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Book Information:
Author:
Christian Meyer
Title:
Modular Calabi-Yau threefolds
Additional book information:
American Mathematical Society,
Providence, RI,
2005,
ix+194 pp.,
ISBN 978-0-8218-3908-9,
US$59.00$
1. G. Bini, C. De Concini, M. Polito, C. Procesi, On the work of Givental relative to mirror symmetry, math.AG/9805097.
Christophe Breuil, Brian Conrad, Fred Diamond, and Richard Taylor, On the modularity of elliptic curves over $\mathbf Q$: wild 3-adic exercises, J. Amer. Math. Soc. 14 (2001), no. 4, 843–939. MR 1839918, DOI 10.1090/S0894-0347-01-00370-8
Philip Candelas, Xenia C. de la Ossa, Paul S. Green, and Linda Parkes, An exactly soluble superconformal theory from a mirror pair of Calabi-Yau manifolds, Phys. Lett. B 258 (1991), no. 1-2, 118–126. MR 1101784, DOI 10.1016/0370-2693(91)91218-K
David A. Cox and Sheldon Katz, Mirror symmetry and algebraic geometry, Mathematical Surveys and Monographs, vol. 68, American Mathematical Society, Providence, RI, 1999. MR 1677117, DOI 10.1090/surv/068
5. P. Deligne, Formes modulaires et représentations 
-adiques, In Sem. Bourbaki 355 (1968/69), Lect. Notes Math. 179 (1971).
Pierre Deligne, La conjecture de Weil. I, Inst. Hautes Études Sci. Publ. Math. 43 (1974), 273–307 (French). MR 340258
Pierre Deligne, La conjecture de Weil. II, Inst. Hautes Études Sci. Publ. Math. 52 (1980), 137–252 (French). MR 601520
Luis Dieulefait and Jayanta Manoharmayum, Modularity of rigid Calabi-Yau threefolds over $\Bbb Q$, Calabi-Yau varieties and mirror symmetry (Toronto, ON, 2001) Fields Inst. Commun., vol. 38, Amer. Math. Soc., Providence, RI, 2003, pp. 159–166. MR 2019150, DOI 10.4310/mrl.2003.v10.n2.a1
Jean-Marc Fontaine and Barry Mazur, Geometric Galois representations, Elliptic curves, modular forms, & Fermat’s last theorem (Hong Kong, 1993) Ser. Number Theory, I, Int. Press, Cambridge, MA, 1995, pp. 41–78. MR 1363495
Alexander B. Givental, Equivariant Gromov-Witten invariants, Internat. Math. Res. Notices 13 (1996), 613–663. MR 1408320, DOI 10.1155/S1073792896000414
Brian Greene, The elegant universe, W. W. Norton & Co. Inc., New York, 1999. Superstrings, hidden dimensions, and the quest for the ultimate theory. MR 1700770
12. K. Hulek, R. Kloosterman, M. Schuett, Modularity of Calabi-Yau varieties, math.AG/0601238.
13. C. Khare and J.-P. Wintenberger, Serre's modularity conjecture (I), (II), preprints.
14. E. Lee, A modular non-rigid Calabi-Yau threefold, math.AG/0508127.
Bong H. Lian, Kefeng Liu, and Shing-Tung Yau, Mirror principle. I, Asian J. Math. 1 (1997), no. 4, 729–763. MR 1621573, DOI 10.4310/AJM.1997.v1.n4.a5
Chiu-Chu Melissa Liu, Kefeng Liu, and Jian Zhou, A proof of a conjecture of Mariño-Vafa on Hodge integrals, J. Differential Geom. 65 (2003), no. 2, 289–340. MR 2058264
Rahul Pandharipande, Rational curves on hypersurfaces (after A. Givental), Astérisque 252 (1998), Exp. No. 848, 5, 307–340. Séminaire Bourbaki. Vol. 1997/98. MR 1685628
Masa-Hiko Saito and Noriko Yui, The modularity conjecture for rigid Calabi-Yau threefolds over $Q$, J. Math. Kyoto Univ. 41 (2001), no. 2, 403–419. MR 1852991, DOI 10.1215/kjm/1250517640
Jean-Pierre Serre, Sur les représentations modulaires de degré $2$ de $\textrm {Gal}(\overline \textbf {Q}/\textbf {Q})$, Duke Math. J. 54 (1987), no. 1, 179–230 (French). MR 885783, DOI 10.1215/S0012-7094-87-05413-5
Richard Taylor, Galois representations, Ann. Fac. Sci. Toulouse Math. (6) 13 (2004), no. 1, 73–119 (English, with English and French summaries). MR 2060030
Shing Tung Yau, Calabi’s conjecture and some new results in algebraic geometry, Proc. Nat. Acad. Sci. U.S.A. 74 (1977), no. 5, 1798–1799. MR 451180, DOI 10.1073/pnas.74.5.1798
- 1.
- G. Bini, C. De Concini, M. Polito, C. Procesi, On the work of Givental relative to mirror symmetry, math.AG/9805097.
- 2.
- C. Breuil, B. Conrad, F. Diamond, R. Taylor, On the modularity of elliptic curves over
: wild 3-adic exercises. J. Amer. Math. Soc. 14 (2001), 843-939. MR 1839918 (2002d:11058)
- 3.
- P. Candelas, X.C. de la Ossa, P.S. Green, L. Parkes, An exactly soluble superconformal theory from a mirror pair of Calabi-Yau manifolds, Phys. Lett. B 258 (1991), no. 1-2, 118-126. MR 1101784 (93b:32028)
- 4.
- D. Cox and S. Katz, Mirror Symmetry and Algebraic Geometry, Mathematical Surveys and Monographs 68, AMS, Providence, RI, 1999. MR 1677117 (2000d:14048)
- 5.
- P. Deligne, Formes modulaires et représentations -adiques, In Sem. Bourbaki 355 (1968/69), Lect. Notes Math. 179 (1971).
- 6.
- P. Deligne, La conjecture de Weil. I, Inst. Hautes Études Sci. Publ. Math. 43 (1974), 273-307. MR 0340258 (49:5013)
- 7.
- P. Deligne, La conjecture de Weil. II. Inst. Hautes Études Sci. Publ. Math. 52 (1980), 137-252. MR 601520 (83c:14017)
- 8.
- L. Dieulefait and J. Manoharmayum, Modularity of rigid Calabi-Yau threefolds over
, In Calabi-Yau Varieties and Mirror Symmetry (Toronto, ON, 2001), Fields Inst. Commun. 38, Amer. Math. Soc., Providence, RI, 2003, 159-166. MR 2019150 (2004m:11081)
- 9.
- J.M. Fontaine and M. Mazur, Geometric Galois representations, In Elliptic Curves, Modular Forms and Fermat's Last Theorem (Hong Kong, 1993), J. Coates, S.-T. Yau (eds.), International Press (1995), 41-78. MR 1363495 (96h:11049)
- 10.
- A. Givental, Equivariant Gromov-Witten invariants, Internat. Math. Res. Notices, 1996, no. 13, 613-663. MR 1408320 (97e:14015)
- 11.
- B. Greene, The Elegant Universe, W. W. Norton
Company, Inc., New York (1999). MR 1700770 (2000f:81001)
- 12.
- K. Hulek, R. Kloosterman, M. Schuett, Modularity of Calabi-Yau varieties, math.AG/0601238.
- 13.
- C. Khare and J.-P. Wintenberger, Serre's modularity conjecture (I), (II), preprints.
- 14.
- E. Lee, A modular non-rigid Calabi-Yau threefold, math.AG/0508127.
- 15.
- B. Lian, K. Liu, S.-T. Yau, Mirror principle I, Asian J. Math. 1 (1997), 729-763. MR 1621573 (99e:14062)
- 16.
- C.-C. Liu, K. Liu, J. Zhou, A proof of a conjecture of Mariño-Vafa on Hodge integrals, J. Differential Geom. 65 (2003), no. 2, 289-340. MR 2058264 (2005m:14104)
- 17.
- R. Pandhripande, Rational curves on hypersurfaces (after A. Givental), Asterisque 252 (1998), 307-340. MR 1685628 (2000e:14094)
- 18.
- M.-H. Saito and N. Yui, The modularity conjecture for rigid Calabi-Yau threefolds over
, J. Math. Kyoto Univ. 41 (2001), 403-419. MR 1852991 (2002k:11102)
- 19.
- J-P. Serre, Sur les représentations modulaires de degré 2 de Gal
, Duke Math. J. 54 (1987), 179-230. MR 885783 (88g:11022)
- 20.
- R. Taylor, Galois representations, Ann. Fac. Sci. Toulouse Math. (6) 13 (2004), 73-119. MR 2060030 (2005a:11071)
- 21.
- S.-T. Yau, Calabi's Conjecture and Some New Results in Algebraic Geometry, Proceedings of the National Academy of Sciences 74 (1977), no. 5, 1798-1799. MR 0451180 (56:9467)
Review Information:
Reviewer:
Kefeng Liu
Affiliation:
Zhejiang University and University of California at Los Angeles
Email:
liu@math.ucla.edu
Journal:
Bull. Amer. Math. Soc.
45 (2008), 163-168
Published electronically:
April 27, 2007
Additional Notes:
The author is supported by NSF.
Review copyright:
© Copyright 2007
American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.
