Book Review
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MathSciNet review:
1567788
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Book Information:
Author:
Serge Lang
Title:
Introduction to Arakelov theory
Additional book information:
Springer-Verlag, New York, Berlin, Heidelberg, 1988, x + 187 pp., $49.95. ISBN 0-387-96793-1.
Allen Altman and Steven Kleiman, Introduction to Grothendieck duality theory, Lecture Notes in Mathematics, Vol. 146, Springer-Verlag, Berlin-New York, 1970. MR 0274461
S. Ju. Arakelov, An intersection theory for divisors on an arithmetic surface, Izv. Akad. Nauk SSSR Ser. Mat. 38 (1974), 1179–1192 (Russian). MR 0472815
S. J. Arakelov, Theory of intersections on the arithmetic surface, Proceedings of the International Congress of Mathematicians (Vancouver, B.C., 1974) Canad. Math. Congress, Montreal, Que., 1975, pp. 405–408. MR 0466150
4. M. Arbarello, M. Cornalba, P. Griffiths, and J. Harris, Geometry of algebraic curves, Springer-Verlag, Berlin and New York, 1985.
T. Chinburg, An introduction to Arakelov intersection theory, Arithmetic geometry (Storrs, Conn., 1984) Springer, New York, 1986, pp. 289–307. MR 861981
T. Chinburg, Minimal models for curves over Dedekind rings, Arithmetic geometry (Storrs, Conn., 1984) Springer, New York, 1986, pp. 309–326. MR 861982
Gerd Faltings, Calculus on arithmetic surfaces, Ann. of Math. (2) 119 (1984), no. 2, 387–424. MR 740897, DOI 10.2307/2007043
William Fulton, Intersection theory, Ergebnisse der Mathematik und ihrer Grenzgebiete (3) [Results in Mathematics and Related Areas (3)], vol. 2, Springer-Verlag, Berlin, 1984. MR 732620, DOI 10.1007/978-3-662-02421-8
Phillip Griffiths and Joseph Harris, Principles of algebraic geometry, Pure and Applied Mathematics, Wiley-Interscience [John Wiley & Sons], New York, 1978. MR 507725
Robin Hartshorne, Algebraic geometry, Graduate Texts in Mathematics, No. 52, Springer-Verlag, New York-Heidelberg, 1977. MR 0463157
Ernst Kunz, Kähler differentials, Advanced Lectures in Mathematics, Friedr. Vieweg & Sohn, Braunschweig, 1986. MR 864975, DOI 10.1007/978-3-663-14074-0
Serge Lang, Diophantine geometry, Interscience Tracts in Pure and Applied Mathematics, No. 11, Interscience Publishers (a division of John Wiley & Sons, Inc.), New York-London, 1962. MR 0142550
Stephen Lichtenbaum, Curves over discrete valuation rings, Amer. J. Math. 90 (1968), 380–405. MR 230724, DOI 10.2307/2373535
I. R. Shafarevich, Lectures on minimal models and birational transformations of two dimensional schemes, Tata Institute of Fundamental Research Lectures on Mathematics and Physics, No. 37, Tata Institute of Fundamental Research, Bombay, 1966. Notes by C. P. Ramanujam. MR 0217068
Paul Vojta, Diophantine approximations and value distribution theory, Lecture Notes in Mathematics, vol. 1239, Springer-Verlag, Berlin, 1987. MR 883451, DOI 10.1007/BFb0072989
- 1.
- S. Altman and S. Kleinman, Introduction to Grothendieck duality theory, Lecture Notes in Math., vol. 146, Springer-Verlag, Berlin and New York, 1970. MR 0274461
- 2.
- S. Arakelov, Intersection theory of divisors on an arithmetic surface, Izv. Akad. Nauk 38(1974), 1179-1192. MR 472815
- 3.
- S. Arakelov, Theory of intersections of divisors on an arithmetic surface, Proc. Internat. Congr. Math., Vancouver, 1974, pp. 405-408. MR 466150
- 4.
- M. Arbarello, M. Cornalba, P. Griffiths, and J. Harris, Geometry of algebraic curves, Springer-Verlag, Berlin and New York, 1985.
- 5.
- T. Chinberg, An introduction to Arakelov intersection theory, Arithmetic Geometry (G. Cornell and J. Silverman, eds. ), Springer-Verlag, Berlin and New York, 1986, pp. 289-308. MR 861981
- 6.
- T. Chinberg, Minimal models for curves over Dedekind rings, Arithmetic Geometry (G. Cornell and J. Silverman, eds. ), Springer-Verlag, Berlin and New York, 1986, pp. 309-326. MR 861982
- 7.
- G. Faltings, Calculus on arithmetic surfaces, Ann. of Math. (2) 119(1984), 387-424. MR 740897
- 8.
- W. Fulton, Intersection theory, Springer-Verlag, Berlin and New York, 1984. MR 732620
- 9.
- P. Griffiths and J. Harris, Principles of algebraic geometry, John Wiley & Sons, New York, 1978. MR 507725
- 10.
- R. Hartshorne, Algebraic geometry, Springer-Verlag, Berlin and New York, 1977. MR 463157
- 11.
- E. Kunz, Kähler differentials, Vieweg Verlag, 1986. MR 864975
- 12.
- S. Lang, Fundamentals of diophantine geometry, Springer-Verlag, Berlin and New York, 1983. MR 142550
- 13.
- S. Lichtenbaum, Curves over discrete valuation rings, Amer. J. Math. 90(1968), 380-405. MR 230724
- 14.
- I. Shafarevich, On minimal models and birational transformations, Tata Institute, Bombay, India, 1966. MR 217068
- 15.
- P. Vojta, Diophantine approximations and value distribution theory, Lecture Notes in Math., vol. 1239, Springer-Verlag, Berlin and New York, 1987. MR 883451
Review Information:
Reviewer:
Joseph H. Silverman
Journal:
Bull. Amer. Math. Soc.
21 (1989), 171-176
DOI:
https://doi.org/10.1090/S0273-0979-1989-15806-0
