Karl Rubin

Karl Rubin, 2009

Karl Rubin (born January 27, 1956 in Urbana , Illinois ) is an American mathematician who works in arithmetic algebraic geometry and number theory .

Life

Rubin studied at Princeton , where he graduated with a bachelor's degree in 1976 . He then completed his master's degree at Harvard in 1977 and received his doctorate there in 1981 with Andrew Wiles with On the arithmetic of CM Elliptic Curves in Zp Extensions (CM Elliptic Curves means those with complex multiplication). In 1982 he was a visiting lecturer at Princeton, 1984 assistant professor at Ohio State University , where he was professor from 1987 to 1999 (1988/9 also at Columbia University ). From 1997 to 2006 he was Professor at Stanford and is currently Thorp Professor at the University of California at Irvine .

Rubin expanded the theory of the Euler systems introduced by Victor Kolyvagin and applied it to arithmetic algebraic geometry. He was the first who (1986) was able to prove the finiteness of the Tate-Shafarevich groups for special elliptic curves over the rational numbers and thus also made progress in the still open conjecture by Birch and Swinnerton-Dyer . He also used these methods to prove the main conjectures of Iwasawa theory in imaginary quadratic number fields.

In addition, he also deals with cryptographic applications of elliptic curves and their generalizations.

He was a Putnam Fellow in 1975, a Sloan Research Fellow in 1985 , a Guggenheim Fellow in 1994 and the National Science Foundation Presidential Young Investigator Award in 1988. In 1992 he received the Cole Prize in number theory. In 2002 he was invited speaker at the ICM in Beijing. In 1999 he received the Humboldt Prize from the Alexander von Humboldt Foundation . He is a fellow of the American Mathematical Society .

Fonts

• Elliptic curves with complex multiplication and the conjecture of Birch and Swinnerton-Dyer . In: Inventiones mathematicae , 64, 1981, pp. 455-470
• Elliptic curves and Z _p -extensions . In: Compositio Mathematica , 56, 1985, pp. 237-250
• Local units, elliptic units, Heegner points and elliptic curves . Inventiones mathematicae , 88, 1987, pp. 405-422
• Global units and ideal class groups . In: Inventiones mathematicae , 89, 1987, pp. 511-526
• Tate-Shafarevich groups and L-functions of elliptic curves with complex multiplication . In: Inventiones mathematicae , 89, 1987, pp. 527-560
• Tate-Shafarevich groups of elliptic curves with complex multiplication in John Coates et al. (Ed.): Algebraic number theory - in honor of K. Iwasawa . Academic Press, Boston 1989, ISBN 0-12-177370-1 , pp. 409-419 (English; Advanced Studies in Pure Mathematics series 17)
• The main conjecture . Appendix in Serge Lang : Cyclotomic Fields I and II , Springer-Verlag, 1990, ISBN 3-540-96671-4 , pp. 397-419 (series Graduate Texts in Mathematics , 121)
• The "main conjectures" of Iwasawa theory for imaginary quadratic fields . In: Inventiones mathematicae , 103, 1991, pp. 25-68
• p-adic L-functions and rational points on elliptic curves with complex multiplication . In: Inventiones mathematicae , 107, 1992, pp. 323-350
• Euler systems and exact formulas in number theory . In: Annual report of the DMV , 98, 1996, pp. 30–39
• Euler systems and modular elliptic curves . (PDF; 243 kB) In: Anthony J. Scholl, RL Taylor (Ed.): Galois representations in arithmetic algebraic geometry . Cambridge University Press, Cambridge 1998, ISBN 0-521-64419-4 , pp. 351-367 (English; London Mathematical Society Lecture Note Series series , 254)
• Euler systems . Princeton University Press, Princeton 2000, ISBN 0-691-05075-9
• with Alice Silverberg: Ranks of Elliptic Curves . In: Bulletin of the AMS , 39, 2002, pp. 455-474
• with Barry Mazur : Elliptic curves and class field theory . arxiv : math / 0304235 . In: Li Tatsien (Ed.): Proceedings of the ICM Beijing 2002 . Vol. II: Invited Lectures . Higher Education Press, Beijing 2002, ISBN 7-04-008690-5 , pp. 185-196
• with Barry Mazur: Kolyvagin Systems . American Mathematical Society, 2004, ISBN 0-8218-3512-2 (Memoirs of the AMS 168 series)

literature

• Bernadette Perrin-Riou : Travaux de Kolyvagin et Rubin . Seminaire Bourbaki 32, 1989–1990, pp. 69–106 (French; Exposé 717)

Web links

• Homepage with preprints