|TITLE:||Splitting of Abelian varieties -- a new local/global problem|
|SPEAKER:||V. Kumar Murty (University of Toronto)|
|DATE:||Vendredi, 13 Février 2004|
Given an irreducible polynomial with integer coefficients, does it remain irreducible when viewed modulo a prime? The answer is clearly 'No'. For example, T^2 +1 becomes the reducible polynomial (T+1)^2 modulo 2. Can this happen modulo every prime? The answer is 'Yes' but this is not obvious and requires algebraic number theory to establish. We shall begin the talk by explaining this phenomenon and then raise a geometric analogue of the question: does a simple Abelian variety over a number field remain simple when viewed modulo a prime? This a new instance of a local/global problem: determining global obstructions to solving a problem that can be solved locally. Such problems have played an important role in the development of number theory. In the case here, the answer is not completely known and seems to be rather subtle. The results we are able to prove require the full force of techniques from arithmetic algebraic geometry. We shall attempt to explain some of these results and the techniques involved.
About the speaker: Professor Vijaya Kumar Murty got his Ph.D. at Harvard in 1982 under the supervision of Professor John Tate. His profound work in number theory touches many aspects of the area: analytic, algebraic, geometric and even applied (cryptography). He was Coxeter James Lecturer in 1991-92, Steacie Fellow in 1995 and Ferran Sunyer Balaguer Prize (joint with Ram Murty) in 1995. He is also Fellow of the Royal Society of Canada since 1995 and Fellow of the Fields Institute of Mathematical Sciences since 2003.
For more information, visit Professor Murty home page .