Speaker: Prof. Marc Hindry ，IMJ-PRG, Université Paris Cité, France

Inviter: 黄治中副研究员

Title: Arithmetic of algebraic surfaces - an anlogue of Brauer-Siegel theorem

Language: French

Time & Venue: 2024.10.12 15:30-17:00 晨兴110

Abstract: We will discuss the following geometric problem, providing an « arithmetic » answer when the base field is finite: given an algebraic surface (smooth, projective), the Néron-Severi group is the group of formal linear combination of curves modulo algebraic deformation; it is known to be finitely generated. Intersection theory gives us an integer valued pairing on this group and we want to estimate asymptotically the regulator (Gram determinant) in terms of simpler quantities like the geometric genus. A concrete example is the family of smooth projective surfaces of degree d in projective space when d goes to infinity. We do not know any answer over an arbitrary field (e.g. the field of complex numbers) but show that when the ground field is finite, the arithmetic situation enables us to bring into play zeta functions. Brauer-Siegel theorem states an asymptotic relation between the three most important invariants of number fields (finite extensions of the field of rational numbers) : discriminant, class number and regulator of units. We show that a partial analogue holds for algebraic surfaces over finite field.