Book Review
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MathSciNet review: 1567735
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Book Information:
Authors: Ph. Cassou-Noguès and M. J. Taylor
Title: Elliptic functions and rings of integers
Additional book information: Progress in Mathematics, vol. 66, Birkhäuser, Boston, Basel and Stuttgart, 1987, xvi + 198 pp., $29.50. ISBN 0-8176-3350-2.
- Jan Brinkhuis, Galois modules and embedding problems, J. Reine Angew. Math. 346 (1984), 141–165. MR 727401, DOI https://doi.org/10.1515/crll.1984.346.141
- Jan Brinkhuis, Normal integral bases and complex conjugation, J. Reine Angew. Math. 375/376 (1987), 157–166. MR 882295, DOI https://doi.org/10.1515/crll.1987.375-376.157
- Ph. Cassou-Noguès and M. J. Taylor, A note on elliptic curves and the monogeneity of rings of integers, J. London Math. Soc. (2) 37 (1988), no. 1, 63–72. MR 921747, DOI https://doi.org/10.1112/jlms/s2-37.121.63
- Jean Cougnard, Conditions nécessaires de monogénéité. Application aux extensions cycliques de degré premier $l\geq 5$ d’un corps quadratique imaginaire, J. London Math. Soc. (2) 37 (1988), no. 1, 73–87 (French). MR 921746, DOI https://doi.org/10.1112/jlms/s2-37.121.73
- Albrecht Fröhlich, Galois module structure of algebraic integers, Ergebnisse der Mathematik und ihrer Grenzgebiete (3) [Results in Mathematics and Related Areas (3)], vol. 1, Springer-Verlag, Berlin, 1983. MR 717033
- Marie-Nicole Gras, Non monogénéité de l’anneau des entiers des extensions cycliques de ${\bf Q}$ de degré premier $l\geq 5$, J. Number Theory 23 (1986), no. 3, 347–353 (French, with English summary). MR 846964, DOI https://doi.org/10.1016/0022-314X%2886%2990079-X D. Hilbert, Mathematical problems, Lecture delivered at the International Congress of Mathematicians in Paris in 1900, Bull. Amer. Math. Soc. 8 (1902), 437-479. L. Kronecker, Werke, Band V, (K. Hensel ed. ), Teubner, Leipzig and Berlin, 1930.
- 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
- R. P. Langlands, Some contemporary problems with origins in the Jugendtraum, Mathematical developments arising from Hilbert problems (Proc. Sympos. Pure Math., Vol. XXVIII, Northern Illinois Univ., De Kalb, Ill., 1974) Amer. Math. Soc., Providence, R. I., 1976, pp. 401–418. MR 0437500
- Heinrich-Wolfgang Leopoldt, Über die Hauptordnung der ganzen Elemente eines abelschen Zahlkörpers, J. Reine Angew. Math. 201 (1959), 119–149 (German). MR 108479, DOI https://doi.org/10.1515/crll.1959.201.119
- Leon R. McCulloh, Galois module structure of abelian extensions, J. Reine Angew. Math. 375/376 (1987), 259–306. MR 882300, DOI https://doi.org/10.1515/crll.1987.375-376.259
- J. S. Milne, Automorphic vector bundles on connected Shimura varieties, Invent. Math. 92 (1988), no. 1, 91–128. MR 931206, DOI https://doi.org/10.1007/BF01393994
- Goro Shimura, Introduction to the arithmetic theory of automorphic functions, Publications of the Mathematical Society of Japan, No. 11. Iwanami Shoten, Publishers, Tokyo; Princeton University Press, Princeton, N.J., 1971. Kanô Memorial Lectures, No. 1. MR 0314766
- H. M. Stark, Hilbert’s twelfth problem and $L$-series, Bull. Amer. Math. Soc. 83 (1977), no. 5, 1072–1074. MR 441923, DOI https://doi.org/10.1090/S0002-9904-1977-14389-9
- H. M. Stark, $L$-functions at $s=1$. III. Totally real fields and Hilbert’s twelfth problem, Advances in Math. 22 (1976), no. 1, 64–84. MR 437501, DOI https://doi.org/10.1016/0001-8708%2876%2990138-9
- Harold M. Stark, $L$-functions at $s=1$. IV. First derivatives at $s=0$, Adv. in Math. 35 (1980), no. 3, 197–235. MR 563924, DOI https://doi.org/10.1016/0001-8708%2880%2990049-3
- John Tate, Les conjectures de Stark sur les fonctions $L$ d’Artin en $s=0$, Progress in Mathematics, vol. 47, Birkhäuser Boston, Inc., Boston, MA, 1984 (French). Lecture notes edited by Dominique Bernardi and Norbert Schappacher. MR 782485
- M. J. Taylor, Formal groups and the Galois module structure of local rings of integers, J. Reine Angew. Math. 358 (1985), 97–103. MR 797677, DOI https://doi.org/10.1515/crll.1985.358.97
- M. J. Taylor, Mordell-Weil groups and the Galois module structure of rings of integers, Illinois J. Math. 32 (1988), no. 3, 428–452. MR 947037
- M. J. Taylor, Relative Galois module structure of rings of integers and elliptic functions, Math. Proc. Cambridge Philos. Soc. 94 (1983), no. 3, 389–397. MR 720789, DOI https://doi.org/10.1017/S0305004100000773
Review Information:
Reviewer: Ted Chinburg
Journal: Bull. Amer. Math. Soc. 20 (1989), 117-121
DOI: https://doi.org/10.1090/S0273-0979-1989-15722-4