American Mathematical Society

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, $49.95$

Greg W. Anderson, $t$-motives, Duke Math. J. 53 (1986), no. 2, 457–502. MR 850546, DOI https://doi.org/10.1215/S0012-7094-86-05328-7

G. ANDERSON, W. D. BROWNAWELL, M. PAPANIKOLAS: Determination of the algebraic relations among special $\Gamma$ -values in positive characteristic, Ann. Math. (to appear).

E. ARTIN: Quadratische Körper im Gebiete der höheren Kongruenzen I, II, Math. Z. 19 (1924) 153-246 (= Coll. Papers, 1-94).

E. Artin and J. Tate, Class field theory, W. A. Benjamin, Inc., New York-Amsterdam, 1968. MR 0223335

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

Gebhard Böckle, Global $L$-functions over function fields, Math. Ann. 323 (2002), no. 4, 737–795. MR 1924278, DOI https://doi.org/10.1007/s002080200325

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/).

L. CARLITZ: On certain functions connected with polynomials in a Galois field, Duke Math. J. 1 (1935) 137-168.

Leo Corry, Modern algebra and the rise of mathematical structures, Science Networks. Historical Studies, vol. 17, Birkhäuser Verlag, Basel, 1996. MR 1391720

R. DEDEKIND: Abriss einer Theorie höheren Congruenzen in Bezug auf einen reelen Rrimzahl-Modulus, J. Reine Angew. Math. 54 (1857) 1-26.

R. DEDEKIND, H. WEBER: Theorie der algebraischen Funktionen einer Verändlichen, J. Reine Angew. Math 92 (1882) 181-290.

Jean Dieudonné, History of algebraic geometry, Wadsworth Mathematics Series, Wadsworth International Group, Belmont, CA, 1985. An outline of the history and development of algebraic geometry; Translated from the French by Judith D. Sally. MR 780183
V. G. Drinfel′d, Elliptic modules, Mat. Sb. (N.S.) 94(136) (1974), 594–627, 656 (Russian). MR 0384707
V. G. Drinfel′d, Elliptic modules. II, Mat. Sb. (N.S.) 102(144) (1977), no. 2, 182–194, 325 (Russian). MR 0439758
David Goss, The algebraist’s upper half-plane, Bull. Amer. Math. Soc. (N.S.) 2 (1980), no. 3, 391–415. MR 561525, DOI https://doi.org/10.1090/S0273-0979-1980-14751-5

D. GOSS: What is a shtuka? Notices of the Amer. Math. Soc. Vol. 50 No. 1 (2003) 36-37.

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).

D. GOSS: Can a Drinfeld module be modular? J. Ramanujan Math. Soc. 17 No. 4 (2002) 221-260.

D. R. Hayes, Explicit class field theory for rational function fields, Trans. Amer. Math. Soc. 189 (1974), 77–91. MR 330106, DOI https://doi.org/10.1090/S0002-9947-1974-0330106-6
Nicholas M. Katz and Peter Sarnak, Zeroes of zeta functions and symmetry, Bull. Amer. Math. Soc. (N.S.) 36 (1999), no. 1, 1–26. MR 1640151, DOI https://doi.org/10.1090/S0273-0979-99-00766-1
Laurent Lafforgue, Chtoucas de Drinfeld et correspondance de Langlands, Invent. Math. 147 (2002), no. 1, 1–241 (French, with English and French summaries). MR 1875184, DOI https://doi.org/10.1007/s002220100174
Jürgen Neukirch, Algebraic number theory, Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], vol. 322, Springer-Verlag, Berlin, 1999. Translated from the 1992 German original and with a note by Norbert Schappacher; With a foreword by G. Harder. MR 1697859

B. RIEMANN: Ueber die Anzahl der Primzahlen unter einer gegebenen Grösse, Monatsberichte der Berliner Akademie (1859); Gesammelte Werke, Teubner, Leipzig (1892).

Peter Roquette, Class field theory in characteristic $p$, its origin and development, Class field theory—its centenary and prospect (Tokyo, 1998) Adv. Stud. Pure Math., vol. 30, Math. Soc. Japan, Tokyo, 2001, pp. 549–631. MR 1846477, DOI https://doi.org/10.2969/aspm/03010549
Dinesh S. Thakur, Gamma functions for function fields and Drinfel′d modules, Ann. of Math. (2) 134 (1991), no. 1, 25–64. MR 1114607, DOI https://doi.org/10.2307/2944332
B. L. van der Waerden, A history of algebra, Springer-Verlag, Berlin, 1985. From al-Khwārizmī to Emmy Noether. MR 803326
Daqing Wan, On the Riemann hypothesis for the characteristic $p$ zeta function, J. Number Theory 58 (1996), no. 1, 196–212. MR 1387735, DOI https://doi.org/10.1006/jnth.1996.0074

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