Algebra Seminar - Prof. Marc Hindry, Paris VII

Title: Arithmetic of surfaces over finite fields

Abstract:

Given a smooth projective surface, we look for an upper bound for the regulator, i.e. the determinant of the intersection pairing on the Néron-Severi group, in terms of the geometric genus of the surface.While this seems out of reach over a general field, we show that, when the field of definition of the surface is finite, such a bound can be attained via the Weil zeta function of the surface, conditional to a well known conjecture of Tate. Precisely, if the surface is defined over a field of cardinality q, we obtain, under some numerical constraints on the invariants of the surface, that the logarithm of the regulator is (essentially) bounded by a constant times the geometric genus, where the constant can be any number > log q. This bound comes together with a bound for the cardinality of the Brauer group of the surface. This bound is essentially optimal, as shown by examples of Griffon (2018). When considering higher dimensional varieties, we also investigate the determinant of the intersection pairing on the groups of algebraic cycles modulo numerical equivalence; in this general case, results are even more conditional.