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Uspekhi Mat. Nauk, 2018, Volume 73, Issue 2(440), Pages 75–140 (Mi umn9814)  

Varieties over finite fields: quantitative theory

S. G. Vlăduţab, D. Yu. Noginb, M. A. Tsfasmancbd

a Aix-Marseille Université, CNRS, Centrale Marseille, Institut de Mathématiques de Marseille (I2M, UMR 7373), Marseille, France
b Institute for Information Transmission Problems of Russian Academy of Sciences, Moscow
c CNRS, Laboratoire de Mathématiques de Versailles (UMR 8100), France
d Independent University of Moscow

Abstract: Algebraic varieties over finite fields are considered from the point of view of their invariants such as the number of points of a variety that are defined over the ground field and its extensions. The case of curves has been actively studied over the last thirty-five years, and hundreds of papers have been devoted to the subject. In dimension two or higher, the situation becomes much more difficult and has been little explored. This survey presents the main approaches to the problem and describes a major part of the known results in this direction.
Bibliography: 102 titles.

Keywords: algebraic varieties over finite fields, zeta functions, points on surfaces, error-correcting codes, arithmetic statistics, explicit formulae in arithmetic.

Funding Agency Grant Number
Russian Science Foundation 14-50-00150
Agence Nationale de la Recherche ANR-17-CE40-0012
The research of the second author was carried out at the Institute for Information Transmission Problems of Russian Academy of Sciences and was supported by the Russian Science Foundation (project no. 14-50-00150). The research of the third author was carried out with the partial financial support of the French National Research Agency project FLAIR (ANR-17-CE40-0012).


DOI: https://doi.org/10.4213/rm9814

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English version:
Russian Mathematical Surveys, 2018, 73:2, 261–322

Bibliographic databases:

UDC: 512.75
MSC: Primary 14G15, 14J20; Secondary 11G25, 14M15, 94B27
Received: 13.11.2017
Revised: 09.01.2018

Citation: S. G. Vlăduţ, D. Yu. Nogin, M. A. Tsfasman, “Varieties over finite fields: quantitative theory”, Uspekhi Mat. Nauk, 73:2(440) (2018), 75–140; Russian Math. Surveys, 73:2 (2018), 261–322

Citation in format AMSBIB
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