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PUBLISHED BY mathematical sciences publishers nonprofit scientific publishing http://msp.org/ © 2014 Mathematical Sciences Publishers ALGEBRA AND NUMBER THEORY 8:10 (2014) msp dx.doi.org/10.2140/ant.2014.8.2297 K3 surfaces and equations for Hilbert modular surfaces Noam Elkies and Abhinav Kumar We outline a method to compute rational models for the Hilbert modular surfaces Y−.D/, which are coarse moduli spaces for principallyp polarized abelian surfaces with real multiplication by the ring of integers in Q. 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. 1. Introduction Hilbert modular surfaces have been objects of extensive investigation in complex and algebraic geometry, and more recently in number theory. Since Hilbert modular varieties are moduli spaces for abelian varieties with real multiplication by an order in a totally real field, they have intrinsic arithmetic content. Their geometry is enriched by the presence of modular subvarieties. In[Hirzebruch 1973; Hirzebruch and van de Ven 1974; Hirzebruch and Zagier 1977] the geometric invariants of many of these surfaces were computed, and they were placed within the Enriques–Kodaira classification. A chief aim of the present work is to compute equations for birational models of some of these surfaces over the field of rational numbers. More precisely, let D be a positive fundamental discriminant,p i.e., the discrimi- nant of the ring of integers OD of the real quadratic field Q. D/. The quotient C − C − PSL2.OD/n.H × H / (where H and H are the complex upper and lower half-planes) parametrizes abelian surfaces with an action of OD. It has a natural Elkies was supported in part by NSF grants DMS-0501029 and DMS-1100511. Kumar was supported in part by NSF grants DMS-0757765 and DMS-0952486, and by a grant from the Solomon Buchsbaum Research Fund. The research was started when Kumar was a postdoctoral researcher at Microsoft Research. He also thanks Princeton University for its hospitality during Fall 2009. MSC2010: primary 11F41; secondary 14G35, 14J28, 14J27. Keywords: elliptic K3 surfaces, moduli spaces, Hilbert modular surfaces, abelian surfaces, real multiplication, genus-2 curves. 2297 2298 Noam Elkies and Abhinav Kumar compactification Y−.D/, obtained by adding finitely many points and desingulariz- ing these cusps. These surfaces Y−.D/ have models defined over Q, and the main goal of this paper is to describe a method to compute explicit equations for these models, as well as to carry out this method for all fundamental discriminants D with 1 < D < 100. This felt like a good place to stop for now, though these calculations may be extended to some higher D, as well as to non-fundamental discriminants. We briefly summarize the method, which we describe in more detail in later sections. The method relies on being able to explicitly parametrize K3 surfaces that are related by Shioda–Inose structure to abelian surfaces with real multiplication by some OD. The K3 surface corresponding to such an abelian surface has Néron– Severi lattice containing L D, a specific indefinite lattice of signature .1; 17/ and discriminant −D. In all our examples, we obtain the moduli space MD of L D- polarized K3 surfaces as a family of elliptic surfaces with a specific configuration of reducible fibers and sections. We then use the 2- and 3-neighbor method to transform to another elliptic fibration, with two reducible fibers of types II∗ and III∗ respectively. This lets us read off the map (generically one-to-one) of moduli spaces from MD into the 3-dimensional moduli space A2 of principally polarized abelian surfaces, using the formulae from[Kumar 2008]. The image of MD is the Humbert surface corresponding to discriminant D. The Hilbert modular surface Y−.D/ itself is a double cover of the Humbert surface, branched along a union of modular curves. We use simple lattice arguments to obtain the branch locus, and pin down the exact twist for the double cover by counting points on reductions of the related abelian surfaces modulo several primes. In all our examples, the Humbert surface happens to be a rational surface (i.e., birational to P2 over Q), and we display the equation of 2 Y−.D/ as a double cover of P branched over a curve of small degree. We analyze these equations in some detail, attempting to produce rational or elliptic curves on them, with the intent of producing several (possibly infinitely many) examples of genus-2 curves whose Jacobians have real multiplication. When Y−.D/ is a K3 surface, it often has very high Picard number (19 or 20), and we attempt to compute generators for the Picard group. When Y−.D/ is an honestly elliptic surface, we analyze the singular fibers and the Mordell–Weil group, and attempt to compute a basis for the sections. To our knowledge, this is the first algebraic description of most of these surfaces by explicit equations. We outline some related work in the literature. Wilson[1998; 2000] obtained equations for the Hilbert modular surface Y−.5/ corresponding to the smallest fundamental discriminant D > 1. Van der Geer[1988] gives a few examples of algebraic equations for Hilbert modular surfaces corresponding to a congruence subgroup of the full modular group (in other words, abelian surfaces K3 surfaces and equations for Hilbert modular surfaces 2299 with some level structure). Humbert surfaces have also been well-studied in the literature, and Runge[1999] and Gruenewald[2008] have obtained equations for some of these. However, these equations are quite complicated, and do not shed as much light on the geometry of Hilbert modular surfaces. While the methods are simpler, involving theta functions and q-expansions, the result is analogous to exhibiting the modular polynomial whose zero locus in A1 ×A1 is a singular model of the complement of the cusps in the modular curve X0.N/. The coefficients of these polynomials can quickly become enormous. We believe that our approach, giving simpler equations for these surfaces together with their maps to A2, is more conducive to an investigation of arithmetic properties. It is our hope that these equations will be of much help in subsequent arithmetic investigation of these surfaces. For instance, they should provide a testing ground for many conjectures in Diophantine geometry, because of the abundance of rational curves and points. Another direction of future investigation is to use these equations to investigate modularity of the corresponding abelian surfaces. Modularity of abelian varieties with real multiplication over Q is now proven, by combining results of Ribet[2004] with the recent proof of Serre’s conjecture by Khare and Wintenberger[2009a; 2009b]. However, unlike the case of dimension 1, where one has modular parametrizations and very good control of the moduli spaces, the situation in dimensions 2 and above is much less clear. For instance, it is not at all clear how to find a modular form corresponding to a given abelian surface with real multiplication.1 We hope that the abundance of examples provided by these equations will help pave the path for a better understanding of the 2-dimensional case.