Units and class groups in number theory and algebraic geometry

Lang, Serge

doi:10.1090/S0273-0979-1982-14997-7

Units and class groups in number theory and algebraic geometry
HTML articles powered by AMS MathViewer

by Serge Lang PDF

Bull. Amer. Math. Soc. 6 (1982), 253-316

References

Greg W. Anderson, Logarithmic derivatives of Dirichlet $L$-functions and the periods of abelian varieties, Compositio Math. 45 (1982), no. 3, 315–332. MR 656608
A. A. Beĭlinson, Higher regulators and values of $L$-functions of curves, Funktsional. Anal. i Prilozhen. 14 (1980), no. 2, 46–47 (Russian). MR 575206
Fedor Alekseivich Bogomolov, Sur l’algébricité des représentations $l$-adiques, C. R. Acad. Sci. Paris Sér. A-B 290 (1980), no. 15, A701–A703 (French, with English summary). MR 574307
Pierrette Cassou-Noguès, $p$-adic $L$-functions for totally real number field, Proceedings of the Conference on $p$-adic Analysis (Nijmegen, 1978) Report, vol. 7806, Katholieke Univ., Nijmegen, 1978, pp. 24–37. MR 522119
J. Coates and S. Lichtenbaum, On $l$-adic zeta functions, Ann. of Math. (2) 98 (1973), 498–550. MR 330107, DOI 10.2307/1970916
J. Coates and W. Sinnott, On $p$-adic $L$-functions over real quadratic fields, Invent. Math. 25 (1974), 253–279. MR 354615, DOI 10.1007/BF01389730
J. Coates and W. Sinnott, Integrality properties of the values of partial zeta functions, Proc. London Math. Soc. (3) 34 (1977), no. 2, 365–384. MR 439815, DOI 10.1112/plms/s3-34.2.365
J. Coates and A. Wiles, On the conjecture of Birch and Swinnerton-Dyer, Invent. Math. 39 (1977), no. 3, 223–251. MR 463176, DOI 10.1007/BF01402975
J. Coates and A. Wiles, On $p$-adic $L$-functions and elliptic units, J. Austral. Math. Soc. Ser. A 26 (1978), no. 1, 1–25. MR 510581
H. Davenport and H. Heilbronn, On the density of discriminants of cubic fields. II, Proc. Roy. Soc. London Ser. A 322 (1971), no. 1551, 405–420. MR 491593, DOI 10.1098/rspa.1971.0075
P. Deligne, Valeurs de fonctions $L$ et périodes d’intégrales, Automorphic forms, representations and $L$-functions (Proc. Sympos. Pure Math., Oregon State Univ., Corvallis, Ore., 1977) Proc. Sympos. Pure Math., XXXIII, Amer. Math. Soc., Providence, R.I., 1979, pp. 313–346 (French). With an appendix by N. Koblitz and A. Ogus. MR 546622
P. Deligne and M. Rapoport, Les schémas de modules de courbes elliptiques, Modular functions of one variable, II (Proc. Internat. Summer School, Univ. Antwerp, Antwerp, 1972) Lecture Notes in Math., Vol. 349, Springer, Berlin, 1973, pp. 143–316 (French). MR 0337993
V. G. Drinfel′d, Two theorems on modular curves, Funkcional. Anal. i Priložen. 7 (1973), no. 2, 83–84 (Russian). MR 0318157
Bruce Ferrero and Lawrence C. Washington, The Iwasawa invariant $\mu _{p}$ vanishes for abelian number fields, Ann. of Math. (2) 109 (1979), no. 2, 377–395. MR 528968, DOI 10.2307/1971116
Georges Gras, Classes d’idéaux des corps abéliens et nombres de Bernoulli généralisés, Ann. Inst. Fourier (Grenoble) 27 (1977), no. 1, ix, 1–66 (French, with English summary). MR 450238
Ralph Greenberg, On $p$-adic $L$-functions and cyclotomic fields, Nagoya Math. J. 56 (1975), 61–77. MR 360536
Ralph Greenberg, On $p$-adic $L$-functions and cyclotomic fields. II, Nagoya Math. J. 67 (1977), 139–158. MR 444614
Benedict H. Gross, $p$-adic $L$-series at $s=0$, J. Fac. Sci. Univ. Tokyo Sect. IA Math. 28 (1981), no. 3, 979–994 (1982). MR 656068
Benedict H. Gross, On the periods of abelian integrals and a formula of Chowla and Selberg, Invent. Math. 45 (1978), no. 2, 193–211. With an appendix by David E. Rohrlich. MR 480542, DOI 10.1007/BF01390273
Kenkichi Iwasawa, On $\Gamma$-extensions of algebraic number fields, Bull. Amer. Math. Soc. 65 (1959), 183–226. MR 124316, DOI 10.1090/S0002-9904-1959-10317-7
Kenkichi Iwasawa, On $p$-adic $L$-functions, Ann. of Math. (2) 89 (1969), 198–205. MR 269627, DOI 10.2307/1970817
Kenkichi Iwasawa, A class number formula for cyclotomic fields, Ann. of Math. (2) 76 (1962), 171–179. MR 154862, DOI 10.2307/1970270
Donald Kersey, Modular units inside cyclotomic units, Ann. of Math. (2) 112 (1980), no. 2, 361–380. MR 592295, DOI 10.2307/1971150
Daniel S. Kubert, The universal ordinary distribution, Bull. Soc. Math. France 107 (1979), no. 2, 179–202 (English, with French summary). MR 545171
Daniel S. Kubert, The $\textbf {Z}/2\textbf {Z}$ cohomology of the universal ordinary distribution, Bull. Soc. Math. France 107 (1979), no. 2, 203–224 (English, with French summary). MR 545172
Daniel S. Kubert and Serge Lang, Modular units, Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], vol. 244, Springer-Verlag, New York-Berlin, 1981. MR 648603
Daniel S. Kubert and Serge Lang, Modular units inside cyclotomic units, Bull. Soc. Math. France 107 (1979), no. 2, 161–178 (English, with French summary). MR 545170
Serge Lang, Cyclotomic fields, Graduate Texts in Mathematics, Vol. 59, Springer-Verlag, New York-Heidelberg, 1978. MR 0485768
Serge Lang, Elliptic functions, Addison-Wesley Publishing Co., Inc., Reading, Mass.-London-Amsterdam, 1973. With an appendix by J. Tate. MR 0409362
Serge Lang, Division points on curves, Ann. Mat. Pura Appl. (4) 70 (1965), 229–234. MR 190146, DOI 10.1007/BF02410091
Serge Lang, Unramified class field theory over function fields in several variables, Ann. of Math. (2) 64 (1956), 285–325. MR 83174, DOI 10.2307/1969975
Serge Lang, Sur les séries $L$ d’une variété algébrique, Bull. Soc. Math. France 84 (1956), 385–407 (French). MR 88777
R. P. Langlands, Modular forms and $\ell$-adic representations, Modular functions of one variable, II (Proc. Internat. Summer School, Univ. Antwerp, Antwerp, 1972) Lecture Notes in Math., Vol. 349, Springer, Berlin, 1973, pp. 361–500. MR 0354617
Heinrich-Wolfgang Leopoldt, Eine Verallgemeinerung der Bernoullischen Zahlen, Abh. Math. Sem. Univ. Hamburg 22 (1958), 131–140 (German). MR 92812, DOI 10.1007/BF02941946
H. W. Leopoldt, Über Einheitengruppe und Klassenzahl reeller abelscher Zahlkörper, Abh. Deutsch. Akad. Wiss. Berlin. Kl. Math. Nat. 1953 (1953), no. 2, 48 pp. (1954) (German). MR 0067927
Heinrich-Wolfgang Leopoldt, Zur Arithmetik in abelschen Zahlkörpern, J. Reine Angew. Math. 209 (1962), 54–71. MR 139602, DOI 10.1515/crll.1962.209.54
B. Mazur, Modular curves and the Eisenstein ideal, Inst. Hautes Études Sci. Publ. Math. 47 (1977), 33–186 (1978). With an appendix by Mazur and M. Rapoport. MR 488287
B. Mazur, Rational isogenies of prime degree (with an appendix by D. Goldfeld), Invent. Math. 44 (1978), no. 2, 129–162. MR 482230, DOI 10.1007/BF01390348
Barry Mazur, Fermat’s last theorem, Selected Lectures in Mathematics, American Mathematical Society, Providence, RI, 1995. A lecture presented in Vancouver, British Columbia, August 1993. MR 1419093
R. James Milgram, Odd index subgroups of units in cyclotomic fields and applications, Algebraic $K$-theory, Evanston 1980 (Proc. Conf., Northwestern Univ., Evanston, Ill., 1980) Lecture Notes in Math., vol. 854, Springer, Berlin-New York, 1981, pp. 269–298. MR 618309, DOI 10.1007/BFb0089526
John Tate and Frans Oort, Group schemes of prime order, Ann. Sci. École Norm. Sup. (4) 3 (1970), 1–21. MR 265368
K. Ramachandra, Some applications of Kronecker’s limit formulas, Ann. of Math. (2) 80 (1964), 104–148. MR 164950, DOI 10.2307/1970494
Michel Raynaud, Schémas en groupes de type $(p,\dots , p)$, Bull. Soc. Math. France 102 (1974), 241–280 (French). MR 419467
Michel Raynaud, Faisceaux amples sur les schémas en groupes et les espaces homogènes, Lecture Notes in Mathematics, Vol. 119, Springer-Verlag, Berlin-New York, 1970 (French). MR 0260758
Kenneth A. Ribet, A modular construction of unramified $p$-extensions of $Q(\mu _{p})$, Invent. Math. 34 (1976), no. 3, 151–162. MR 419403, DOI 10.1007/BF01403065
Gilles Robert, Unités elliptiques et formules pour le nombre de classes des extensions abéliennes d’un corps quadratique imaginaire, Société Mathématique de France, Paris, 1973 (French). Bull. Soc. Math. France, Mém. No. 36; Supplément au Bull. Soc. Math. France, Tome 101. MR 0469889
Gilles Robert, Nombres de Hurwitz et unités elliptiques, Ann. Sci. École Norm. Sup. (4) 11 (1978), no. 3, 297–389 (French). MR 521636
Jean-Pierre Serre, Propriétés galoisiennes des points d’ordre fini des courbes elliptiques, Invent. Math. 15 (1972), no. 4, 259–331 (French). MR 387283, DOI 10.1007/BF01405086
Goro Shimura, Introduction to the arithmetic theory of automorphic functions, Kanô Memorial Lectures, No. 1, Iwanami Shoten Publishers, Tokyo; Princeton University Press, Princeton, N.J., 1971. Publications of the Mathematical Society of Japan, No. 11. MR 0314766
Takuro Shintani, On a Kronecker limit formula for real quadratic fields, J. Fac. Sci. Univ. Tokyo Sect. IA Math. 24 (1977), no. 1, 167–199. MR 460283
Takuro Shintani, On certain ray class invariants of real quadratic fields, J. Math. Soc. Japan 30 (1978), no. 1, 139–167. MR 498490, DOI 10.2969/jmsj/03010139
Carl Ludwig Siegel, Lectures on advanced analytic number theory, Tata Institute of Fundamental Research Lectures on Mathematics, No. 23, Tata Institute of Fundamental Research, Bombay, 1965. Notes by S. Raghavan. MR 0262150
W. Sinnott, On the Stickelberger ideal and the circular units of a cyclotomic field, Ann. of Math. (2) 108 (1978), no. 1, 107–134. MR 485778, DOI 10.2307/1970932
W. Sinnott, On the Stickelberger ideal and the circular units of an abelian field, Invent. Math. 62 (1980/81), no. 2, 181–234. MR 595586, DOI 10.1007/BF01389158
H. M. Stark, Class fields and modular forms of weight one, Modular functions of one variable, V (Proc. Second Internat. Conf., Univ. Bonn, Bonn, 1976) Lecture Notes in Math., Vol. 601, Springer, Berlin, 1977, pp. 277–287. MR 0450243
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
John T. Tate, Algebraic cycles and poles of zeta functions, Arithmetical Algebraic Geometry (Proc. Conf. Purdue Univ., 1963) Harper & Row, New York, 1965, pp. 93–110. MR 0225778
John T. Tate, The arithmetic of elliptic curves, Invent. Math. 23 (1974), 179–206. MR 419359, DOI 10.1007/BF01389745
H. S. Vandiver, Fermat’s last theorem. Its history and the nature of the known results concerning it, Amer. Math. Monthly 53 (1946), 555–578. MR 18660, DOI 10.2307/2305236
C. T. C. Wall, Classification of Hermitian Forms. VI. Group rings, Ann. of Math. (2) 103 (1976), no. 1, 1–80. MR 432737, DOI 10.2307/1971019
Lawrence C. Washington, The non-$p$-part of the class number in a cyclotomic $\textbf {Z}_{p}$-extension, Invent. Math. 49 (1978), no. 1, 87–97. MR 511097, DOI 10.1007/BF01399512
Andrew Wiles, Modular curves and the class group of $\textbf {Q}(\zeta _{p})$, Invent. Math. 58 (1980), no. 1, 1–35. MR 570872, DOI 10.1007/BF01402272
K\B{o}ichi Yamamoto, The gap group of multiplicative relationships of Gaussian sums, Symposia Mathematica, Vol. XV (Convegno di Strutture in Corpi Algebrici, INDAM, Rome, 1973) Academic Press, London, 1975, pp. 427–440. MR 0382188
Koichi Yamamoto, On a conjecture of Hasse concerning multiplicative relations of Gaussian sums, J. Combinatorial Theory 1 (1966), 476–489. MR 213311
Jing Yu, A cuspidal class number formula for the modular curves $X_{1}(N)$, Math. Ann. 252 (1980), no. 3, 197–216. MR 593633, DOI 10.1007/BF01420083
Horst G. Zimmer, Computational problems, methods, and results in algebraic number theory, Lecture Notes in Mathematics, Vol. 262, Springer-Verlag, Berlin-New York, 1972. MR 0323751