Volume 4, Issue 11
Special volume AJAX in honor of René Schoof
Abstract:
Given a smooth projective surface, it is well known that the Néron–Severi group is finitely generated. We look for estimates of its complexity, which we can quantify either by its discriminant or its regulator. Thus we search, under some conditions, for an upper bound of the determinant of the intersection pairing on the Néron–Severi group, in terms of the geometric genus . While this seems out of reach over a general field, we show that, when the field of definition of the surface is finite, such an upper bound can be attained via the Weil zeta function of the surface, conditional on a well-known conjecture of Artin and Tate. Precisely, if the surface is defined over a field of cardinality , we obtain, under some numerical constraints on the invariants of the surface, that the logarithm of the discriminant is bounded by , where can be any constant . This bound comes together with a bound for the cardinality of the Brauer group of the surface. This upper 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.
Keywords: Algebraic surfaces and varieties over finite fields, algebraic cycles, zeta functions, Brauer group, Néron–Severi group.
Citation:
Marc Hindry. Asymptotic arithmetic of varieties over finite fields. Polynesian Journal of Mathematics, Volume 4, Issue 11, Pages 1–33. (Sept. 2026) DOI: 10.69763/polyjmath.4.11
Milestones:
Received 26 Feb 2026
Revised 2 Sep 2026
Accepted 2 Sep 2026
Published 28 Sep 2026
Communicated by Valerio Talamanca
The above content is published under the terms of the Creative Commons Attribution 4.0 International license.