Cohomology of units and 𝐿-values at zero

Ritter, Jürgen; Weiss, Alfred

doi:10.1090/S0894-0347-97-00234-8

Cohomology of units and $L$-values at zero
HTML articles powered by AMS MathViewer

by Jürgen Ritter and Alfred Weiss

J. Amer. Math. Soc. 10 (1997), 513-552

DOI: https://doi.org/10.1090/S0894-0347-97-00234-8

PDF | Request permission

References

Armand Brumer, On the units of algebraic number fields, Mathematika 14 (1967), 121–124. MR 220694, DOI 10.1112/S0025579300003703
T. Chinburg, On the Galois structure of algebraic integers and $S$-units, Invent. Math. 74 (1983), no. 3, 321–349. MR 724009, DOI 10.1007/BF01394240
T. Chinburg, Derivatives of $L$-functions at $s=0$, Compositio Math. 48 (1983), no. 1, 119–127. MR 700583
Ted Chinburg, Exact sequences and Galois module structure, Ann. of Math. (2) 121 (1985), no. 2, 351–376. MR 786352, DOI 10.2307/1971177
Albrecht Fröhlich, Galois module structure of algebraic integers, Ergebnisse der Mathematik und ihrer Grenzgebiete (3) [Results in Mathematics and Related Areas (3)], vol. 1, Springer-Verlag, Berlin, 1983. MR 717033, DOI 10.1007/978-3-642-68816-4
A. Fröhlich, $L$-values at zero and multiplicative Galois module structure (also Galois Gauss sums and additive Galois module structure), J. Reine Angew. Math. 397 (1989), 42–99. MR 993218, DOI 10.1515/crll.1989.397.42
Cornelius Greither, Class groups of abelian fields, and the main conjecture, Ann. Inst. Fourier (Grenoble) 42 (1992), no. 3, 449–499 (English, with English and French summaries). MR 1182638, DOI 10.5802/aif.1299
Greither, C. and Holland, D., Chinburg’s third invariant in the defect class group for abelian extensions of $\Q$. Preprint December 1993
Ralph Greenberg, On $p$-adic $L$-functions and cyclotomic fields. II, Nagoya Math. J. 67 (1977), 139–158. MR 444614, DOI 10.1017/S0027763000022583
Ralph Greenberg, On $p$-adic Artin $L$-functions, Nagoya Math. J. 89 (1983), 77–87. MR 692344, DOI 10.1017/S0027763000020250
Ralph Greenberg, On the structure of certain Galois groups, Invent. Math. 47 (1978), no. 1, 85–99. MR 504453, DOI 10.1007/BF01609481
K. W. Gruenberg and A. Weiss, Galois invariants for units, Proc. London Math. Soc. (3) 70 (1995), no. 2, 264–284. MR 1309230, DOI 10.1112/plms/s3-70.2.264
Kenkichi Iwasawa, On $\textbf {Z}_{l}$-extensions of algebraic number fields, Ann. of Math. (2) 98 (1973), 246–326. MR 349627, DOI 10.2307/1970784
Kolster, M., $K$-theory and Arithmetic. Fields Institute Monographs, AMS; in preparation
Stephen Lichtenbaum, On the values of zeta and $L$-functions. I, Ann. of Math. (2) 96 (1972), 338–360. MR 360527, DOI 10.2307/1970792
Stephen Lichtenbaum, Values of zeta and $L$-functions at zero, Journées Arithmétiques de Bordeaux (Conf., Univ. Bordeaux, Bordeaux, 1974) Astérisque, Nos. 24-25, Soc. Math. France, Paris, 1975, pp. 133–138. MR 0401711
J. Martinet, Character theory and Artin $L$-functions, Algebraic number fields: $L$-functions and Galois properties (Proc. Sympos., Univ. Durham, Durham, 1975) Academic Press, London, 1977, pp. 1–87. MR 0447187
B. Mazur and A. Wiles, Class fields of abelian extensions of $\textbf {Q}$, Invent. Math. 76 (1984), no. 2, 179–330. MR 742853, DOI 10.1007/BF01388599
J. Ritter and A. Weiss, On the local Galois structure of $S$-units, Algebra and number theory (Essen, 1992) de Gruyter, Berlin, 1994, pp. 229–245. MR 1285369
—, A Tate sequence for global units. Compositio Math. 102 (1996), 147-178
Jean-Pierre Serre, Corps locaux, Publications de l’Institut de Mathématique de l’Université de Nancago, VIII, Hermann, Paris, 1962 (French). MR 0150130
Solomon, D., On Lichtenbaum’s conjecture in the case of number fields. Ph.D. thesis, Brown University 1988
H. M. Stark, Values of $L$-functions at $s=1$. I. $L$-functions for quadratic forms, Advances in Math. 7 (1971), 301–343 (1971). MR 289429, DOI 10.1016/S0001-8708(71)80009-9
J. Tate, The cohomology groups of tori in finite Galois extensions of number fields, Nagoya Math. J. 27 (1966), 709–719. MR 207680, DOI 10.1017/S0027763000026490
John Tate, Les conjectures de Stark sur les fonctions $L$ d’Artin en $s=0$, Progress in Mathematics, vol. 47, Birkhäuser Boston, Inc., Boston, MA, 1984 (French). Lecture notes edited by Dominique Bernardi and Norbert Schappacher. MR 782485
Lawrence C. Washington, Introduction to cyclotomic fields, Graduate Texts in Mathematics, vol. 83, Springer-Verlag, New York, 1982. MR 718674, DOI 10.1007/978-1-4684-0133-2
Weiss, A., Multiplicative Galois module structure. Fields Institute Monographs 5, AMS (1996)
A. Wiles, The Iwasawa conjecture for totally real fields, Ann. of Math. (2) 131 (1990), no. 3, 493–540. MR 1053488, DOI 10.2307/1971468
A. Wiles, On a conjecture of Brumer, Ann. of Math. (2) 131 (1990), no. 3, 555–565. MR 1053490, DOI 10.2307/1971470