Hostname: page-component-78c5997874-dh8gc Total loading time: 0 Render date: 2024-11-04T02:25:51.187Z Has data issue: false hasContentIssue false
  • English
  • Français

A Modular Quintic Calabi–Yau Threefold of Level 55

Published online by Cambridge University Press:  20 November 2018

Edward Lee
Edward Lee*
Affiliation:
School of Mathematics, Aras Na Laoi, University College Cork, Cork, Ireland email: [email protected], [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.

In this note we search the parameter space of Horrocks–Mumford quintic threefolds and locate a Calabi–Yau threefold that is modular, in the sense that the $L$-function of its middle-dimensional cohomology is associated with a classical modular form of weight 4 and level 55.

Keywords

14J1511F2314J3211G40Calabi-Yau threefold, non-rigid Calabi-Yau threefold, two-dimensional Galois representation, modular variety, Horrocks-Mumford vector bundle
Type
Research Article
Information
Canadian Journal of Mathematics , Volume 63 , Issue 3 , 01 June 2011 , pp. 616 - 633
Copyright
Copyright © Canadian Mathematical Society 2011

References

[1] Aure, A., Surfaces on quintic threefolds associated to the Horrocks-Mumford bundle. In: Arithmetic of complex manifolds (Erlangen, 1988), Lecture Notes in Math., 1399, Springer-Verlag, Berlin, 1989, pp. 1–9.Google Scholar
[2] Breuil, C., Conrad, B., Diamond, F., and Taylor, R., On the modularity of elliptic curves over Q: Wild 3-adic exercises. J. Amer. Math. Soc. 14(2001), no. 4, 843–939. doi:10.1090/S0894-0347-01-00370-8 Google Scholar
[3] Barth, W., Hulek, K., and Moore, R., Degenerations of Horrocks-Mumford surfaces. Math. Ann. 277(1987), no. 4, 735–755. doi:10.1007/BF01457868 Google Scholar
[4] Barth, W. and Moore, R., Geometry in the space of Horrocks-Mumford surfaces. Topology 28(1989), no. 2, 231–245. doi:10.1016/0040-9383(89)90023-2 Google Scholar
[5] Cynk, S. and Meyer, C., Modularity of some non-rigid double octic Calabi–Yau threefolds. Rocky Mountain J. Math. 38(2008), no. 6, 1937–1958. doi:10.1216/RMJ-2008-38-6-1937 Google Scholar
[6] Fisher, T., Some examples of 5 and 7 descent for elliptic curves over Q. J. Eur. Math. Soc. (JEMS) 3(2001), no. 2, 169–201. doi:10.1007/s100970100030 Google Scholar
[7] Freitag, E. and Kiehl, R., Étale cohomology and the Weil conjecture. Ergebnisse der Mathematik und ihrer Grenzgebiete (3), 13, Springer-Verlag, Berlin, 1988.Google Scholar
[8] Griffiths, P. and Harris, J., Principles of algebraic geometry. Pure and Applied Mathematics, Wiley-Interscience, New York, 1978.Google Scholar
[9] Gross, M. and Popescu, S., Calabi–Yau threefolds and moduli of abelian surfaces. I. CompositioMath. 127(2001), 169–228. doi:10.1023/A:1012076503121 Google Scholar
[10] Gouvea, F. and Yui, N., Rigid Calabi–Yau threefolds over Q are modular. arXiv:0902.1466v3 [math.NT].Google Scholar
[11] Hulek, K., The Horrocks-Mumford bundle. In: Vector bundles and algebraic geometry (Durham 1993), London Math. Soc. Lecture Note Ser., 208, Cambridge University Press, Cambridge, 1993, pp. 139–177.Google Scholar
[12] Horrocks, G. and Mumford, D., A rank 2 vector bundle on P4 with 15, 000 symmetries. Topology 12(1973), 63–81. doi:10.1016/0040-9383(73)90022-0 Google Scholar
[13] Hulek, K. and Lange, H., The Hilbert modular surface for the ideal (p5) and the Horrocks-Mumford bundle. Math. Z. 198(1988), 95–116. doi:10.1007/BF01183042 Google Scholar
[14] 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), 103–146.Google Scholar
[15] Jones, J., Tables of Number Fields With Prescribed Ramification, http://hobbes.la.asu.edu/NFDB/. Google Scholar
[16] Khare, C. and Wintenberger, J.-P., On Serre's conjecture for 2-dimensional mod p representations of Gal(Q/Q). Ann. of Math. (2) 169(2009), no. 1, 229–253. doi:10.4007/annals.2009.169.229 Google Scholar
[17] Lee, E., A Modular non-rigid Calabi–Yau threefold. In: Mirror symmetry. V, AMS/IP Stud. Adv. Math. 38, American Mathematical Society, Providence, RI, 2006, pp. 89–122.Google Scholar
[18] Livné, R., Cubic exponential sums and Galois representations. In: Current trends in arithmetical algebraic geometry (Arcata, Calif., 1985), Contemporary Mathematics, 67, American Mathematical Society, Providence, RI, 1987, pp. 247–261.Google Scholar
[19] Meyer, C., Modular Calabi–Yau threefolds. Fields Institute Monographs, 22, American Mathematical Society, Providence, RI, 2005.Google Scholar
[20] Milne, J., Lectures on étale cohomology. 1998, http://www.jmilne.org/math/CourseNotes/. Google Scholar
[21] Moore, R., Heisenberg-invariant quintic threefolds and sections of the Horrocks-Mumford bundle, preprint, Canberra 1985.Google Scholar
[22] Munkres, J., Elements of algebraic topology. Addison-Wesley, Menlo Park, CA, 1984.Google Scholar
[23] Schoen, C., On the geometry of a special determinantal hypersurface associated to the Mumford-Horrocks vector bundle. J. Reine. Angew. Math. 364(1986), 85–111.Google Scholar
[24] Schütt, M., On the modularity of three Calabi–Yau threefolds with bad reduction at 11. Canad. Math. Bull. 49(2006), no. 2, 296–312. doi:10.4153/CMB-2006-031-9 Google Scholar
[25] Serre, J.-P., Resumé des cours de 1984-1985. Annuaire du College de France, 1985, 85–90.Google Scholar
[26] Serre, J.-P., Abelian l-adic representations and elliptic curves. Research Notes in Mathematics, 7, Peters, A. K., Wellesley, MA, 1998.Google Scholar
[27] Stein, W., The modular forms database. http://modular.math.washington.edu/Tables/ Google Scholar
[28] Taylor, R. and Wiles, A., Ring-theoretic properties of certain Hecke algebras. Ann. of Math. (2) 141(1995), no. 3, 553–572. doi:10.2307/2118560 Google Scholar
[29] Werner, J. and van Geemen, B. , New examples of threefolds with c1 = 0. Math. Z. 203(1990), 211–225. doi:10.1007/BF02570731 Google Scholar