Colloquium Mathematics - Timo Keller, University of Groningen

Title: The conjecture of Birch and Swinnerton-Dyer

Abstract:

The conjecture of Birch and Swinnerton-Dyer (BSD) relates in a surprising way algebraic and analytic, local and global invariants of elliptic curves (smooth plane cubics with a rational point) over the rational numbers or more generally abelian varieties over finitely

generated fields. Its weak form, the rank conjecture, would give an efficient algorithm to decide whether a cubic equation has finitely or infinitely many rational solutions, or more generally an algorithm to determine a basis of the Mordell–Weil group. Starting with a classical, elementary and yet unsolved question, the congruent number problem, I will motivate the invariants occurring in the BSD conjecture and state the conjecture itself. Then I will report on known results, especially on the strong form of the BSD conjecture.

Finally, I will indicate recent theoretical and computational developments on strong BSD for:

(1) abelian surfaces over the rationals,

(2) the p-part of BSD for Eisenstein primes, and

(3) general abelian varieties over finitely generated fields of positive characteristic.