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15
ELLIPTIC FIBRATIONS OF SOME EXTREMAL K3 SURFACES
, 2005
"... This paper is concerned with the construction of extremal elliptic K3 surfaces. It gives a complete treatment of those fibrations which can be derived from rational elliptic surfaces by easy manipulations of their Weierstrass equations. In particular, this approach enables us to find explicit equati ..."
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Cited by 11 (8 self)
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This paper is concerned with the construction of extremal elliptic K3 surfaces. It gives a complete treatment of those fibrations which can be derived from rational elliptic surfaces by easy manipulations of their Weierstrass equations. In particular, this approach enables us to find explicit equations for 38 semistable extremal elliptic K3 fibrations, 32 of which are indeed defined over Q. They are realized as pullback of nonsemistable extremal rational elliptic surfaces via base change. This is related to work of J. Top and N. Yui which exhibited the same procedure for the semistable extremal rational elliptic surfaces.
Modularity of CalabiYau Varieties
 GLOBAL ASPECTS OF COMPLEX GEOMETRY
, 2006
"... In this paper we discuss recent progress on the modularity of CalabiYau varieties. We focus mostly on the case of surfaces and threefolds. We will also discuss some progress on the structure of the Lfunction in connection with mirror symmetry. Finally, we address some questions and open problems. ..."
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Cited by 7 (1 self)
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In this paper we discuss recent progress on the modularity of CalabiYau varieties. We focus mostly on the case of surfaces and threefolds. We will also discuss some progress on the structure of the Lfunction in connection with mirror symmetry. Finally, we address some questions and open problems.
The modularity of K3 surfaces with nonsymplectic group actions
, 2009
"... We consider complex K3 surfaces with a nonsymplectic group acting trivially on the algebraic cycles. Vorontsov and Kondō classified those K3 surfaces with transcendental lattice of minimal rank. The purpose of this note is to study the Galois representations associated to these K3 surfaces. The ra ..."
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Cited by 7 (4 self)
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We consider complex K3 surfaces with a nonsymplectic group acting trivially on the algebraic cycles. Vorontsov and Kondō classified those K3 surfaces with transcendental lattice of minimal rank. The purpose of this note is to study the Galois representations associated to these K3 surfaces. The rank of the transcendental lattices is even and varies from 2 to 20, excluding 8 and 14. We show that these K3 surfaces are dominated by Fermat surfaces, and hence they are all of CM type. We will establish the modularity of the Galois representations associated to them. Also we discuss mirror symmetry for these K3 surfaces in the sense of Dolgachev, and show that a mirror K3 surface exists with one exception.
Arithmetic of a singular K3 surface
 Michigan J. of Math., preprint
"... This paper is concerned with the arithmetic of the elliptic K3 surface with configuration [1,1,1,12,3*]. We determine the newforms and zetafunctions associated to X and its twists. We verify conjectures of Tate, Artin and Shioda for the reductions of X at 2 and 3. ..."
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Cited by 6 (5 self)
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This paper is concerned with the arithmetic of the elliptic K3 surface with configuration [1,1,1,12,3*]. We determine the newforms and zetafunctions associated to X and its twists. We verify conjectures of Tate, Artin and Shioda for the reductions of X at 2 and 3.
K3 surfaces with Picard rank 20 over Q
"... We compute all K3 surfaces with Picard rank 20 over Q. Our proof uses modularity, the ArtinTate conjecture and class group theory. With different techniques, the result has been established by Elkies to show that MordellWeil rank 18 over Q is impossible for an elliptic K3 surface. We also apply ou ..."
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Cited by 4 (1 self)
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We compute all K3 surfaces with Picard rank 20 over Q. Our proof uses modularity, the ArtinTate conjecture and class group theory. With different techniques, the result has been established by Elkies to show that MordellWeil rank 18 over Q is impossible for an elliptic K3 surface. We also apply our methods to general singular K3 surfaces, i.e. with geometric Picard rank 20, but not necessarily over Q.
K3 surfaces with Picard rank 20
, 2008
"... We determine all complex K3 surfaces with Picard rank 20 over Q. Here the NéronSeveri group has rank 20 and is generated by divisors which are defined over Q. Our proof uses modularity, the ArtinTate conjecture and class group theory. With different techniques, the result has been established by E ..."
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Cited by 2 (0 self)
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We determine all complex K3 surfaces with Picard rank 20 over Q. Here the NéronSeveri group has rank 20 and is generated by divisors which are defined over Q. Our proof uses modularity, the ArtinTate conjecture and class group theory. With different techniques, the result has been established by Elkies to show that MordellWeil rank 18 over Q is impossible for an elliptic K3 surface. We also apply our methods to general singular K3 surfaces, i.e. with NéronSeveri group of rank 20, but not necessarily generated by divisors over Q.
Arithmetic of K3 surfaces
 Jahresber. Deutsch. Math. Verein
"... We review recent developments in the arithmetic of K3 surfaces. Our focus lies on aspects of modularity, Picard number and rational points. Throughout we emphasise connections to geometry. ..."
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Cited by 2 (0 self)
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We review recent developments in the arithmetic of K3 surfaces. Our focus lies on aspects of modularity, Picard number and rational points. Throughout we emphasise connections to geometry.