The arithmetic of the product of two algebraic curves over a finite field
type: Item
instance of: scholarly article
title: The arithmetic of the product of two algebraic curves over a finite field
zbMATH Open document ID: 0606.14026
DOI: 10.1016/0021-8693(86)90018-9
published in: Journal of Algebra
publication date: 1986-01-01T00:00:00Z
review text: Let \(X=C_ 1\times C_ 2\) be the product of two nonsingular projective curves defined over a finite field \({\mathbb{F}}_ q\). The author computes the order of the Brauer group of X using the Artin-Tate formula. The arithmetical invariants appearing in the formula are explicitly determined. At some stage, the author assumes that the formal Brauer group of X is representable by a formal group over \(F_ q\). Under this hypothesis, she proves the Artin-Mazur conjecture for \(X=C_ 1\times C_ 2\), as a byproduct of her calculation. It is impossible to describe all the results of this long paper. We only mention that in the generic case, the order of the Brauer group of X is the resultant of the characteristic polynomials of the Frobenius endomorphism of the Jacobian variety \(J(C_ i)\), \(i=1,2\), divided by a certain power of q. In the last section some selected examples are discussed.
Mathematics Subject Classification ID: 14H99
Mathematics Subject Classification ID: 14F22
Mathematics Subject Classification ID: 14G15
Mathematics Subject Classification ID: 14H25
zbMATH DE Number: 3979245
zbMATH Keywords: product of two nonsingular projective curves defined over a finite field
zbMATH Keywords: order of the Brauer group
zbMATH Keywords: Artin-Tate formula
zbMATH Keywords: Artin-Mazur conjecture
author: Noriko Yui
reviewed by: M.-A. Knus
MaRDI profile type: Publication
full work available at URL: 0021-8693(86)90018-9
OpenAlex ID: W2066255377
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description: scientific article; zbMATH DE number 3979245