Transcendental Lattices of Some K3-surfaces

In a previous paper, [S2], we described six families of K3-surfaces with Picardnumber 19, and we identified surfaces with Picard-number 20. In these notes we classify some of the surfaces by computing their transcendental lattices. Moreover we show that the surfaces with Picard-number 19 are birational to a Kummer surface which is the quotient of a non-product type abelian surface by an involution. 0. Introduction Given a K3-surface an important step toward its classification in view of the Torelli theorem is to compute the Picard lattice and the transcendental lattice. When the rank of the Picard lattice (i.e. the Picard-number, which we denote by ρ) of the K3-surface is 20, the maximal possible, the transcendental lattice has rank two. These K3-surfaces are called by Shioda and Inose singular. In [SI], Shioda and Inose classified such surfaces in terms of their transcendental lattice, more precisely they show the following: Theorem 0.1. [SI, Theorem 4, §4] There is a natural one-to-one correspondence from the set of singular K3-surfaces to the set of equivalence classes of positive-definite even integral binary quadratic forms with respect to SL2(Z). When the Picard-number is 19 the transcendental lattice has rank three and by results of Morrison, [M], and Nikulin, [N], the embedding in the K3-lattice Λ := −E8⊕−E8⊕U⊕U⊕U is unique, hence it identifies the moduli curve classifying the K3-surfaces. In general however it seems to be difficult to compute explicitly the transcendental lattice. In [S2] we describe six families of K3-surfaces with Picard-number 19 and we identify in each family four surfaces with Picard-number 20. The aim of these notes is to compute their transcendental lattice and to classify them. In [S2] we describe completely the Picard lattice of the general surface in two of the families and of the special surfaces and we describe the Picard lattice of six surfaces with Picard-number 20 in the other families. Here by using lattice-theory and results on quadratic forms we compute the transcendental lattices of these surfaces. The methods are similar as the methods used by Barth in [B] for describing the K3-surfaces of [BS]. By a result of Morrison, [M, Cor. 6.4], K3-surfaces with ρ = 19 and 20 have a Shioda-Inose structure, in particular this means that there is a birational map from the K3-surface to a Kummer surface. It is well known (cf. [SI]) that if ρ = 20, then the Kummer surface is the quotient by an involution of a product-type abelian variety. When ρ = 19 this is not always the case. In fact we use the transcendental lattices to show that in our cases the abelian variety is not a product of two elliptic curves. In this case we call the Shioda-Inose structure simple. The paper is organized as follows: in section 1 we recall some basic facts about lattices and quadratic forms and the construction of the families of K3-surfaces. Then section 2 1991 Mathematics Subject Classification. 14J28, 14C22.