The arithmetic of the product of two algebraic curves over a finite field: Volume 98, number 1, January 1986: Noriko Yui, pp. 102–142

Abstract

Let X= C, x Cz be the product of two nonsingular projective curves defined over a tinite field k = IF,. We compute the order of the Brauer group Br(X) of X. In the generic case, it is given by the resultant of the characteristic polynomials of the Frobenius endomorphism of the Jacobian variety J(C,) (I = 1, 2). divided by a certain power of 4. In particular, if at least one component C, has ordinary Jacobian variety J(C,), then the order of Br(X) is equal to the resultant divided by y”u, where pq is the geometric genus of X. (’ 1986 Academic Pres. Inc