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Bulletin of the American Mathematical Society

The Bulletin publishes expository articles on contemporary mathematical research, written in a way that gives insight to mathematicians who may not be experts in the particular topic. The Bulletin also publishes reviews of selected books in mathematics and short articles in the Mathematical Perspectives section, both by invitation only.

ISSN 1088-9485 (online) ISSN 0273-0979 (print)

The 2020 MCQ for Bulletin of the American Mathematical Society is 0.84.

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Book Review

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Full text of review: PDF   This review is available free of charge.
Book Information:

Author: Karl Rubin
Title: Euler systems
Additional book information: Ann. of Math. Stud., vol. 147, Princeton Univ. Press, Princeton, NJ, 2000, xii+227 pp., ISBN 0-691-05075-9, $69.50$

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

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Flach, Matthias. A finiteness theorem for the symmetric square of an elliptic curve. Invent. Math. 109 (1992), no. 2, 307-327. MR 1172693
Gross, Benedict H. Kolyvagin's work on modular elliptic curves. $L$-functions and arithmetic (Durham, 1989), 235-256, London Math. Soc. Lecture Note Ser., 153, Cambridge Univ. Press, Cambridge, 1991. MR 1110395
Gross, Benedict H.; Zagier, Don B. Heegner points and derivatives of $L$-series. Invent. Math. 84 (1986), no. 2, 225-320. MR 0833192
Kolyvagin, V. A. The Mordell-Weil and Shafarevich-Tate groups for Weil elliptic curves. (Russian) Izv. Akad. Nauk SSSR Ser. Mat. 52 (1988), no. 6,1154-1180, 1327 translation in Math. USSR-Izv. 33 (1989), no. 3, 473-499.
Kolyvagin, V. A. Finiteness of $E(\mathbb{Q})$ and $\underline{III}(E,\mathbb{Q})$ for a subclass of Weil curves. (Russian) Izv. Akad. Nauk SSSR Ser. Mat. 52 (1988), no. 3, 522-540, 670-671; translation in Math. USSR-Izv. 32 (1989), no. 3, 523-541. MR 0954295
Kolyvagin, V. A. Euler systems. The Grothendieck Festschrift, Vol. II, 435-483, Progr. Math., 87, Birkhäuser Boston, Boston, MA, 1990. MR 1106906
Kolyvagin, V. A.; Logachev, D. Yu. Finiteness of $\underline{III}$ over totally real fields. (Russian) Izv. Akad. Nauk SSSR Ser. Mat. 55 (1991), no. 4, 851-876; translation in Math. USSR-Izv. 39 (1992), no. 1, 829-853. MR 1137589
Nekovár, Jan. Kolyvagin's method for Chow groups of Kuga-Sato varieties. Invent. Math. 107 (1992), no. 1, 99-125. MR 1135466
Rubin, Karl. Tate-Shafarevich groups and $L$-functions of elliptic curves with complex multiplication. Invent. Math. 89 (1987), no. 3, 527-559. MR 0903383
Rubin, Karl. Kolyvagin's system of Gauss sums. Arithmetic algebraic geometry (Texel, 1989), 309-324, Progr. Math., 89, Birkhäuser Boston, Boston, MA, 1991. MR 1085265
Rubin, Karl. Appendix in Lang, Serge, Cyclotomic fields I and II. Combined second edition, Graduate Texts in Mathematics, 121, Springer-Verlag, New York, 1990. MR 1029028
Rubin, Karl. The ``main conjectures" of Iwasawa theory for imaginary quadratic fields. Invent. Math. 103 (1991), no. 1, 25-68. MR 1079839
Scholl, A. J. An introduction to Kato's Euler systems. Galois representations in arithmetic algebraic geometry (Durham, 1996), 379-460, London Math. Soc. Lecture Note Ser., 254, Cambridge Univ. Press, Cambridge, 1998. MR 1696501
Thaine, Francisco. On the ideal class groups of real abelian number fields. Ann. of Math. (2) 128 (1988), no. 1, 1-18. MR 0951505
Taylor, Richard; Wiles, Andrew. Ring-theoretic properties of certain Hecke algebras. Ann. of Math. (2) 141 (1995), no. 3, 553-572. MR 1333036
Wiles, Andrew. Modular elliptic curves and Fermat's last theorem. Ann. of Math. (2) 141 (1995), no. 3, 443-551. MR 1333035
Zhang, Shouwu. Heights of Heegner cycles and derivatives of $L$-series. Invent. Math. 130 (1997), no. 1, 99-152. MR 1471887
Zhang, Shouwu. Heights of Heegner points on Shimura curves. Ann. of Math. (2) 153 (2001), no. 1, 27-147.

Review Information:

Reviewer: Henri Darmon
Affiliation: McGill University
Journal: Bull. Amer. Math. Soc. 39 (2002), 407-414
Published electronically: April 11, 2002
Review copyright: © Copyright 2002 American Mathematical Society