Wojciech J. Gajda

Abstract: According to Northcott’s theorem, each set of algebraic numbers whose height and degree are bounded is finite. Analogous finiteness properties are also satisfied by many other heights, such as Faltings’s celebrated height. Given the many (expected and proven) links between heights and special values of L-functions (with the BSD conjecture as the most remarkable example), it is natural to ask whether the special values of an L-function have properties similar to the conclusions of Northcott's theorem or of Bogomolov's conjecture. In this talk, based on a joint work with Fabien Pazuki, and on another joint work with Jerson Caro and Fabien Pazuki, we will explain how these Diophantine properties are often satisfied for special values taken at the left of the critical strip, and not satisfied on the right. We will also overview the links between these Diophantine properties and those of the motivic heights defined by Kato, and also some effective aspects of our work, which aim at giving some explicit bounds for the cardinality of the finite sets that we come across.