K3 surfaces and equations for Hilbert modular surfaces

Abstract

We outline a method to compute rational models for the Hilbert modular surfaces Y_{-}(D), which are coarse moduli spaces for principally polarized abelian surfaces with real multiplication by the ring of integers in Q(sqrt{D}), via moduli spaces of elliptic K3 surfaces with a Shioda-Inose structure. In particular, we compute equations for all thirty fundamental discriminants D with 1 < D < 100, and analyze rational points and curves on these Hilbert modular surfaces, producing examples of genus-2 curves over Q whose Jacobians have real multiplication over Q.Comment: 83 pages. Final versio

The Geometry and Moduli of K3 Surfaces

“…16, we see that if S is projective then Amp(S) = K(S) ∩ NS(S) ⊗ R. Given this, most of the results on the ample cone that we saw in Sect. Remark 14.…”

Examples of abelian surfaces with everywhere good reduction

“…An equation for the Hilbert modular surface Y − (8) is given in [17] (see 2.2 for a quick review of the results we need here). As a double-cover of P 2 r,s , it is given by…”

“…The corresponding algorithms have been implemented in the Hilbert Modular Forms Package in Magma [3]). Recently, Elkies and the second author [17] computed explicit birational models over Q for these Hilbert modular surfaces for all the fundamental discriminants D less than 100, by identifying the Humbert surface H D with a moduli space of elliptic K3 surfaces, which may be computed explicitly. For the fundamental discriminants in the range 1 < D < 100, the Humbert surface is a rational surface, i.e.…”

“…In the comprehensive reference [50], arithmetic models for some of them are also described. However, for our work, we need explicit equations for these surfaces along with the map to A 2 , and this does not seem to be available for any discriminant other than 5 except in [17] (though it could be worked out in principle using Hilbert and Siegel modular forms). Consequently, we will use the equations from [17] throughout.…”

“…4.1.2, using the results from [17]. The surface Y − (17) is a double-cover of the (weighted) projective space P 2 g,h /Q given by…”