Explicit Equations of Some Elliptic Modular Surfaces

Top, Jaap; Yui, Noriko

doi:10.1216/rmjm/1181068772

Cited by 13 publications

(26 citation statements)

References 14 publications

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“…Because of the fibration over P 1 C , the j-function will explicitly depend on the projective coordinate of this base Riemann sphere and can thus be treated as a rational map from the base P 1 C to its range P 1 . All of these j-functions are given in [10] and certain simplifications, in [7,19]. Generalizing [7,11], we find the nice fact that Proposition 2.1.…”

Section: Modular Surfaces Ramification Data and J-functionsmentioning

confidence: 73%

Modular subgroups, dessins d’enfants and elliptic K3 surfaces

He¹,

McKay²,

Read³

2013

LMS J. Comput. Math.

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We consider the 33 conjugacy classes of genus zero, torsion-free modular subgroups, computing ramification data and Grothendieck's dessins d'enfants. In the particular case of the index 36 subgroups, the corresponding Calabi-Yau threefolds are identified, in analogy with the index 24 cases being associated to K3 surfaces. In a parallel vein, we study the 112 semi-stable elliptic fibrations over P 1 as extremal K3 surfaces with six singular fibres. In each case, a representative of the corresponding class of subgroups is identified by specifying a generating set for that representative.

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Section: Modular Surfaces Ramification Data and J-functionsmentioning

confidence: 73%

Modular subgroups, dessins d’enfants and elliptic K3 surfaces

He¹,

McKay²,

Read³

2013

LMS J. Comput. Math.

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“…The recurrence for this example is (16) (n + 1) 2 u n+1 = (10n 2 + 2)u n−1 − 9(n − 1) 2 u n−3 u 0 = 1, u 1 = 0, u 2 = 3, u 3 = 0, u 4 = 15.…”

Section: Further Boundsmentioning

confidence: 98%

“…Next, consider the elliptic family associated to Γ(8; 4, 1, 2). From [16] we know that the defining equation is y 2 = x 3 − 2(8t 4 − 16t 3 + 16t 2 − 8t + 1)x 2 + (2t − 1) 4 (8t 2 − 8t + 1)x.…”

Section: Asymptotic Behaviormentioning

confidence: 99%

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Arithmetic properties of Picard–Fuchs equations and holonomic recurrences

Walker

2013

Journal of Number Theory

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The coefficient series of the holomorphic Picard-Fuchs differential equation associated with the periods of elliptic curves often have surprising number-theoretic properties. These have been widely studied in the case of the torsion-free, genus zero congruence subgroups of index 6 and 12 (e.g. the Beauville families). Here, we consider arithmetic properties of the Picard-Fuchs solutions associated to general elliptic families, with a particular focus on the index 24 congruence subgroups. We prove that elliptic families with rational parameters admit linear reparametrizations such that their associated Picard-Fuchs solutions lie in Z [[t]]. A sufficient condition is given such that the same holds for holomorphic solutions at infinity. An Atkin-Swinnerton-Dyer congruence is proven for the coefficient series attached to Γ1(7). We conclude with a consideration of asymptotics, wherein it is proved that many coefficient series satisfy asymptotic expressions of the form un ∼ ℓλ n /n. Certain arithmetic results extend to the study of general holonomic recurrences.

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“…We also include a reference, but naturally the given models are far from unique. Other models may be found in [Elkies 2008b;Schütt 2007a;Top and Yui 2007] [Elkies 2008b, §5] For d = −8, −12, it was shown in [Schütt 2007a, §7] that the named fibrations are defined over ‫.ޑ‬ To obtain Picard rank 20 over ‫,ޑ‬ it suffices to apply a quadratic twist as in Example 8 such that the fibre of type I 4 or I 3 , respectively, becomes split-multiplicative.…”

Section: Existence Of K3 Surfaces Of Picard Rank 20 Over ‫ޑ‬mentioning

confidence: 99%

K3 surfaces with Picard rank 20

Schütt

2010

ANT

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We determine all complex K3 surfaces with Picard rank 20 over ‫.ޑ‬ Here the Néron-Severi group has rank 20 and is generated by divisors which are defined over ‫.ޑ‬ Our proof uses modularity, the Artin-Tate conjecture and class group theory. With different techniques, the result has been established by Elkies to show that Mordell-Weil rank 18 over ‫ޑ‬ is impossible for an elliptic K3 surface. We apply our methods to general singular K3 surfaces, that is, those with Néron-Severi group of rank 20, but not necessarily generated by divisors over ‫.ޑ‬

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Explicit Equations of Some Elliptic Modular Surfaces

Cited by 13 publications

References 14 publications

Modular subgroups, dessins d’enfants and elliptic K3 surfaces

Modular subgroups, dessins d’enfants and elliptic K3 surfaces

Arithmetic properties of Picard–Fuchs equations and holonomic recurrences

K3 surfaces with Picard rank 20

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