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Book Review

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Book Information:

Author: Haruzo Hida
Title: $ p$-Adic automorphic forms on Shimura varieties
Additional book information: Springer-Verlag, New York, 2004, xii+390 pp., ISBN 0-387-20711-2, US$99.00

References [Enhancements On Off] (What's this?)

  • [B] A. I. Badulescu, Unitary representations and global Jacquet-Langlands correspondence, preprint.
  • [BM] M. V. Borovoĭ, Conjugation of Shimura varieties, Proceedings of the International Congress of Mathematicians, Vol. 1, 2 (Berkeley, Calif., 1986) Amer. Math. Soc., Providence, RI, 1987, pp. 783–790. MR 934281
    J. S. Milne, Introduction to Shimura varieties, Harmonic analysis, the trace formula, and Shimura varieties, Clay Math. Proc., vol. 4, Amer. Math. Soc., Providence, RI, 2005, pp. 265–378. MR 2192012
  • [BW] Armand Borel and Nolan R. Wallach, Continuous cohomology, discrete subgroups, and representations of reductive groups, Annals of Mathematics Studies, vol. 94, Princeton University Press, Princeton, N.J.; University of Tokyo Press, Tokyo, 1980. MR 554917
  • [F] 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
    Henri Darmon, Fred Diamond, and Richard Taylor, Fermat’s last theorem, Elliptic curves, modular forms & Fermat’s last theorem (Hong Kong, 1993) Int. Press, Cambridge, MA, 1997, pp. 2–140. MR 1605752
  • [FI] S. Ramanan and Arnaud Beauville (eds.), Proceedings of the Indo-French Conference on Geometry, Hindustan Book Agency, Delhi, 1993. Held in Bombay, 1989. MR 1274489
  • [FP] J-M. Fontaine and B. Perrin-Riou, Cohomologie galoisienne et valeurs de fonctions $ L$, [Mo].
  • [Gr] Ralph Greenberg, Iwasawa theory for 𝑝-adic representations, Algebraic number theory, Adv. Stud. Pure Math., vol. 17, Academic Press, Boston, MA, 1989, pp. 97–137. MR 1097613
  • [GS] Phillip Griffiths and Wilfried Schmid, Recent developments in Hodge theory: a discussion of techniques and results, Discrete subgroups of Lie groups and applicatons to moduli (Internat. Colloq., Bombay, 1973) Oxford Univ. Press, Bombay, 1975, pp. 31–127. MR 0419850
  • [Ha] Günter Harder, Eisensteinkohomologie und die Konstruktion gemischter Motive, Lecture Notes in Mathematics, vol. 1562, Springer-Verlag, Berlin, 1993 (German). MR 1285354
  • [Hi] Haruzo Hida, Congruences of cusp forms and Hecke algebras, Séminaire de théorie des nombres, Paris 1983–84, Progr. Math., vol. 59, Birkhäuser Boston, Boston, MA, 1985, pp. 133–146. MR 902830,
    Haruzo Hida, Galois representations into 𝐺𝐿₂(𝑍_{𝑝}[[𝑋]]) attached to ordinary cusp forms, Invent. Math. 85 (1986), no. 3, 545–613. MR 848685,
  • [Kh] Chandrasekhar Khare, Serre's modularity conjecture, preprint.
  • [Ki] M. Kisin, Moduli of finite flat group schemes and modularity, preprint, Univ. of Chicago.
  • [KS] Willem Kuyk and J.-P. Serre (eds.), Modular functions of one variable. III, Lecture Notes in Mathematics, Vol. 350, Springer-Verlag, Berlin-New York, 1973. MR 0323724
  • [L] Haruzo Hida, Dinakar Ramakrishnan, and Freydoon Shahidi (eds.), Contributions to automorphic forms, geometry, and number theory, Johns Hopkins University Press, Baltimore, MD, 2004. Papers from the conference in honor of Joseph Shalika held at Johns Hopkins University, Baltimore, MD, May 14–17, 2002. MR 2058599
  • [LM] J. S. Milne, Arithmetic duality theorems, Perspectives in Mathematics, vol. 1, Academic Press, Inc., Boston, MA, 1986. MR 881804
  • [LS] R. P. Langlands, Automorphic representations, Shimura varieties, and motives. Ein Märchen, Automorphic forms, representations and 𝐿-functions (Proc. Sympos. Pure Math., Oregon State Univ., Corvallis, Ore., 1977) Proc. Sympos. Pure Math., XXXIII, Amer. Math. Soc., Providence, R.I., 1979, pp. 205–246. MR 546619
    Pierre Deligne, James S. Milne, Arthur Ogus, and Kuang-yen Shih, Hodge cycles, motives, and Shimura varieties, Lecture Notes in Mathematics, vol. 900, Springer-Verlag, Berlin-New York, 1982. MR 654325
  • [Mo] Uwe Jannsen, Steven Kleiman, and Jean-Pierre Serre (eds.), Motives, Proceedings of Symposia in Pure Mathematics, vol. 55, American Mathematical Society, Providence, RI, 1994. MR 1265518
  • [MW] B. Mazur and A. Wiles, Class fields of abelian extensions of 𝑄, Invent. Math. 76 (1984), no. 2, 179–330. MR 742853,
    John Coates, The work of Mazur and Wiles on cyclotomic fields, Bourbaki Seminar, Vol. 1980/81, Lecture Notes in Math., vol. 901, Springer, Berlin-New York, 1981, pp. 220–242. MR 647499
  • [R] Karl Rubin, The “main conjectures” of Iwasawa theory for imaginary quadratic fields, Invent. Math. 103 (1991), no. 1, 25–68. MR 1079839,
  • [T] J. T. Tate, 𝑝-divisible groups, Proc. Conf. Local Fields (Driebergen, 1966) Springer, Berlin, 1967, pp. 158–183. MR 0231827
  • [Ta] Richard Taylor, Galois representations, Ann. Fac. Sci. Toulouse Math. (6) 13 (2004), no. 1, 73–119 (English, with English and French summaries). MR 2060030
  • [Ti] J. Tilouine, Galois representations congruent to those coming from Shimura varieties, [Mo].
  • [W] André Weil, On a certain type of characters of the idèle-class group of an algebraic number-field, Proceedings of the international symposium on algebraic number theory, Tokyo & Nikko, 1955, Science Council of Japan, Tokyo, 1956, pp. 1–7. MR 0083523

Review Information:

Reviewer: R. P. Langlands
Affiliation: Institute for Advanced Study
Journal: Bull. Amer. Math. Soc. 44 (2007), 291-308
Published electronically: October 18, 2006
Review copyright: © Copyright 2006 R. P. Langlands