Normal forms, K3 surface moduli, and modular
                    parametrizations

Clingher, Adrian; Doran, Charles F.; Lewis, Jacob; Whitcher, Ursula

doi:10.1090/crmp/047/06

Cited by 15 publications

(26 citation statements)

References 23 publications

(54 reference statements)

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“…Clingher, Doran, Lewis and Whitcher [CDLW09] have shown that M -polarized K3 surfaces have a coarse moduli space given by the locus d = 0 in the weighted projective space WP(2, 3, 6) with weighted coordinates (a, b, d). Thus, by normalizing d = 1, we may associate a pair of complex numbers (a, b) to an M -polarized K3 surface (X, i).…”

Section: Undoing the Kummer Constructionmentioning

confidence: 99%

Families of Lattice Polarized K3 Surfaces with Monodromy

Doran¹,

Harder²,

Novoseltsev³

et al. 2015

Int Math Res Notices

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Abstract. We extend the notion of lattice polarization for K3 surfaces to families over a (not necessarily simply connected) base, in a way that gives control over the action of monodromy on the algebraic cycles, and discuss the uses of this new theory in the study of families of K3 surfaces admitting fibrewise symplectic automorphisms. We then give an application of these ideas to the study of Calabi-Yau threefolds admitting fibrations by lattice polarized K3 surfaces.

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Section: Undoing the Kummer Constructionmentioning

confidence: 99%

Families of Lattice Polarized K3 Surfaces with Monodromy

Doran¹,

Harder²,

Novoseltsev³

et al. 2015

Int Math Res Notices

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“…In this case F L ⊥ is isomorphic to the symmetric product of two copies of the classical modular curve; see [10] or [11] for more details.…”

Section: 1mentioning

confidence: 99%

The Geometry and Moduli of K3 Surfaces

Harder

Thompson

2015

Calabi-Yau Varieties: Arithmetic, Geometry and Physics

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Abstract. These notes will give an introduction to the theory of K3 surfaces. We begin with some general results on K3 surfaces, including the construction of their moduli space and some of its properties. We then move on to focus on the theory of polarized K3 surfaces, studying their moduli, degenerations and the compactification problem. This theory is then further enhanced to a discussion of lattice polarized K3 surfaces, which provide a rich source of explicit examples, including a large class of lattice polarizations coming from elliptic fibrations. Finally, we conclude by discussing the ample and Kähler cones of K3 surfaces, and give some of their applications.

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“…In order to calculate I d,i 's using Theorem 3 we can proceed as follows: We use the Gröbner basis algorithm and find the irreducible components of the affine variety given by the ideal J d,1 , J d,2 , J d, 3 and among them identify the variety V d . In deg(J 2,1 ) = 42, deg(J 2,2 ) = 40, deg(J 2,3 ) = 69 and calculating the Gröbner basis of the ideal J d,1 , J d,2 , J d, 3 is a huge amount of computations.…”

Section: Proof Of Theoremmentioning

confidence: 99%

“…In deg(J 2,1 ) = 42, deg(J 2,2 ) = 40, deg(J 2,3 ) = 69 and calculating the Gröbner basis of the ideal J d,1 , J d,2 , J d, 3 is a huge amount of computations. We use the q-expansion of t i 's and we calculate I d,i , i = 1, 2, 3, d = 2, 3.…”

Section: Proof Of Theoremmentioning

confidence: 99%

“…In the present article we describe Hecke operators for such quasi-modular forms, and certain differential ideals related to modular curves. In [3] the authors describe a differential equation in the j-invariant of two elliptic curves which is tangent to all modular curves of degree d isogenies of elliptic curves. This differential equation can be derived from the Schwarzian differential equation of the j-function and the latter can be calculated from the Ramanujan differential equation between Eisenstein series.…”

Section: Introductionmentioning

confidence: 99%

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Quasi-modular forms attached to elliptic curves: Hecke operators

Movasati

2015

Journal of Number Theory

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In this article we describe Hecke operators on the differential algebra of geometric quasi-modular forms. As an application for each natural number d we construct a vector field in six dimensions which determines uniquely the polynomial relations between the Eisenstein series of weight 2, 4 and 6 and their transformation under multiplication of the argument by d, and in particular, it determines uniquely the modular curve of degree d isogenies between elliptic curves.

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Normal forms, K3 surface moduli, and modular parametrizations

Cited by 15 publications

References 23 publications

Families of Lattice Polarized K3 Surfaces with Monodromy

Families of Lattice Polarized K3 Surfaces with Monodromy

The Geometry and Moduli of K3 Surfaces

Quasi-modular forms attached to elliptic curves: Hecke operators

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