American Mathematical Society

Book Review

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MathSciNet review: 1568092
Full text of review: PDF This review is available free of charge.
Book Information:

Author: Joseph H. Silverman and John T. Tate
Title: Rational points on elliptic curves
Additional book information: Undergraduate Texts in Mathematics, Springer-Verlag, New York and Berlin, 1992 (first ed.\ 1989), x+281 pp., US$29.95. ISBN 0-387-97825-9.

J. W. S. Cassels, Diophantine equations with special reference to elliptic curves, J. London Math. Soc. 41 (1966), 193–291. MR 199150, DOI 10.1112/jlms/s1-41.1.193

J. W. S. Cassels, Lectures on elliptic curves, London Mathematical Society Student Texts, vol. 24, Cambridge University Press, Cambridge, 1991. MR 1144763, DOI 10.1017/CBO9781139172530

J. S. Chahal, Topics in number theory, The University Series in Mathematics, Plenum Press, New York, 1988. MR 955797, DOI 10.1007/978-1-4899-0439-3

David A. Cox, Primes of the form $x^2 + ny^2$, A Wiley-Interscience Publication, John Wiley & Sons, Inc., New York, 1989. Fermat, class field theory and complex multiplication. MR 1028322

J. E. Cremona, Algorithms for modular elliptic curves, Cambridge University Press, Cambridge, 1992. MR 1201151

Stéfane Fermigier, Un exemple de courbe elliptique définie sur $\textbf {Q}$ de rang $\geq 19$, C. R. Acad. Sci. Paris Sér. I Math. 315 (1992), no. 6, 719–722 (French, with English and French summaries). MR 1183810

Gerhard Frey, Links between stable elliptic curves and certain Diophantine equations, Ann. Univ. Sarav. Ser. Math. 1 (1986), no. 1, iv+40. MR 853387

Andrew Granville, The Kummer-Wieferich-Skula approach to the first case of Fermat’s last theorem, Advances in number theory (Kingston, ON, 1991) Oxford Sci. Publ., Oxford Univ. Press, New York, 1993, pp. 479–497. MR 1368443

Andrew Granville and Michael B. Monagan, The first case of Fermat’s last theorem is true for all prime exponents up to 714,591,416,091,389, Trans. Amer. Math. Soc. 306 (1988), no. 1, 329–359. MR 927694, DOI 10.1090/S0002-9947-1988-0927694-5

Benedict H. Gross, Kolyvagin’s work on modular elliptic curves, $L$-functions and arithmetic (Durham, 1989) London Math. Soc. Lecture Note Ser., vol. 153, Cambridge Univ. Press, Cambridge, 1991, pp. 235–256. MR 1110395, DOI 10.1017/CBO9780511526053.009

Benedict H. Gross and Don B. Zagier, Heegner points and derivatives of $L$-series, Invent. Math. 84 (1986), no. 2, 225–320. MR 833192, DOI 10.1007/BF01388809

Dale Husemoller, Elliptic curves, Graduate Texts in Mathematics, vol. 111, Springer-Verlag, New York, 1987. With an appendix by Ruth Lawrence. MR 868861, DOI 10.1007/978-1-4757-5119-2

Kenneth Ireland and Michael Rosen, A classical introduction to modern number theory, 2nd ed., Graduate Texts in Mathematics, vol. 84, Springer-Verlag, New York, 1990. MR 1070716, DOI 10.1007/978-1-4757-2103-4

[14]

A. W. Knapp, Elliptic curves, Math. Notes, vol. 40, Princeton Univ. Press, Princeton, NJ, 1992.

Neal Koblitz, Introduction to elliptic curves and modular forms, Graduate Texts in Mathematics, vol. 97, Springer-Verlag, New York, 1984. MR 766911, DOI 10.1007/978-1-4684-0255-1

V. A. Kolyvagin, Euler systems, The Grothendieck Festschrift, Vol. II, Progr. Math., vol. 87, Birkhäuser Boston, Boston, MA, 1990, pp. 435–483. MR 1106906

Serge Lang, Elliptic curves: Diophantine analysis, Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], vol. 231, Springer-Verlag, Berlin-New York, 1978. MR 518817

H. W. Lenstra Jr., Factoring integers with elliptic curves, Ann. of Math. (2) 126 (1987), no. 3, 649–673. MR 916721, DOI 10.2307/1971363

B. Mazur, Modular curves and the Eisenstein ideal, Inst. Hautes Études Sci. Publ. Math. 47 (1977), 33–186 (1978). With an appendix by Mazur and M. Rapoport. MR 488287

B. Mazur, Number theory as gadfly, Amer. Math. Monthly 98 (1991), no. 7, 593–610. MR 1121312, DOI 10.2307/2324924

K. A. Ribet, On modular representations of $\textrm {Gal}(\overline \textbf {Q}/\textbf {Q})$ arising from modular forms, Invent. Math. 100 (1990), no. 2, 431–476. MR 1047143, DOI 10.1007/BF01231195

Kenneth A. Ribet, From the Taniyama-Shimura conjecture to Fermat’s last theorem, Ann. Fac. Sci. Toulouse Math. (5) 11 (1990), no. 1, 116–139 (English, with English and French summaries). MR 1191476

J.-P. Serre, Lettre à J.-F. Mestre, Current trends in arithmetical algebraic geometry (Arcata, Calif., 1985) Contemp. Math., vol. 67, Amer. Math. Soc., Providence, RI, 1987, pp. 263–268 (French). MR 902597, DOI 10.1090/conm/067/902597

Jean-Pierre Serre, Sur les représentations modulaires de degré $2$ de $\textrm {Gal}(\overline \textbf {Q}/\textbf {Q})$, Duke Math. J. 54 (1987), no. 1, 179–230 (French). MR 885783, DOI 10.1215/S0012-7094-87-05413-5

Joseph H. Silverman, The arithmetic of elliptic curves, Graduate Texts in Mathematics, vol. 106, Springer-Verlag, New York, 1986. MR 817210, DOI 10.1007/978-1-4757-1920-8

Jonathan W. Tanner and Samuel S. Wagstaff Jr., New bound for the first case of Fermat’s last theorem, Math. Comp. 53 (1989), no. 188, 743–750. MR 982371, DOI 10.1090/S0025-5718-1989-0982371-4

John T. Tate, The arithmetic of elliptic curves, Invent. Math. 23 (1974), 179–206. MR 419359, DOI 10.1007/BF01389745

M. Waldschmidt, P. Moussa, J. M. Luck, and C. Itzykson (eds.), From number theory to physics, Springer-Verlag, Berlin, 1992. Papers from the Meeting on Number Theory and Physics held in Les Houches, March 7–16, 1989. MR 1221099, DOI 10.1007/978-3-662-02838-4

André Weil, Über die Bestimmung Dirichletscher Reihen durch Funktionalgleichungen, Math. Ann. 168 (1967), 149–156 (German). MR 207658, DOI 10.1007/BF01361551

[30]

A. Wiles, Modular elliptic curves and Fermat's Last Theorem (to appear).

Review Information:

Reviewer: William R. Hearst III
Reviewer: Kenneth A. Ribet

Journal: Bull. Amer. Math. Soc. 30 (1994), 248-252
DOI: https://doi.org/10.1090/S0273-0979-1994-00465-3