Finding rational solutions of polynomial equations is one of the most difficult questions in arithmetic geometry. Surprisingly, the qualitative behaviour of these solutions is the subject of a completely general conjecture, the Tamagawa number conjecture of Beilinson, Bloch and Kato. It asserts roughly that all important invariants of an algebraic variety are encoded in its associated L-function. One of the simplest cases of this general conjecture is the, already very difficult, Birch-Swinnerton-Dyer conjecture (one of the millennium problems), which deals with the case of elliptic curves, i.e. Riemann surfaces of genus one.

In this talk we want to give a gentle introduction to the Birch-Swinnerton-Dyer conjecture, avoiding all technicalities and starting with a review of elliptic curves and the problem of counting solutions over finite fields. At the very end we will mention some recent progress obtained by Bertolini, Darmon, Rotgers on the one hand and by Loeffer, Zerbes and the speaker on the other hand. This talk is aimed at non-experts.

Guido Kings was born in 1965 in Cologne, Germany. He got his Ph.D. in 1994 in Munster, under the direction of Christopher Deninger, (Higher regulators, Hilbert-Blumenthal surfaces and special values of L-functions). He is currently Professor at the the University of Regensburg. In 2002, he was Invited Speaker at the ICM in Beijing.

Kings works on the arithmetic of motives and L-functions, with particular interest in the Birch and Swinnerton-Dyer conjecture, a Millennium Problem, and the Equivariant Tamagawa Number Conjecture. Kings obtained several fundamental results in this area, leading to important breakthroughs in our understanding of the Birch and Swinnerton-Dyer conjecture and the Equivariant Tamagawa Number Conjecture.

More infos are available here:
https://de.wikipedia.org/wiki/Guido_Kings