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JSTOR is a not-for-profit service that helps scholars, researchers, and students discover, use, and build upon a wide range of content in a trusted digital archive. We use information technology and tools to increase productivity and facilitate new forms of scholarship. For more information about JSTOR, please contact support@jstor.org.. The Johns Hopkins University Press is collaborating with JSTOR to digitize, preserve and extend access to American Journal of Mathematics.Dedicated to Professor J. Igusa for his 60th birthday 1. Introduction. The Picard number p(X) of a nonsingular projective variety X over an algebraically closed field k is defined to be the rank of the Neron-Severi group NS(X), which is the group of divisors on X modulo algebraic equivalence and which is known to be a finitely generated abelian group. When k = C (the field of complex numbers), p(X) is equivalently defined as the largest number of homologically independent divisors on X, and it is equal to the dimension of the space of rational cohomology classes of type (1, 1) by Lefschetz' criterion.It is desirable to be able to compute the Picard number of a given variety, especially because of its close connection to many important arithmetic problems such as the Tate Conjecture. Even in the case of surfaces (dimX = 2) over C, this is not so easy. In fact, as Zariski [19, p. 110] remarked almost 40 years ago, "the evaluation of p for a given surface presents in general grave difficulties." Of course, there has been considerable progress since then. For example, the behavior of Picard numbers is fairly well understood in the case of algebraic surfaces of special types such as abelian surfaces, K3 surfaces or elliptic surfaces. Also some purely characteristic p phenomena such as supersingular surfaces with positive geometric genus have attracted the interest of many geometers. Nevertheless the above remark of Zariski is "still very valid" (cf. [19, p. 124]). Therefore it will be of some interest if one can find an explicit algorithm for computing the Picard number of surfaces in some restricted classes. Instead of the Picard number, one may consider the Lefschetz number )(X) = b2(X) -p(X), b2(X) being the second Betti number of X; the definition of b2 (and the fact X(X) _ 0) in arbitrary characteristic was first given by Igusa [41 (cf. [19, p. 122]). The Lefschetz number is a bira-

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