexander Grothendieck and Maurice Auslanderinfluenced. From 1977 to 1991 he was Maître de Recherches and then Research Director of the CNRS at the École Normale Supérieure in Paris (Rue d'Ulm), then Research Director of the Classe Exceptionelle at the University of Paris in Orsay. From 1999 he was a professor at the CUNY Graduate Center of the City University of New York , where he led a joint New York number theory seminar with Dorian Goldfeld and Peter Sarnak . Szpiro worked a. a. with Arakelov theory. Szpiro made a conjecture about the asymptotic behavior of the (minimal) discriminants with regard to the size of the leader of an elliptic curve : ${\displaystyle D}$ ${\displaystyle N}$

${\displaystyle |D|={\mathcal {O}}(N^{6+\epsilon })}$.

In other words: for each there is a positive constant such that . Leader and discriminant are two important arithmetic invariants of elliptical curves. ${\displaystyle \epsilon >0}$ ${\displaystyle K_{\epsilon }}$ ${\displaystyle |D|<K_{\epsilon }N^{6+\epsilon }}$

The abc conjecture follows from this conjecture . For modular elliptic curves, Szpiro's conjecture is also equivalent to a conjecture about the asymptotic behavior of the order of the Tate-Shafarevich groups of the elliptic curve, as Szpiro and Goldfeld showed.

Szpiro's conjecture, which he presented in Hanover in 1983, was also one of the motivations for the a b c conjecture (Masser, Oesterlé 1985).

He also dealt with the theory of perfect complexes, geometry and dynamics of heights in number theory, geometry of surfaces that are fiberized over a curve, commutative algebra and projective geometry.

Emmanuel Ullmo is one of his PhD students .

He was visiting professor or visiting scholar at Columbia University , MSRI , Tata Institute for Fundamental Research in Bombay, IMPA in Rio de Janeiro, Institute for Advanced Study , Max Planck Institute for Mathematics in Bonn, at lunchtime -Leffler Institute , at Aarhus University, Brandeis University and in Tokyo. From 1972 to 1992 he organized an Oberwolfach seminar every year.

Szpiro received the Prix ​​Paul Doistau-Émile Blutet of the Académie des Sciences in 1987 for his work in commutative algebra and algebraic geometry and his contribution to Faltings ' proof of the Mordell presumption .

He was a fellow of the American Mathematical Society . In 2014 he was accepted as an external member of the Academia Europaea . From 1991 to 1993 he was chief editor of Astérisque .

Szpiro died of heart failure in Paris in April 2020 at the age of 78.

Fonts

• with C. Peskine: Dimension projective finie et cohomologie locale. Applications à la demonstration de conjectures de M. Auslander, H. Bass et A. Grothendieck, Inst. Hautes Études Sci. Publ. Math., Vol. 42, 1973, pp. 47-119.
• with C. Peskine: Liaison des variétés algébriques, Invent. Math., Vol. 26, 1974, pp. 271-302.
• Sur le théorème de rigidité de Parsin et Arakelov , Journées de Géométrie Algébrique de Rennes (Rennes 1978), Volume 2, Astérisque, Volume 64, 1979, pp. 169-202.
• Sur les propriétés numériques du dualisant relatif d'une surface arithmétique, in: The Grothendieck Festschrift, Volume 3, Progress in Mathematics 88, Birkhäuser 1990, pp. 229–246
• with E. Ullmo, S. Zhang: Equirépartition des petits points , Invent. Math., Vol. 127, 1997, pp. 337-347.
• with J. Pesenti: Inégalité du discriminant pour les pinceaux elliptiques à réductions quelconques, Compositio Math., Volume 120, 2000, pp. 83-117.
• with T. Tucker: A Shafarevich-Faltings Theorem for rational functions, Quarterly Journal of Pure and Applied Mathematics, for the occasion of the sixtieth birthday of Fedor Bogomolov, Volume 4, No. 3, 2008, pp. 715-728.
• Szpiro “Conjecture de Mordell”, Bourbaki Seminar 1983/4

Web links

• Lucien Szpiro in the Mathematics Genealogy Project (English)
• Homepage
• Brief biography

References

1. ↑ Lucien Szpiro 1941-2020
2. ^ "Discriminant et conducteur", Seminaire sur les pinceaux des courbes elliptiques, Astérisque vol. 183, 1990. p. 7
3. ↑ The guide is a product of prime numbers p that divide the discriminants ("bad primes", "bad primes"), each provided with an exponent that indicates the type of singularity of the elliptic curve for the "bad prime numbers". The elliptic curve is singular if the discriminant vanishes, e.g. B. for elliptic curves mod p where p is a bad prime number.
4. ↑ z. B. Oesterlé, Seminar Bourbaki, No. 694, 1988, William Stein in commentary on Matt Baker's lecture ( memento of the original from April 7, 2015 in the Internet Archive ) Info: The archive link has been inserted automatically and has not yet been checked. Please check the original and archive link according to the instructions and then remove this notice. 
5. ↑ Szpiro, Goldfeld “Bounds on the order of the Tate-Shafarevich Group”, Compositio Mathematica, Vol. 97, 1995, p. 71, online here 1995 97 1-2 71 0 ( Memento of the original from April 7, 2015 on the Internet Archive ) Info: The archive link was inserted automatically and has not yet been checked. Please check the original and archive link according to the instructions and then remove this notice. 
6. ^ Bombieri, Gubler Heights in Diophantine Geometry , Cambridge University Press 2007, p. 442
7. ↑ Laudation in his CV, see his homepage