Book Review
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MathSciNet review:
3020832
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Book Information:
Author:
Umberto Zannier
Title:
Some problems of unlikely intersections in arithmetic and geometry
Additional book information:
Annals of Mathematics Studies, 181,
Princeton University Press,
Princeton, New Jersey,
2012,
xiv+160 pp.,
ISBN 978-0-691-15371-1,
US $75.00
Yves André, $G$-functions and geometry, Aspects of Mathematics, E13, Friedr. Vieweg & Sohn, Braunschweig, 1989. MR 990016, DOI 10.1007/978-3-663-14108-2
Yves André, Finitude des couples d’invariants modulaires singuliers sur une courbe algébrique plane non modulaire, J. Reine Angew. Math. 505 (1998), 203–208 (French, with English summary). MR 1662256, DOI 10.1515/crll.1998.118
Matthew Baker and Laura DeMarco, Preperiodic points and unlikely intersections, Duke Math. J. 159 (2011), no. 1, 1–29. MR 2817647, DOI 10.1215/00127094-1384773
E. Bombieri, D. Masser, and U. Zannier, Anomalous subvarieties—structure theorems and applications, Int. Math. Res. Not. IMRN 19 (2007), Art. ID rnm057, 33. MR 2359537, DOI 10.1093/imrn/rnm057
V. A. Dem′janenko, Rational points of a class of algebraic curves, Izv. Akad. Nauk SSSR Ser. Mat. 30 (1966), 1373–1396 (Russian). MR 0205991
P. Habegger, Intersecting subvarieties of abelian varieties with algebraic subgroups of complementary dimension, Invent. Math. 176 (2009), no. 2, 405–447. MR 2495768, DOI 10.1007/s00222-008-0170-6
P. Habegger, On the bounded height conjecture, Int. Math. Res. Not. IMRN 5 (2009), 860–886. MR 2482128, DOI 10.1093/imrn/rnn149
Ju. I. Manin, The $p$-torsion of elliptic curves is uniformly bounded, Izv. Akad. Nauk SSSR Ser. Mat. 33 (1969), 459–465 (Russian). MR 0272786
D. Masser and U. Zannier, Torsion anomalous points and families of elliptic curves, Amer. J. Math. 132 (2010), no. 6, 1677–1691. MR 2766181
D. W. Masser, Specializations of finitely generated subgroups of abelian varieties, Trans. Amer. Math. Soc. 311 (1989), no. 1, 413–424. MR 974783, DOI 10.1090/S0002-9947-1989-0974783-6
Guillaume Maurin, Courbes algébriques et équations multiplicatives, Math. Ann. 341 (2008), no. 4, 789–824 (French, with English summary). MR 2407327, DOI 10.1007/s00208-008-0212-9
Frans Oort, Canonical liftings and dense sets of CM-points, Arithmetic geometry (Cortona, 1994) Sympos. Math., XXXVII, Cambridge Univ. Press, Cambridge, 1997, pp. 228–234. MR 1472499
Jonathan Pila, O-minimality and the André-Oort conjecture for $\Bbb C^n$, Ann. of Math. (2) 173 (2011), no. 3, 1779–1840. MR 2800724, DOI 10.4007/annals.2011.173.3.11
R. Pink. A common generalization of the conjectures of André–Oort, Manin–Mumford, and Mordell–Lang. unpublished manuscript dated 17th April 2005.
Joseph H. Silverman, Heights and the specialization map for families of abelian varieties, J. Reine Angew. Math. 342 (1983), 197–211. MR 703488, DOI 10.1515/crll.1983.342.197
Boris Zilber, Exponential sums equations and the Schanuel conjecture, J. London Math. Soc. (2) 65 (2002), no. 1, 27–44. MR 1875133, DOI 10.1112/S0024610701002861
References
- Y. André. $G$-functions and geometry. Aspects of Mathematics, E13. Friedr. Vieweg & Sohn, Braunschweig, 1989. MR 990016
- Y. André. Finitude des couples d’invariants modulaires singuliers sur une courbe algébrique plane non modulaire. J. Reine Angew. Math., 505:203–208, 1998. MR 1662256
- M. Baker and L. DeMarco. Preperiodic points and unlikely intersections. Duke Math. J., 159(1):1–29, 2011. MR 2817647
- E. Bombieri, D. Masser, and U. Zannier. Anomalous subvarieties—structure theorems and applications. Int. Math. Res. Not. IMRN, (19):Art. ID rnm057, 33, 2007. MR 2359537
- V. A. Dem’janenko. Rational points of a class of algebraic curves. Izv. Akad. Nauk SSSR Ser. Mat., 30:1373–1396, 1966. MR 0205991
- P. Habegger. Intersecting subvarieties of abelian varieties with algebraic subgroups of complementary dimension. Invent. Math., 176(2):405–447, 2009. MR 2495768
- P. Habegger. On the bounded height conjecture. Int. Math. Res. Not. IMRN, (5):860–886, 2009. MR 2482128
- J. Manin. The $p$-torsion of elliptic curves is uniformly bounded. Izv. Akad. Nauk SSSR Ser. Mat., 33:450–465, 1969. MR 0272786
- D. Masser and U. Zannier. Torsion anomalous points and families of elliptic curves. Amer. J. Math., 132(6):1677–1691, 2010. MR 2766181
- D. W. Masser. Specializations of finitely generated subgroups of abelian varieties. Trans. Amer. Math. Soc., 311(1):413–424, 1989. MR 974783
- G. Maurin. Courbes algébriques et équations multiplicatives. Math. Ann., 341(4):789–824, 2008. MR 2407327
- F. Oort. Canonical liftings and dense sets of CM-points. In Arithmetic geometry (Cortona, 1994), Sympos. Math., XXXVII, pages 228–234. Cambridge Univ. Press, Cambridge, 1997. MR 1472499
- J. Pila. $O$-minimality and the André–Oort conjecture for $\mathbb {C}^n$. Ann. of Math. (2), 173:1779–1840, 2011. MR 2800724
- R. Pink. A common generalization of the conjectures of André–Oort, Manin–Mumford, and Mordell–Lang. unpublished manuscript dated 17th April 2005.
- J. H. Silverman. Heights and the specialization map for families of abelian varieties. J. Reine Angew. Math., 342:197–211, 1983. MR 703488
- B. Zilber. Exponential sums equations and the Schanuel conjecture. J. London Math. Soc. (2), 65(1):27–44, 2002. MR 1875133
Review Information:
Reviewer:
Joseph H. Silverman
Affiliation:
Mathematics Department, Brown University, Providence, Rhode Island 02912
Journal:
Bull. Amer. Math. Soc.
50 (2013), 353-358
DOI:
https://doi.org/10.1090/S0273-0979-2012-01386-1
Keywords:
unlikely intersection
Published electronically:
July 24, 2012
Review copyright:
© Copyright 2012
American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.
