On elliptic modular surfaces

Shioda, Tetsuji

doi:10.2969/jmsj/02410020

Cited by 366 publications

(289 citation statements)

References 18 publications

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“…For detail, see [Sho1] and [Sho2]. Let S be a compact complex surface and let C be an algebraic curve.…”

Section: The Mordell-weil Group Of Sectionsmentioning

confidence: 99%

Period Differential Equations for the Families of K3 Surfaces With Two Parameters Derived From the Reflexive Polytopes

Nagano¹

2012

Kyushu J. Math.

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Abstract. In this paper, we study the period mappings for the families of K3 surfaces derived from the three-dimensional reflexive polytopes with five vertices. We determine the lattice structures, the period differential equations and the projective monodromy groups. Moreover, we show that one of our period differential equations coincides with the uniformizing differential equation of the Hilbert modular orbifold for the field Q( √ 5). IntroductionA K3 surface S is characterized by the condition K S = 0 and simply connectedness. It means that a K3 surface is a two-dimensional Calabi-Yau manifold. Batyrev [Ba] introduced the notion of the reflexive polytope for the study of the Calabi-Yau manifold.In this article, we use the three-dimensional reflexive polytopes with at most terminal singularities. These polytopes with five-vertices are given as where the column vectors correspond to the coordinates of the vertices (see [Ot] or [KS]).

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“…For detail, see [Sho1] and [Sho2]. Let S be a compact complex surface and let C be an algebraic curve.…”

Section: The Mordell-weil Group Of Sectionsmentioning

confidence: 99%

Period Differential Equations for the Families of K3 Surfaces With Two Parameters Derived From the Reflexive Polytopes

Nagano¹

2012

Kyushu J. Math.

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“…In order to try and find a seven-square magic square, (12) was searched for solutions also satisfying at least one of the equations (13). The only solution (up to symmetry) occurred at λ = 13 with (p, q) = (9, 2), giving rise to the square at (1).…”

Section: Remarkmentioning

confidence: 99%

“…Denote the Néron-Severi group of the surface S over C by NS(S, C). Then NS(S, C) is a finitely generated Z-module, and it follows from Shioda [13] that rank NS(S, C) = rank E λ (C(λ)) + 2 + 4 · (2 − 1) + 4 · (4 − 1).…”

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confidence: 99%

On squares of squares II

Bremner

2001

Acta Arith.

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“…The surface E is an elliptic modular surface associated to P SL 2 (Z) (see [Sh1]), up to an isomorphism of the base. The formula was proved by Stiller [St] in the case where r = 12n, in general, by Shioda [Sh2], and later, in the author's Ph.D. thesis [F2].…”

Section: Remarkmentioning

confidence: 99%

Computing Mordell-Weil ranks of cyclic covers of elliptic surfaces

Fastenberg

2001

Proc. Amer. Math. Soc.

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Abstract. We give explicit formulas for computing the Mordell-Weil ranks of the elliptic surfaces Er : Y 2 = X 3 + a(t r )X + b(t r ) subject to some restrictions on the surface E 1 .Let π : E → P 1 be a smooth complex relatively minimal elliptic surface with section. For each positive integer r, define π r : E r → P 1 to be the relatively minimal compactification of the Neron model of the generic fiber of E × P 1 P 1 of the pullback of E by the morphism of P 1 defined by t → t r . The main result of [F1], [F2] is that under certain conditions on E, the rank of the Mordell-Weil group of sections of E r is bounded independently of r. The purpose of this paper is to use this result to compute or give explicit upper bounds on the rank of E r (P 1 ) for several examples.We and assume that γ < 1. For each positive integer r let φ r : P 1 → P 1 be the morphism defined by φ r (t) = t r , and let E r be the pullback of E via φ r . LetIt is not known whether the rank of an elliptic surface over P 1 is bounded. The largest known rank for an elliptic surface over P 1 is 56 (see Example 1, and [Sh2], [St]). We will see that we can often use a variant on the proof of Theorem 1 to get an explicit formula for the rank of the surfaces E r , so that the technique outlined below can be used to look for elliptic surfaces with large rank. We have the following corollary to Theorem 1.

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On elliptic modular surfaces

Cited by 366 publications

References 18 publications

Period Differential Equations for the Families of K3 Surfaces With Two Parameters Derived From the Reflexive Polytopes

Period Differential Equations for the Families of K3 Surfaces With Two Parameters Derived From the Reflexive Polytopes

On squares of squares II

Computing Mordell-Weil ranks of cyclic covers of elliptic surfaces

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