中国科学院数学与系统科学研究院

数学研究所

中科院华罗庚数学重点实验室

综合报告会

（Colloquium）

报告人：Prof. Marc Hindry(IMJ-PRG, Université Paris Cité, France)

题  目：Arithmetic of algebraic surfaces - an anlogue of Brauer-Siegel theorem.

时  间：2024.10.12（星期六）15:30-17:00

地  点：晨兴110

报告人简介：Marc Hindry是巴黎西岱大学的教授。他于1987年在Michel Laurent 教授的指导下获得博士学位。他的主要研究领域为丢番图几何，有理点密度，阿贝尔簇算术等。他是著名的教科书《Diophantine geometry》GTM201的作者之一。

摘要：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.