Book Review
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Book Information:
Author:
Michael Rosen
Title:
Number theory in function fields
Additional book information:
Springer-Verlag,
New York,
2002,
xii+358 pp.,
ISBN 0-387-95335-3,
$49.95$
[A1] G. ANDERSON:
-motives, Duke Math. J. 53 (1986) 457-502. MR 0850546
[ABP1] G. ANDERSON, W. D. BROWNAWELL, M. PAPANIKOLAS: Determination of the algebraic relations among special
-values in positive characteristic, Ann. Math. (to appear).
[Ar1] E. ARTIN: Quadratische Körper im Gebiete der höheren Kongruenzen I, II, Math. Z. 19 (1924) 153-246 (= Coll. Papers, 1-94).
[AT1] E. ARTIN, J. TATE: Class Field Theory, Benjamin, New York-Amsterdam (1968). MR 0223335
[AW1] E. ARTIN, G. WHAPLES: Axiomatic characterization of fields by the product formula for valuations, Bull. Amer. Math. Soc. 51 (1945) 469-492. MR 7:111f
[Boc1] G. B¨OCKLE: Global
-functions over function fields, Math. Ann. 323 (2002) 737-795. MR 1924278
[Boc2] G. B¨OCKLE: An Eichler-Shimura isomorphism over function fields between Drinfeld modular forms and cohomology classes of crystals, Lecture Notes in Math. (to appear) (preprint, available at http://www.math.ethz.ch/~boeckle/
).
[Ca1] L. CARLITZ: On certain functions connected with polynomials in a Galois field, Duke Math. J. 1 (1935) 137-168.
[Co1] L. CORRY: Modern Algebra and the Rise of Mathematical Structures, Birkhäuser, Basel (1996). MR 1391720
[D1] R. DEDEKIND: Abriss einer Theorie höheren Congruenzen in Bezug auf einen reelen Rrimzahl-Modulus, J. Reine Angew. Math. 54 (1857) 1-26.
[DW1] R. DEDEKIND, H. WEBER: Theorie der algebraischen Funktionen einer Verändlichen, J. Reine Angew. Math 92 (1882) 181-290.
[Di1] J. DIEUDONNÉ: History of Algebraic Geometry, Wadsworth, Monterey (1985). MR 0780183
[Dr1] V.G. DRINFELD: Elliptic modules, Math. Sbornik 94 (1974) 594-627; English transl.: Math. U.S.S.R. Sbornik 23 (1976) 561-592. MR 0384707
[Dr2] V.G. DRINFELD: Elliptic modules II, Math. U.S.S.R. Sbornik 31 (1977) 159-170. MR 0439758
[Go1] D. GOSS: The Algebraist's Upper Half Plane, Bull. Amer. Math. Soc. 2 No. 3 (May 1980) 391-415. MR 0561525
[Go2] D. GOSS: What is a shtuka? Notices of the Amer. Math. Soc. Vol. 50 No. 1 (2003) 36-37.
[Go3] D. GOSS: The impact of the infinite primes on the Riemann hypothesis for characteristic
valued
-series, in: Algebra, Arithmetic, and Geometry with Applications Papers from Shreeram S. Abhyankar's 70th Birthday Conference, Springer (to appear).
[Go4] D. GOSS: Can a Drinfeld module be modular? J. Ramanujan Math. Soc. 17 No. 4 (2002) 221-260.
[H1] D. HAYES: Explicit class field theory for rational function fields, Trans Amer. Math. Soc. 189 (1974) 77-91. MR 0330106
[KS1] N. KATZ, P. SARNAK: Zeroes of zeta functions and symmetry, Bull. Amer. Math. Soc. (N.S.) 36 (1999) 1-26. MR 1640151
[Laf1] L. LAFFORGUE: Chtoucas de Drinfeld et correspondance de Langlands, Invent. Math. 147 (2002) 1-241. MR 1875184
[N1] J. NEUKIRCH: Algebraic Number Theory, Springer, Berlin-Heidelberg-New York (1999). MR 1697859
[R1] B. RIEMANN: Ueber die Anzahl der Primzahlen unter einer gegebenen Grösse, Monatsberichte der Berliner Akademie (1859); Gesammelte Werke, Teubner, Leipzig (1892).
[Ro1] P. ROQUETTE: Class field theory in characteristic
, its origin and development. Class field theory--its centenary and prospect (Tokyo, 1998) Adv. Stud. Pure Math. 30, Math. Soc. Japan, Tokyo (2001). MR 1846477
[Th1] D. THAKUR: Gamma functions for function fields and Drinfeld modules, Ann. Math. (2) 134 (1991) 25-64. MR 1114607
[v1] B.L. VAN DER WAERDEN: A History of Algebra, Springer, Berlin-Heidelberg-New York (1985). MR 0803326
[Wa1] D. WAN: On the Riemann hypothesis for the characteristic
zeta function, J. Number Theory 58 (1996) 196-212. MR 1387735
[We1] A. WEIL: Variétés Abéliennes et Courbes Algébriques, Hermann (1971). MR 10:621d
- [A1]
- G. ANDERSON:
-motives, Duke Math. J. 53 (1986) 457-502. MR 0850546
- [ABP1]
- G. ANDERSON, W. D. BROWNAWELL, M. PAPANIKOLAS: Determination of the algebraic relations among special
-values in positive characteristic, Ann. Math. (to appear).
- [Ar1]
- E. ARTIN: Quadratische Körper im Gebiete der höheren Kongruenzen I, II, Math. Z. 19 (1924) 153-246 (= Coll. Papers, 1-94).
- [AT1]
- E. ARTIN, J. TATE: Class Field Theory, Benjamin, New York-Amsterdam (1968). MR 0223335
- [AW1]
- E. ARTIN, G. WHAPLES: Axiomatic characterization of fields by the product formula for valuations, Bull. Amer. Math. Soc. 51 (1945) 469-492. MR 7:111f
- [Boc1]
- G. B¨OCKLE: Global
-functions over function fields, Math. Ann. 323 (2002) 737-795. MR 1924278
- [Boc2]
- G. B¨OCKLE: An Eichler-Shimura isomorphism over function fields between Drinfeld modular forms and cohomology classes of crystals, Lecture Notes in Math. (to appear) (preprint, available at
http://www.math.ethz.ch/~boeckle/
).
- [Ca1]
- L. CARLITZ: On certain functions connected with polynomials in a Galois field, Duke Math. J. 1 (1935) 137-168.
- [Co1]
- L. CORRY: Modern Algebra and the Rise of Mathematical Structures, Birkhäuser, Basel (1996). MR 1391720
- [D1]
- R. DEDEKIND: Abriss einer Theorie höheren Congruenzen in Bezug auf einen reelen Rrimzahl-Modulus, J. Reine Angew. Math. 54 (1857) 1-26.
- [DW1]
- R. DEDEKIND, H. WEBER: Theorie der algebraischen Funktionen einer Verändlichen, J. Reine Angew. Math 92 (1882) 181-290.
- [Di1]
- J. DIEUDONNÉ: History of Algebraic Geometry, Wadsworth, Monterey (1985). MR 0780183
- [Dr1]
- V.G. DRINFELD: Elliptic modules, Math. Sbornik 94 (1974) 594-627; English transl.: Math. U.S.S.R. Sbornik 23 (1976) 561-592. MR 0384707
- [Dr2]
- V.G. DRINFELD: Elliptic modules II, Math. U.S.S.R. Sbornik 31 (1977) 159-170. MR 0439758
- [Go1]
- D. GOSS: The Algebraist's Upper Half Plane, Bull. Amer. Math. Soc. 2 No. 3 (May 1980) 391-415. MR 0561525
- [Go2]
- D. GOSS: What is a shtuka? Notices of the Amer. Math. Soc. Vol. 50 No. 1 (2003) 36-37.
- [Go3]
- D. GOSS: The impact of the infinite primes on the Riemann hypothesis for characteristic
valued
-series, in: Algebra, Arithmetic, and Geometry with Applications Papers from Shreeram S. Abhyankar's 70th Birthday Conference, Springer (to appear).
- [Go4]
- D. GOSS: Can a Drinfeld module be modular? J. Ramanujan Math. Soc. 17 No. 4 (2002) 221-260.
- [H1]
- D. HAYES: Explicit class field theory for rational function fields, Trans Amer. Math. Soc. 189 (1974) 77-91. MR 0330106
- [KS1]
- N. KATZ, P. SARNAK: Zeroes of zeta functions and symmetry, Bull. Amer. Math. Soc. (N.S.) 36 (1999) 1-26. MR 1640151
- [Laf1]
- L. LAFFORGUE: Chtoucas de Drinfeld et correspondance de Langlands, Invent. Math. 147 (2002) 1-241. MR 1875184
- [N1]
- J. NEUKIRCH: Algebraic Number Theory, Springer, Berlin-Heidelberg-New York (1999). MR 1697859
- [R1]
- B. RIEMANN: Ueber die Anzahl der Primzahlen unter einer gegebenen Grösse, Monatsberichte der Berliner Akademie (1859); Gesammelte Werke, Teubner, Leipzig (1892).
- [Ro1]
- P. ROQUETTE: Class field theory in characteristic
, its origin and development. Class field theory--its centenary and prospect (Tokyo, 1998) Adv. Stud. Pure Math. 30, Math. Soc. Japan, Tokyo (2001). MR 1846477
- [Th1]
- D. THAKUR: Gamma functions for function fields and Drinfeld modules, Ann. Math. (2) 134 (1991) 25-64. MR 1114607
- [v1]
- B.L. VAN DER WAERDEN: A History of Algebra, Springer, Berlin-Heidelberg-New York (1985). MR 0803326
- [Wa1]
- D. WAN: On the Riemann hypothesis for the characteristic
zeta function, J. Number Theory 58 (1996) 196-212. MR 1387735
- [We1]
- A. WEIL: Variétés Abéliennes et Courbes Algébriques, Hermann (1971). MR 10:621d
Review Information:
Reviewer:
David Goss
Affiliation:
The Ohio State University
Email:
goss@math.ohio-state.edu
Journal:
Bull. Amer. Math. Soc.
41 (2004), 127-133
Published electronically:
October 29, 2003
Review copyright:
© Copyright 2003
American Mathematical Society