| L. Adleman/M. Huang: Primality testing and abelian varieties over finite fields. SLN Math. 1512 (1992). 5375 G. Belyi: On Galois extensions of the maximal cyclotomic field. Math. USSR Izvestiya 14 (1980), 247-256. Massimo Bertolini/Giuseppe Canuto: La congettura di Shimura-Taniyama-Weil. Boll. UMI 10-A (1996), 213-247. This expository paper outlines the proof of the conjecture of Shimura-Taniyama-Weil for semistable elliptic curves by Wiles and illustrates some consequences of this work on Fermat's last theorem and the conjecture of Birch and Swinnerton-Dyer. 8336 Amnon Besser: Euler systems for higher-weight modular forms. Internet 1996, 6p. G. Billing/K. Mahler: On exceptional points on cubic curves. J. London Math. Soc. 15 (1940), 32-43. The authors show that on an elliptic curve defined over Q there don't exist rational points of order 11. 1990 S. Bloch: The proof of the Mordell conjecture. Math. Intell. 6/2 (1984), 41-47. Enrico Bombieri: The Mordell conjecture revisited. Annali di Pisa 17 (1990), 615-640. 3453 A. Brumer/O. McGuiness: The behaviour of the Mordell-Weil group of elliptic curves. Bull. AMS 23 (1990), 375-382. A. Buium: Differential algebra and diophantine geometry. Hermann 1994, 190p. 2-705-66226-X. FFR 130. "The book develops differential algebraic geometry, a geometry in which local theory is provided by classical differential algebra ... This theory has intriguing applications to diophantine geometry: the author gives new proofs of the conjectures of Lang and Mordell over function fields of characteristic zero." (EMS Newsletter). Fabrizio Catanese (ed.): Arithmetic geometry. Symp. Math. 37 (1997), 300p. 2681 J.S. Chahal: Topics in number theory. Plenum Press 1988. 7806 Barry Cipra: Fermat prover points to next challenges. Science 22 March 1996, 1668-1669. 3277 John Coates: Elliptic curves with complex multiplication and Iwasawa theory. Bull. London Math.Soc. 23 (1991), 321-350. R. Coleman: Effective Chabauty. Duke Math. J. 52 (1985), 765-770. Very sharp upper estimates for the number of rational points in special cases. 5678 Jean-Louis Colliot-Thelene/Dimitri Kanevsky/Jean-Jacques Sansuc: Arithmetique des surfaces cubiques diagonales. 1938 Wstholz, 1-108. 6324 Jean-Louis Colliot-Thelene/Kazuya Kato/Paul Vojta (ed.): Arithmetic algebraic geometry. SLN Math. 1553 (1993), 220p. 3-540-57110-8. DM 82. 1744 Gary Cornell/Joseph Silverman (ed.): Arithmetic geometry. Springer 1986. Standard reference. Gary Cornell/Joseph Silverman/Glenn Stevens (ed.): Modular forms and Fermat's last theorem. Springer 1997, 3-540-94609-8. $50. 3445 Pierre Deligne: Preuve des conjectures de Tate et de Shafarevich. Asterisque 121/122 (1985, 25-41. B. Edixhoven/J.-H. Evertse: Diophantine approximation and abelian varieties. SLN Math. 1566 (1993). 3-540-57528-6. DM 34. Fabiano/G. Pucci/A. Yger: Effective Nullstellensatz and geometric degree for zero-dimensional ideals. Acta Arithm. 78 (1996), 165-187. 1858 Gerd Faltings: Die Vermutungen von Tate und Mordell. Jber. DMV 86 (1984), 1-13. 1859 Gerd Faltings: Endlichkeitssaetze fuer abelsche Varietaeten ueber Zahlkoerpern. Inv. Math. 73 (1983), 349-366. Gerd Faltings: Lectures on the arithmetic Riemann-Roch theorem. Annals of Mathematics Studies 1993. Paperback ISBN 0-691-02544-4. $15. The arithmetic Riemann-Roch theorem has been shown recently by Bismut, Gillet and Soule'. The proof mixes algebra, arithmetic and analysis. "This book contains very deep and quite recent results. ... In contrast to the very interesting contents the style of presentation seems rather problematica to me ... There is more or less no motivation for definitions and results, and it is also not indicated what the results could be used for ... " (A. Cap). 4784 Gerd Faltings: Recent progress in diophantine geometry. 4727 Casacuberta/Castellet, 78-86. Gerd Faltings: Calculus on arithmetic surfaces. Annals Math. 118 (1984), 387-424. 1844 Gerd Faltings/Gisbert Wuestholz (ed.): Rational points. Vieweg 1986. 3604 Eberhard Freitag/Reinhardt Kiel: Etale cohomology and the Weil conjecture. Springer 1988. Gerhard Frey: Links between solutions of A-B=C and elliptic curves. SLN Math. 1380 (1989), 31-62. Gerhard Frey: Rationale Punkte auf Fermatkurven und getwisteten Modulkurven. J. reine u. angew. Math. 331 (1982), 185-191. Gerhard Frey: Links between stable elliptic curves and certain diophantine equations. Ann. Univ. Saraviensis 1 (1986), 1-40. Gerhard Frey: On Artin's conjecture for odd 2-dimensional representations. SLN Math. 1585 (1994). 3-540-58387-4. 3716 G. van der Geer/F. Oort/J. Steenbrink (ed.): Arithmetic algebraic geometry. Birkhaeuser 1991. Fernando Gouvea/Noriko Yui: Arithmetic of diagonal hypersurfaces over finite fields. Cambridge UP 1995, 180p. 0-521-49834-1. $33. This book deals with the arithmetic of diagonal hypersurfaces over finite fields, with special focus on the Tate conjecture and the Lichtenbaum-Milne formula for the central value of the L-function. Gu''nter Harder: Eisensteinkohomologie und die Konstruktion gemischter Motive. SLN Math. 1562 (1993), 180p. 3-540-57408-5. 9663 Gu''nter Harder: Wittvektoren. Jber. DMV 99 (1997), 18-48. Yves Hellegouarch: Courbes elliptiques et quations de Fermat. These, Besancon 1972 (?). Yves Hellegouarch: Invitation aux mathematiques de Fermat-Wiles. Masson 1997, 400p. ISBN 2-225-83008-8 (pb) (or ISSN 1269-7842). 5363 John Horgan: Fermat's MacGuffin. Scientific American September 1993, 14-15. In June 1993 Andrew Wiles proposed a proof of Fermat's last theorem, although the complete paper, 200 pages long, has still to be examined in detail, most experts believe the proof should be true. For seven years, after that Frey and Ribet had reduced the problem to a (difficult!) problem about elliptic curves, Wiles virtually stopped writing papers, attending conferences or even reading anything unrelated to his goal. 4732 Wilfred Hulsbergen: Conjectures in arithmetic algebraic geometry. Vieweg 1992. 2718 Horst Knoerrer a.o.: Arithmetik und Geometrie. Birkhaeuser 1986. 1850 Neal Koblitz (ed.): Number theory related to Fermat's last theorem. Birkhaeuser 1982. 2054 V. Kolyvagin: On the Mordell-Weil group and the Shafarevich-Tate group of modular elliptic curves. MPI Mathematik Bonn 69/1990. 1848 Hanspeter Kraft: Algebraische Kurven und diophantische Gleichungen. 1847 Borho, 93-114. 3450 Gerhard Kramarz: All congruent number less than 2000. Math. Ann. 273 (1986), 337-340. 1885 Serge Lang: Integral points on curves. Publ. IHES 6 (1960), 27-43. Serge Lang: Higher dimensional diophantine problems. Bull. AMS 80 (1974), 779-788. 1889 Serge Lang: Hyperbolic and diophantine analysis. Bull. AMS 14 (1986), 159-205. 5207 Serge Lang: Vojta's conjecture. SLN Math. 1111 (1985), 407-419. 2015 Serge Lang: Fundamentals of diophantine geometry. Springer 1983. Serge Lang: Number theory III. Diophantine geometry. Springer 1991, 300p. DM 128. "Das vorliegende Buch gibt einen hervorragenden und geschmackvollen Ueberblick ueber die diophantische Geometrie." (G. Wuestholz). 4652 Serge Lang: Introduction to Arakelov theory. Springer 1988. 3607 Serge Lang: Elliptic curves - diophantine analysis. Springer 1978. Michael Larsen: Unitary groups and l-adic representations. Thesis. Princeton UP 1988. Michael Larsen: Arthmetic compactification of some Shimura surfaces. See Zentralblatt 760 (1993), 57. Qing Liu: Algebraic geometry and arithmetic curves. Oxford UP 2002, 460p. Pds 40. 3427 David Masser: Counting points of small height on elliptic curves. Bull. Soc.Math. France 117 (1989), 247-265. 3428 David Masser/Gisbert Wuestholz: Estimating isogenies on elliptic curves. Inv. Math. 100 (1990), 1-24. 3093 Barry Mazur: Number theory as gadfly. Am. Math. Monthly 98 (1991), 593-610. Predicts the key role of Taniyama's conjecture in the proof of Fermat's theorem. At the same time an introduction to Riemann surfaces for beginners! Very beautiful. 4935 Barry Mazur: Arithmetic on curves. Bull. AMS 14 (1986), 207-259. Barry Mazur: Modular curves and the Eisenstein ideal. Publ. Math. IHES 47 (1977), 33-186. Barry Mazur: Rational isogenies of prime degree. Inv. Math. 44 (1978), 129-162. Barry Mazur/Andrew Wiles: Class fields of abelian extensions of Q. Inv. Math. 76 (1984), 179-330. J.-F. Mestre: Construction of an elliptic curve of rank ³ 12. Comptes Rendus 295 (1982), 643-644. J. Mestre: Formules explicites et minorations de conducteurs de varietes algebriques. Comp. Math. circa 58 (1986), 209-232. On the rank of the group of rational points of an elliptic curve. Carlos Moreno: Algebraic curves over finite fields. Cambridge UP 1990, 270p. 0-521-34252-x. Pds. 30. Should be somewhat difficult to read. J. Oesterle': Nouvelles approches du theoreme de Fermat. Asterisque 161-162 (1988), 165-186. Explains the link between Fermat's problem and the associated elliptic curve introduced by Hellegouarch and Frey. The proofs make essential use of the arithmetic theory of modular forms. A. Parshin: Algebraic curves over function fields I. Izv. Ak. Nauk SSSR 32 (1968), 1145-1170. A. Parshin: Quelques conjectures de finitude en geometrie diophantienne. Actes Congr. Int. Math. 1 (1970), 467-471. E. Peyre/Y. Tschinkel (ed.): Rational points on algebraic varieties. Birkha''user 2001, 450p. Eur 85. 4888 Christoph Poeppe: Der Beweis der Fermatschen Vermutung. Spektrum 1993/8, 14-16. Alf van der Poorten: Notes on Fermat's last theorem. Wiley 1996, 220p. 0-471-06261-8. Paulo Ribenboim: Fermat's last theorem for amateurs. Springer 1999. 3-540-98508-5. $40. Kenneth Ribet: On modular representations of Gal(A/Q) arising from modular forms. Inv. Math. 100 (1990), 431-476. [A=algebraic numbers.] Kenneth Ribet: Twists of modular forms and endomorphisms of abelian varieties. Math. Annalen 253 (1980), 43-62. Kenneth Ribet: From the Taniyama-Shimura conjecture to Fermat's last theorem. Ann. Fac. Sci. Toulouse Math. 11 (1990), 116-139. 5641 Kenneth Ribet: Wiles proves Taniyama's conjecture; Fermat's last theorem follows. Notices AMS 40 (1993), 575-576. 7178 Kenneth Ribet: Galois representations and modular forms. Bull. AMS 32 (1995), 375-402. 8338 Karl Rubin: Modularity of mod 5 representations. Internet 1995, 9p. An elliptic curve defined over Q and semistable at 3 and 5 is modular. 7768 Karl Rubin: Euler systems and exact formulas in number theory. Jber. DMV 98 (1996), 30-39. 3449 P. Satge': Un analogue du calcul de Heegner. Inv.Math. 87 (1987), 425-439. 1962 S. Schanuel: Heights in number fields. Bull. SMF 107 (1979), 433-449. 4801 Claus-Guenther Schmidt: Die Fermat-Kurve und ihre Jacobi-Mannigfaltigkeit.2718 Knoerrer, 9-28. 3594 Claus-Guenther Schmidt: Arithmetik abelscher Varietaeten mit komplexer Multiplikation. SLN Math. 1082 (1984). 3430 R. Schoof: Elliptic curves over finite fields and the computation of square roots mod p. Math. Comp. 44 (1985), 483-494. Jean-Pierre Serre: Lectures on the Mordell-Weil theorem. Vieweg 1989, 220p. DM 52. Jean-Pierre Serre: Proprietes galoisiennes des points d'ordre fini des courbes elliptiques. Inv. Math. 15 (1972), 259-331. Jean-Pierre Serre: Sur les representations modulaires de degre' ? de Gal(A/Q). Duke Math. J. 54 (1987), 179-230. [A=algebraic numbers.] Goro Shimura: Correspondances modulaires et les fonctions zeta de courbes algebriques. J. Math. Soc. Japan 10 (1958), 1-28. Goro Shimura: On the factors of the Jacobian variety of a modular function field. J. Math. Soc. Japan 25 (1973), 523-544. Goro Shimura: Class fields over real quadratic fields and Hecke operators. Annals Math. 95 (1972), 130-190. Goro Shimura: On elliptic curves with complex multiplication as factors of the jacobians of modular function fields. Nagoya Math. J. 43 (1971), 199-208. 3150 T. Shioda: Mordell-Weil lattices and sphere packings. Am. J. Math. 113 (1991), 931-948. 1919 Joseph Silverman: Lower bound for the canonical height on elliptic curves. Duke Math. J. 48 (1981), 633-648. 11731 Simon Singh/Kenneth Ribet: Die Lo''sung des Fermatschen Ra''tsels. Spektrum 1998/1, 96-103. C. Soule'/D. Abramovich/J.-F. Burnol/J. Kramer: Lectures on Arakelov geometry. Cambridge UP, 190p. 0-521-41669-8. Pds. 30. S. Stepanov: Arithmetic of algebraic curves. Consultants Bureau 1994. 0-306-11036-9. G. Stevens: Stickelberger elements and modular parametrizations of elliptic curves. Inv. Math. 98 (1989), 75-106. On uniformization of elliptic curves by modular curves. 3560 N. Suwa: Fermat motives and the Artin-Tate formula II. Proc.Japan Ac. 67A (1991), 135-138. 8706 Peter Swinnerton-Dyer: Diophantine equations - the geometric approach. Jber. DMV 98 (1996), 146-164. 3444 L. Szpiro: La conjecture de Mordell. Asterisque 121/122 (1985), 83-103. 3322 L. Szpiro (ed.): Seminaire sur les pinceaux arithmetique: la conjecture de Mordell. Asterisque 127 (1985). J. Tunnell: Artin's conjecture for representations of octahedral type. Bull. AMS 5 (1981), 173-175. V. Voevodsky/G. Shabat: Equilateral triangulations of Riemann surfaces and curves over algebraic number fields. Circa 1990. 1737 Paul Vojta: Diophantine approximations and value distribution theory. SLN Math. 1239 (1987). Paul van Wamelen: On the CM character of the curves y^2=x^q-1. J. Number Theory 64 (1997), 59-83. Andre' Weil: L'arithmetique sur les courbes algebriques. Acta Math. 52 (1928), 281-315. Andre' Weil: The field of definition of a variety. Am. J. Math. 78 (1956), 509-524. Andrew Wiles: Modular elliptic curves and Fermat's last theorem. Ann. Math. 141 (1995), 443-551. 1938 Gisbert Wuestholz (ed.): Diophantine approximation and transcendence theory. SLN Math. 1290 (1987). J. Zarhin: Isogenies of abelian varieties over fields of finite characteristics. Mat. Sb. 95/137/3 (1974), 451-461. 1853 Horst Gu''nter Zimmer: Zur Arithmetik der elliptischen Kurven. Grazer Berichte 271 (1986), 110p. | |
|
