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Bulletin of the American Mathematical Society

ISSN 1088-9485(online) ISSN 0273-0979(print)

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

References [Enhancements On Off] (What's this?)

  • [A1] G. ANDERSON: $t$-motives, Duke Math. J. 53 (1986) 457-502. MR 87j:11042
  • [ABP1] G. ANDERSON, W. D. BROWNAWELL, M. PAPANIKOLAS: Determination of the algebraic relations among special $\Gamma$-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 36:6383
  • [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 $L$-functions over function fields, Math. Ann. 323 (2002) 737-795. MR 2003e:11052
  • [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 97i:01023
  • [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 86h:01004
  • [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 52:5580
  • [Dr2] V.G. DRINFELD: Elliptic modules II, Math. U.S.S.R. Sbornik 31 (1977) 159-170. MR 55:12644
  • [Go1] D. GOSS: The Algebraist's Upper Half Plane, Bull. Amer. Math. Soc. 2 No. 3 (May 1980) 391-415. MR 81g:10042
  • [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 $p$ valued $L$-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 48:8444
  • [KS1] N. KATZ, P. SARNAK: Zeroes of zeta functions and symmetry, Bull. Amer. Math. Soc. (N.S.) 36 (1999) 1-26. MR 2000f:11114
  • [Laf1] L. LAFFORGUE: Chtoucas de Drinfeld et correspondance de Langlands, Invent. Math. 147 (2002) 1-241. MR 2002m:11039
  • [N1] J. NEUKIRCH: Algebraic Number Theory, Springer, Berlin-Heidelberg-New York (1999). MR 2000m:11104
  • [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 $p$, its origin and development. Class field theory--its centenary and prospect (Tokyo, 1998) Adv. Stud. Pure Math. 30, Math. Soc. Japan, Tokyo (2001). MR 2002g:11156
  • [Th1] D. THAKUR: Gamma functions for function fields and Drinfeld modules, Ann. Math. (2) 134 (1991) 25-64. MR 92g:11058
  • [v1] B.L. VAN DER WAERDEN: A History of Algebra, Springer, Berlin-Heidelberg-New York (1985). MR 87e:01001
  • [Wa1] D. WAN: On the Riemann hypothesis for the characteristic $p$ zeta function, J. Number Theory 58 (1996) 196-212. MR 97c:11064
  • [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
MSC (2000): Primary 11R58, 11G09, 11R60
Published electronically: October 29, 2003
Review copyright: © Copyright 2003 American Mathematical Society
American Mathematical Society