From dynamics on surfaces to rational points on curves
HTML articles powered by AMS MathViewer
- by Curtis T. McMullen PDF
- Bull. Amer. Math. Soc. 37 (2000), 119-140 Request permission
References
- S. Ju. Arakelov, Families of algebraic curves with fixed degeneracies, Izv. Akad. Nauk SSSR Ser. Mat. 35 (1971), 1269–1293 (Russian). MR 0321933
- Lipman Bers, An extremal problem for quasiconformal mappings and a theorem by Thurston, Acta Math. 141 (1978), no. 1-2, 73–98. MR 477161, DOI 10.1007/BF02545743
- Jeffrey F. Brock, Iteration of mapping classes on a Bers slice: examples of algebraic and geometric limits of hyperbolic $3$-manifolds, Lipa’s legacy (New York, 1995) Contemp. Math., vol. 211, Amer. Math. Soc., Providence, RI, 1997, pp. 81–106. MR 1476982, DOI 10.1090/conm/211/02815
- Peter Buser, Geometry and spectra of compact Riemann surfaces, Progress in Mathematics, vol. 106, Birkhäuser Boston, Inc., Boston, MA, 1992. MR 1183224
- Gary Cornell and Joseph H. Silverman (eds.), Arithmetic geometry, Springer-Verlag, New York, 1986. Papers from the conference held at the University of Connecticut, Storrs, Connecticut, July 30–August 10, 1984. MR 861969, DOI 10.1007/978-1-4613-8655-1
- Henri Darmon, Fred Diamond, and Richard Taylor, Fermat’s last theorem, Current developments in mathematics, 1995 (Cambridge, MA), Int. Press, Cambridge, MA, 1994, pp. 1–154. MR 1474977
- Pierre Deligne, Preuve des conjectures de Tate et de Shafarevitch (d’après G. Faltings), Astérisque 121-122 (1985), 25–41 (French). Seminar Bourbaki, Vol. 1983/84. MR 768952
- P. Deligne, Un théorème de finitude pour la monodromie, Discrete groups in geometry and analysis (New Haven, Conn., 1984) Progr. Math., vol. 67, Birkhäuser Boston, Boston, MA, 1987, pp. 1–19 (French). MR 900821, DOI 10.1007/978-1-4899-6664-3_{1}
- G. Faltings, Endlichkeitssätze für abelsche Varietäten über Zahlkörpern, Invent. Math. 73 (1983), no. 3, 349–366 (German). MR 718935, DOI 10.1007/BF01388432
- Travaux de Thurston sur les surfaces, Astérisque, vol. 66, Société Mathématique de France, Paris, 1979 (French). Séminaire Orsay; With an English summary. MR 568308
- Frederick P. Gardiner, Teichmüller theory and quadratic differentials, Pure and Applied Mathematics (New York), John Wiley & Sons, Inc., New York, 1987. A Wiley-Interscience Publication. MR 903027
- Hans Grauert, Mordells Vermutung über rationale Punkte auf algebraischen Kurven und Funktionenkörper, Inst. Hautes Études Sci. Publ. Math. 25 (1965), 131–149 (German). MR 222087, DOI 10.1007/BF02684399
- Phillip Griffiths and Joseph Harris, Principles of algebraic geometry, Pure and Applied Mathematics, Wiley-Interscience [John Wiley & Sons], New York, 1978. MR 507725
- Yôichi Imayoshi and Hiroshige Shiga, A finiteness theorem for holomorphic families of Riemann surfaces, Holomorphic functions and moduli, Vol. II (Berkeley, CA, 1986) Math. Sci. Res. Inst. Publ., vol. 11, Springer, New York, 1988, pp. 207–219. MR 955842, DOI 10.1007/978-1-4613-9611-6_{1}5
- Y. Imayoshi and M. Taniguchi, An introduction to Teichmüller spaces, Springer-Verlag, Tokyo, 1992. Translated and revised from the Japanese by the authors. MR 1215481, DOI 10.1007/978-4-431-68174-8
- Nicholas M. Katz, An overview of Deligne’s proof of the Riemann hypothesis for varieties over finite fields, Mathematical developments arising from Hilbert problems (Proc. Sympos. Pure Math., Northern Illinois Univ., De Kalb, Ill., 1974) Amer. Math. Soc., Providence, R.I., 1976, pp. 275–305. MR 0424822
- K. Kodaira, A certain type of irregular algebraic surfaces, J. Analyse Math. 19 (1967), 207–215. MR 216521, DOI 10.1007/BF02788717
- A. Gloden, Sur la résolution en nombres entiers du système $A^{kx}+B^{kx}+C^{kx}+D^{kx}=E^x+F^x+G^x+H^x$ ($x=1$, $2$ et $3$), Bol. Mat. 12 (1939), 118–122 (French). MR 24
- Serge Lang, Introduction to Arakelov theory, Springer-Verlag, New York, 1988. MR 969124, DOI 10.1007/978-1-4612-1031-3
- Olli Lehto, Univalent functions and Teichmüller spaces, Graduate Texts in Mathematics, vol. 109, Springer-Verlag, New York, 1987. MR 867407, DOI 10.1007/978-1-4613-8652-0
- J. E. Littlewood, Littlewood’s miscellany, Cambridge University Press, Cambridge, 1986. Edited and with a foreword by Béla Bollobás. MR 872858
- Ju. I. Manin, Proof of an analogue of Mordell’s conjecture for algebraic curves over function fields, Dokl. Akad. Nauk SSSR 152 (1963), 1061–1063 (Russian). MR 0154868
- Barry Mazur, Notes on étale cohomology of number fields, Ann. Sci. École Norm. Sup. (4) 6 (1973), 521–552 (1974). MR 344254, DOI 10.24033/asens.1257
- Barry Mazur, Arithmetic on curves, Bull. Amer. Math. Soc. (N.S.) 14 (1986), no. 2, 207–259. MR 828821, DOI 10.1090/S0273-0979-1986-15430-3
- Curt McMullen, Rational maps and Kleinian groups, Proceedings of the International Congress of Mathematicians, Vol. I, II (Kyoto, 1990) Math. Soc. Japan, Tokyo, 1991, pp. 889–899. MR 1159274
- Curtis T. McMullen, Renormalization and 3-manifolds which fiber over the circle, Annals of Mathematics Studies, vol. 142, Princeton University Press, Princeton, NJ, 1996. MR 1401347, DOI 10.1515/9781400865178
- Shinichi Mochizuki, The intrinsic Hodge theory of $p$-adic hyperbolic curves, Proceedings of the International Congress of Mathematicians, Vol. II (Berlin, 1998), 1998, pp. 187–196. MR 1648069
- David Mumford, A remark on Mahler’s compactness theorem, Proc. Amer. Math. Soc. 28 (1971), 289–294. MR 276410, DOI 10.1090/S0002-9939-1971-0276410-4
- Subhashis Nag, The complex analytic theory of Teichmüller spaces, Canadian Mathematical Society Series of Monographs and Advanced Texts, John Wiley & Sons, Inc., New York, 1988. A Wiley-Interscience Publication. MR 927291
- M. S. Narasimhan and M. V. Nori, Polarisations on an abelian variety, Proc. Indian Acad. Sci. Math. Sci. 90 (1981), no. 2, 125–128. MR 653950, DOI 10.1007/BF02837283
- Jean-Pierre Otal, Le théorème d’hyperbolisation pour les variétés fibrées de dimension 3, Astérisque 235 (1996), x+159 (French, with French summary). MR 1402300
- A. N. Paršin, Algebraic curves over function fields, Dokl. Akad. Nauk SSSR 183 (1968), 524–526 (Russian). MR 0236180
- Tamio Hara, Equivariant bordism of $(G,{\scr E})$-manifolds, Kyushu J. Math. 48 (1994), no. 2, 427–439. MR 1294540, DOI 10.2206/kyushujm.48.427
- A. Reznikov. Hakenness and $b_1$. Preprint, 1998.
- Kenneth A. Ribet, Galois representations and modular forms, Bull. Amer. Math. Soc. (N.S.) 32 (1995), no. 4, 375–402. MR 1322785, DOI 10.1090/S0273-0979-1995-00616-6
- K. Rubin and A. Silverberg, A report on Wiles’ Cambridge lectures, Bull. Amer. Math. Soc. (N.S.) 31 (1994), no. 1, 15–38. MR 1256978, DOI 10.1090/S0273-0979-1994-00512-9
- Leila Schneps (ed.), The Grothendieck theory of dessins d’enfants, London Mathematical Society Lecture Note Series, vol. 200, Cambridge University Press, Cambridge, 1994. Papers from the Conference on Dessins d’Enfant held in Luminy, April 19–24, 1993. MR 1305390, DOI 10.1017/CBO9780511569302
- C. Soulé, Lectures on Arakelov geometry, Cambridge Studies in Advanced Mathematics, vol. 33, Cambridge University Press, Cambridge, 1992. With the collaboration of D. Abramovich, J.-F. Burnol and J. Kramer. MR 1208731, DOI 10.1017/CBO9780511623950
- Lucien Szpiro, La conjecture de Mordell (d’après G. Faltings), Astérisque 121-122 (1985), 83–103 (French). Seminar Bourbaki, Vol. 1983/84. MR 768955
- Richard Taylor and Andrew Wiles, Ring-theoretic properties of certain Hecke algebras, Ann. of Math. (2) 141 (1995), no. 3, 553–572. MR 1333036, DOI 10.2307/2118560
- William P. Thurston, On the geometry and dynamics of diffeomorphisms of surfaces, Bull. Amer. Math. Soc. (N.S.) 19 (1988), no. 2, 417–431. MR 956596, DOI 10.1090/S0273-0979-1988-15685-6
- W. P. Thurston. Hyperbolic structures on 3-manifolds II: Surface groups and 3-manifolds which fiber over the circle. Preprint, 1986.
- Paul Vojta, Diophantine approximations and value distribution theory, Lecture Notes in Mathematics, vol. 1239, Springer-Verlag, Berlin, 1987. MR 883451, DOI 10.1007/BFb0072989
- Paul Vojta, Siegel’s theorem in the compact case, Ann. of Math. (2) 133 (1991), no. 3, 509–548. MR 1109352, DOI 10.2307/2944318
- André Weil, On discrete subgroups of Lie groups, Ann. of Math. (2) 72 (1960), 369–384. MR 137792, DOI 10.2307/1970140
- A. Weil. Sur l’analogie entre les corps de nombres algébriques et les corps de fonctions algébriques [1939a]. In Oeuvres Scient., volume I, pages 236–240. Springer-Verlag, 1980.
- Andrew Wiles, Modular elliptic curves and Fermat’s last theorem, Ann. of Math. (2) 141 (1995), no. 3, 443–551. MR 1333035, DOI 10.2307/2118559
- Scott A. Wolpert, Geodesic length functions and the Nielsen problem, J. Differential Geom. 25 (1987), no. 2, 275–296. MR 880186
Additional Information
- Curtis T. McMullen
- Affiliation: Mathematics Department, Harvard University, 1 Oxford Street, Cambridge, MA 02138-2901
- Received by editor(s): February 19, 1999
- Received by editor(s) in revised form: September 22, 1999
- Published electronically: December 21, 1999
- Additional Notes: Research partially supported by the NSF
- © Copyright 2000 American Mathematical Society
- Journal: Bull. Amer. Math. Soc. 37 (2000), 119-140
- MSC (2000): Primary 11G30; Secondary 32G15, 57Mxx
- DOI: https://doi.org/10.1090/S0273-0979-99-00856-3
- MathSciNet review: 1713286