Explicit Equations of Some Elliptic Modular Surfaces

We compute explicit rational models for some Hilbert modular surfaces corresponding to square discriminants, by connecting them to moduli spaces of elliptic K3 surfaces. Since they parametrize decomposable principally polarized abelian surfaces, they are also moduli spaces for genus-2 curves covering elliptic curves via a map of fixed degree. We thereby extend classical work of Jacobi, Hermite, Bolza etc., and more recent work of Kuhn, Frey, Kani, Shaska, Völklein, Magaard and others, producing explicit families of reducible Jacobians. In particular, we produce a birational model for the moduli space of pairs (C, E) of a genus 2 curve C and elliptic curve E with a map of degree n from C to E, as well as a tautological family over the base, for 2 ≤ n ≤ 11. We also analyze the resulting models from the point of view of arithmetic geometry, and produce several interesting curves on them.

Analytic Number Theory, 2007

Archiv der Mathematik, 2007

In this paper, we study the structure, deformations and the moduli spaces of complex projective surfaces admitting genus two fibrations over elliptic curves. We observe that, a surface admitting a smooth fibration as above is elliptic and we employ results on the moduli of polarized elliptic surfaces, to construct moduli spaces of these smooth fibrations. In the case of nonsmooth fibrations, we relate the moduli spaces to the Hurwitz schemes H(1, X(d), n) of morphisms of degree n from elliptic curves to the modular curve X(d), d ≥ 3. Ultimately, we show that the moduli spaces in the nonsmooth case are fiber spaces over the affine line A 1 with fibers determined by the components of H(1, X(d), n). 1

2006

Mathematics of Computation, 2002

We prove that there are exactly 149 genus two curves C defined over Q such that there exists a nonconstant morphism π : X 1 (N) → C defined over Q and the jacobian of C is Q-isogenous to the abelian variety A f attached by Shimura to a newform f ∈ S 2 (Γ 1 (N)). We determine the corresponding newforms and present equations for all these curves.

Illinois Journal of Mathematics, 2007

We study differential equations satisfied by modular forms of two variables associated to Γ 1 × Γ 2 , where Γ i (i = 1, 2) are genus zero subgroups of SL 2 (R) commensurable with SL 2 (Z), e.g., Γ 0 (N) or Γ 0 (N) * for some N. In some examples, these differential equations are realized as the Picard-Fuchs differential equations of families of K3 surfaces with large Picard numbers, e.g., 19, 18, 17, 16. Our method rediscovers some of the Lian-Yau examples of "modular relations" involving power series solutions to the second and the third order differential equations of Fuchsian type in [14], [15].

Mathematical Research Letters, 2007

Michigan Mathematical Journal

The combinatorial type of an elliptic K3 surface with a zero section is the pair of the ADE-type of the singular fibers and the torsion part of the Mordell-Weil group. We determine the set of connected components of the moduli of elliptic K3 surfaces with fixed combinatorial type. Our method relies on the theory of Miranda and Morrison on the structure of a genus of even indefinite lattices, and computer-aided calculations of p-adic quadratic forms.

1982

Chapter O. Historical background, motivations and outline of the contents. Chapter I. Hodge structures attached to primitive forms of weight O. Definitions and notations. 1. Hodge structures attached to Hilbert modular surfaces. Hodge structures attached to primitive forms of weight 2. Nonholomorphic involutive automorphisms of Hilbert modular surfaces. Period relation of Riemann-Hodge. Chapter II. Abelian varieties attached to primitive forms. Abelian varieties attached to Clifford algebras. 6. The period moduli of the isogeny classes A}-(J) and A}CI(J). 7. Main theorem A and its corollaries. 8. Selfconjugate forms and algebraic cycles. 9. Abelian varieties attached to non-selfconjugate forms. Chapter III.Correspondence between real Nebentype elliptic modular forms and Hilbert modular forms. 10. Weil representation and theta series. 11. Construction of real Nebentype elliptic modular forms. 12. Construction of Hilbert modular forms. 13. The adjointness formula. Chapter IV. Period relation for the lifting of modular forms and ix 2. 21 41 transcendental cycles. 14. Hodge structures attached to real Nebentype elliptic modular forms of weight 2. 15. Construction of 2-cycles. 16. Arithmetic index theorems. 17. Period relation for the Doi-Naqanuma lifting and Main Theorem B. 18. Selfconju9ate forms and transcendental cycles. 19. Notes on i-adic cohomology groups of certain Hilbert modular surfaces. 20. Remarks. Bibliography.

Using some theory of (rational) elliptic surfaces plus elementary properties of a Mordell-Weil group regarded as module over the endomorphism ring of a (CM) elliptic curve, we present examples of such surfaces with j-invariant zero. In particular, a K3 example is given with the maximal possible rank 18, plus that many explicit independent sections on it. Several other high rank cases are discussed; for instance Shioda's example with rank 68.