The arithmetic of the product of two algebraic curves over a finite field
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.
product of two nonsingular projective curves defined over a finite field
scientific article; zbMATH DE number 3979245
scientific article