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Book Review

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Book Information

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

References:

[DDT]
Darmon, Henri; Diamond, Fred; Taylor, Richard. Fermat's last theorem. Elliptic curves, modular forms and Fermat's last theorem (Hong Kong, 1993), 2-140, Internat. Press, Cambridge, MA, 1997. MR 99d:11067b

[Fl]
Flach, Matthias. A finiteness theorem for the symmetric square of an elliptic curve. Invent. Math. 109 (1992), no. 2, 307-327. MR 93g:11066

[Gr]
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 93c:11039

[GZ]
Gross, Benedict H.; Zagier, Don B. Heegner points and derivatives of $L$-series. Invent. Math. 84 (1986), no. 2, 225-320. MR 87j:11057

[Ko88a]
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.

[Ko88b]
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 89m:11056

[Ko90]
Kolyvagin, V. A. Euler systems. The Grothendieck Festschrift, Vol. II, 435-483, Progr. Math., 87, Birkhäuser Boston, Boston, MA, 1990. MR 92g:11109

[KL]
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 93d:11063

[Ne92]
Nekovár, Jan. Kolyvagin's method for Chow groups of Kuga-Sato varieties. Invent. Math. 107 (1992), no. 1, 99-125. MR 93b:11076

[Ru87]
Rubin, Karl. Tate-Shafarevich groups and $L$-functions of elliptic curves with complex multiplication. Invent. Math. 89 (1987), no. 3, 527-559. MR 89a:11065

[Ru89]
Rubin, Karl. Kolyvagin's system of Gauss sums. Arithmetic algebraic geometry (Texel, 1989), 309-324, Progr. Math., 89, Birkhäuser Boston, Boston, MA, 1991. MR 92a:11121

[Ru90]
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 91c:11001

[Ru91]
Rubin, Karl. The ``main conjectures" of Iwasawa theory for imaginary quadratic fields. Invent. Math. 103 (1991), no. 1, 25-68. MR 92f:11151

[Sch]
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 2000g:11057

[Th88]
Thaine, Francisco. On the ideal class groups of real abelian number fields. Ann. of Math. (2) 128 (1988), no. 1, 1-18. MR 89m:11099

[TW]
Taylor, Richard; Wiles, Andrew. Ring-theoretic properties of certain Hecke algebras. Ann. of Math. (2) 141 (1995), no. 3, 553-572. MR 96d:11072

[W]
Wiles, Andrew. Modular elliptic curves and Fermat's last theorem. Ann. of Math. (2) 141 (1995), no. 3, 443-551. MR 96d:11071

[Zh97]
Zhang, Shouwu. Heights of Heegner cycles and derivatives of $L$-series. Invent. Math. 130 (1997), no. 1, 99-152. MR 98i:11044

[Zh01]
Zhang, Shouwu. Heights of Heegner points on Shimura curves. Ann. of Math. (2) 153 (2001), no. 1, 27-147.

Additional Information:

Reviewer(s):
Henri Darmon
Affiliation: McGill University
Email: darmon@math.mcgill.ca

Review Information:
Journal: Bull. Amer. Math. Soc. 39 (2002), 407-414.

MSC (2000): Primary 11R34; Secondary 11R39, 11M41
DOI: 10.1090/S0273-0979-02-00939-4
PII: S 0273-0979(02)00939-4
Posted: April 11, 2002
Copyright of article: Copyright 2002, American Mathematical Society

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