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Higher-Dimensional Modular Calabi–Yau Manifolds

Published online by Cambridge University Press:  20 November 2018

S. Cynk
Affiliation:
Instytut Mathematyki, Uniwersytetu Jagiellońskiego, Ul. Reymonta 4, 30-059 Kraków, Poland e-mail: slawomir.cynk@im.uj.edu.pl
K. Hulek
Affiliation:
Institut für algebraische Geometrie, Leibniz Universität Hannover, Welfengarten 1, 30167 Hannover, Germany e-mail: hulek@math.uni-hannover.de
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Abstract

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We construct several examples of higher-dimensional Calabi–Yau manifolds and prove their modularity.

Keywords

Type
Research Article
Copyright
Copyright © Canadian Mathematical Society 2007

References

[1] Ahlgren, S., The points of a certain fivefold over finite fields and the twelfth power of the eta function. Finite Fields Appl. 8(2002), no. 1, 1833.Google Scholar
[2] Ahlgren, A., Ono, K., and Penniston, D., Zeta functions of an infinite family of K3 surfaces. Amer. J. Math. 124(2002), no. 2, 353368.Google Scholar
[3] Borcea, C., Calabi–Yau threefolds and complex multiplication. In: Essays on Mirror Manifolds. Int. Press, Hong Kong, 1992, pp. 489502.Google Scholar
[4] Borcea, C., K3 surfaces with involution and mirror pairs of Calabi–Yau manifolds. In: Mirror Symmetry, AMS/IP Stud. Adv. Math. 1, American Mathematical Society, Providence, RI, 1997, pp. 717743.Google Scholar
[5] Cynk, S. and Meyer, C., Geometry and arithmetic of certain double octic Calabi–Yau manifolds. Canad. Math. Bull. 48(2005), no. 2, 180194.Google Scholar
[6] Dieulefait, L. and Manoharmayum, J., Modularity of rigid Calabi–Yau threefolds over . In: Calabi–Yau Varieties and Mirror Symmetry, Fields Inst. Commun. 38, American Mathematical Society, Providence, RI, 2003, pp. 159166.Google Scholar
[7] Fontaine, J.-M. and Mazur, B., Geometric Galois representations. In: Elliptic Curves, Modular Forms, & Fermat's Last Theorem, Ser. Number Theory 1, Int. Press, Cambridge, MA, 1995, pp. 4178.Google Scholar
[8] Hulek, K. and Verrill, H., On modularity of rigid and nonrigid Calabi–Yau varieties associated to the root lattice A4 . Nagoya Math. J. 179(2005), 103146.Google Scholar
[9] Hulek, K. and Verrill, H., On the modularity of Calabi–Yau threefolds containing elliptic ruled surfaces, In: Mirror Symmetry 5, AMS/IP Studies in Advanced Mathematics 38, American Mathematical Society, Providence, RI, 2006, pp. 1934.Google Scholar
[10] Kim, H. H. and Shahidi, F., Functorial products for GL2 ×GL3 and the symmetric cube for GL2 . Ann. of Math. 155(2002), no. 3, 837893.Google Scholar
[11] Livné, R., Motivic orthogonal two-dimensional representations of . Israel J. Math. 92(1995), no. 1–3, 149156.Google Scholar
[12] Livné, R. and Yui, N., The modularity of certain non-rigid Calabi–Yau threefolds. J. Math. Kyoto Univ. 45(2005), no. 4, 645665.Google Scholar
[13] Meyer, C., Modular Calabi–Yau Threefolds, Fields Institute Monograph 22, American Mathematical Society, Providence, RI, 2005.Google Scholar
[14] Miranda, R. and Persson, U., On extremal rational elliptic surfaces. Math. Z. 193(1986), no. 4, 537558.Google Scholar
[15] Ribet, K., Galois representations attached to eigenforms with Nebentypus. In: Modular Functions of One Variable, V, Lecture Notes in Math. 601, Springer, Berlin, 1977. pp. 1751.Google Scholar
[16] Serre, J.-P., Abelian l-adic Representations and Elliptic Curves. Research Notes in Mathematics 7, A K Peters, Wellesley, MA, 1998.Google Scholar
[17] Shioda, T. and Inose, H., On singular K3surface. In: Complex Analysis and Algebraic Geometry, Iwanami Shoten, Tokyo, 1977, pp. 119136.Google Scholar
[18] Ueno, K., Classification Theory of Algebraic Varieties and Compact Complex Spaces. Lecture Notes in Mathematics 439, Springer-Verlag, Berlin, 1975.Google Scholar
[19] Voisin, C., Miroirs et involutions sur les surfaces K3. In: Journées de Géométrie Algébrique d’Orsay. Astérisque 218, 1993, 273323.Google Scholar
[20] Wiles, A., Modular elliptic curves and Fermat's last theorem. Ann. of Math. 141(1995), no. 3, 443551.Google Scholar
[21] Yui, N., Update on the modularity of Calabi–Yau varieties. In: Calabi–Yau Varieties and Mirror Symmetry, Fields Inst. Commun. 38, American Mathematical Society, Providence, RI, 2003, pp. 307362.Google Scholar