Hostname: page-component-cd9895bd7-hc48f Total loading time: 0 Render date: 2024-12-24T00:06:00.622Z Has data issue: false hasContentIssue false
  • English
  • Français

On modularity of rigid and nonrigid Calabi-Yau varieties associated to the Root Lattice A4

Published online by Cambridge University Press:  11 January 2016

Klaus Hulek  and
Helena Verrill
Klaus Hulek
Affiliation:
Institut für Mathematik (C) Universität Hannover, Welfengarten 1 30060 Hannover, [email protected]
Helena Verrill
Affiliation:
Department of Mathematics, Louisiana State University, Baton Rouge, LA 70803-4918, [email protected]
Rights & Permissions [Opens in a new window]

Abstract

Core share and HTML view are not available for this content. However, as you have access to this content, a full PDF is available via the ‘Save PDF’ action button.

We prove the modularity of four rigid and three nonrigid Calabi-Yau threefolds associated with the A4 root lattice.

Keywords

14J3214G2511F0311F2311F32
Type
Research Article
Information
Nagoya Mathematical Journal , Volume 179 , 2005 , pp. 103 - 146
Copyright
Copyright © Editorial Board of Nagoya Mathematical Journal 2005

References

[Ba] Batyrev, V. V., Dual polyhedra and mirror symmetry for Calabi-Yau hypersurfaces in toric varieties, Journal of Algebraic Geometry, 3 (1994), 493–535.Google Scholar
[Bo] Borcea, C., Polygon spaces, tangents to quadrics and special Lagrangians, Ober-wolfach Report, 42 (2004), 2181–2183.Google Scholar
[BCDT] Breuil, C., Conrad, B., Diamond, F. and Taylor, R., On the modularity of elliptic curves over : wild 3-adic exercises, J. Am. Math. Soc., 14 (2001), no. 4, 843–939.Google Scholar
[BCP] Bosma, W., Cannon, J. and Playoust, C., The Magma algebra system. I. The user language, J. Symbolic Comput., 24 (1997), no. 3-4, 235–265.CrossRefGoogle Scholar
[C] Clemens, H., Double solids, Adv. in Math., 47 (1983), 107–230.Google Scholar
[CS] Consani, C. and Scholten, J., Arithmetic on a quintic threefold, International Journal of Mathematics, 12 (2001), no. 8, 943–972.CrossRefGoogle Scholar
[CM] Cynk, S. and Meyer, C., Geometry and Arithmetic of certain Double Octic Calabi-Yau Manifolds, preprint, math.AG/0304121, Canad. Math. Bull., 48 (2005), no. 2, 180–194.Google Scholar
[DM] Dieulefait, L. and Manoharmayum, J., Modularity of rigid Calabi-Yau threefolds over , Calabi-Yau varieties and mirror symmetry (Toronto, ON, 2001) (Yui, N. and James, J. D., eds.), Fields Institute Communications 38, Amer. Math. Soc., Providence, RI (2003), pp. 159–166.Google Scholar
[DL] Dolgachev, I. and Lunts, V., A character formula for the representation of a Weyl group in the cohomology of the associated toric variety, J. Algebra, 168 (1994), no. 3, 741–772.CrossRefGoogle Scholar
[EH] Eisenbud, D. and Harris, J., The geometry of schemes, Graduate Texts in Mathematics 197, Springer-Verlag, New York, 2000.Google Scholar
[FM] Fontaine, J.-M. and Mazur, B., Geometric Galois representations, Elliptic curves, modular forms, & Fermat’s last theorem, Series in Number Theory, I, Internat. Press, Cambridge, MA (1995), pp. 41–78.Google Scholar
[FK] Freitag, E. and Kiehl, R., Ètale cohomology and the Weil conjecture, Ergeb. Math. Grenzgeb. (3) 3, Springer Verlag, 1988.Google Scholar
[F] Fulton, W., Intersection Theory, Second Edition, Springer Verlag, 1998.Google Scholar
[GW] Geemen, B. van and Werner, J., Nodal quintics in ℙ4, Arithmetic of complex manifolds, Proceedings Erlangen 1988 (Barth, W.-P. and Lange, H., eds.), Springer Lecture Notes 1399 (1989), pp. 48–59.Google Scholar
[Ha] Hartshorne, R., Algebraic geometry, Graduate Texts in Mathematics 52, Springer-Verlag, New York, 1977.Google Scholar
[H] Hirschfeld, J. W. P., Projective geometries over finite fields, Oxford Mathematical Monographs, The Clarendon Press, Oxford University Press, New York, 1979.Google Scholar
[Ko] Kollár, J., Flops, Nagoya Math. J., 113 (1989), 15–36.CrossRefGoogle Scholar
[J] Jones, J., Tables of number fields with prescribed ramification. http://math.la.asu.edu/~jj/numberfields/.Google Scholar
[Li] Livné, R., Cubic exponential sums and Galois representations, Contemporary Mathematics, 67 (1987), 247–261.CrossRefGoogle Scholar
[LY] Livné, R. and Yui, N., The modularity of certain non-rigid Calabi-Yau threefolds, math.AG/0304497, to appear, J. Math. Kyoto Univ. Google Scholar
[Lu] Ludwig, K., Torische Varietäten und Calabi-Yau-Mannigfaltigkeiten, Diplomarbeit, Institut für Mathematik, Universität Hannover (2003). (available from http://www-ifm.math.uni-hannover.de/~hulek/AG/data/DiplomarbeitLudwig.pdf).Google Scholar
[P] Procesi, C., the toric variety associated to the Weyl chambers, in “mots” (Lothaire, M., ed.), Hermés, Paris (1990), pp. 153–161.Google Scholar
[Se1] Serre, J.-P., Cours d’arithmétique, Deuxiéme édition revue et corrigée, Le Mathématicien, No. 2, Presses Universitaires de France, Paris, 1977.Google Scholar
[Se2] Serre, J.-P., Représentations l-adiques, Kyoto Int. Symposium on algebraic number theory, Japan Soc. for the Promotion of Science (1977), pp. 177–193. (oeuvres, t. III, 384–400).Google Scholar
[Sc] Schoen, C., On fiber products of rational elliptic surfaces with section, Math. Z., 197 (1988), no. 2, 177–199.Google Scholar
[St] Stein, W. A., Explicit approaches to modular abelian varieties, U. C. Berkeley Ph.D. thesis (2000).Google Scholar
[V] Verrill, H. A., Root lattices and pencils of varieties, J. Math. Kyoto Univ., 36 (1996), 423–446.Google Scholar
[We] Werner, J., Kleine Auflösungen spezieller dreidimensionaler Varietäten, Bonner Mathematische Schriften 186, 1987.Google Scholar
[Wi] Wiles, A., Modular elliptic curves and Fermat’s Last Theorem, Ann. Math. (2), 141 (1995), no. 3, 443–551.CrossRefGoogle Scholar
[Y1] Yui, N., The arithmetic of certain Calabi-Yau varieties over number fields, The arithmetic and geometry of algebraic cycles (Banff, AB, 1998), NATO Sci. Ser. C, Math. Phys. Sci., 548, Kluwer Acad. Publ., Dordrecht (2000), pp. 515–560.Google Scholar
[Y2] Yui, N., Update on the modularity of Calabi-Yau varieties, With an appendix by Helena Verrill, Calabi-Yau varieties and mirror symmetry (Toronto, ON, 2001) (Yui, N. and James, J. D., eds.), Fields Institute Communications 38, Amer. Math. Soc., Providence, RI (2003), pp. 159–166.Google Scholar