Iwasawa theory and Fitting ideals. (English) Zbl 1056.11063

Kurihara, Masato

J. Reine Angew. Math. 561, 39-86 (2003).

Let \(F/{\mathbb Q}\) be an imaginary abelain extension of finite degree and let \(\text{Cl}'(F)\) denote the class group of \(F\) considered over the ring \({\mathbb Z}':={\mathbb Z}[1/2]\), so that it is viewed as a \({\mathbb Z}'[\text{Gal}(F/{\mathbb Q}]\)-module. For any module \(M\) over this group ring, \(M^-\) denotes the submodule on which the complex conjugation acts by \(-1\). Conjectures which go under the general rubric of Main Conjectures, relate algebraic invariants associated to such Galois modules with analytic or arithmetic invariants related to \(p\)-adic \(L\)-functions. The author considers the minus part of the initial Fitting ideal of \(\text{Cl}'(F)^-\) (which is an algebraic invariant) and the minus part of a ‘Stickelberger element’ \(\Theta_{F/{\mathbb Q}} \in {\mathbb Z}'[\text{Gal}(F/{\mathbb Q})]\), which is to be viewed as an arithmetic or analytic object as it is related to zeta functions. He conjectures that these two invariants are actually equal, in the spirit of the Main Conjectures. Of course, this conjecture is equivalent to the two elements being equal when extended to \({\mathbb Z}_p\) for each odd prime \(p\).
The author proves the latter conjecture in the following three cases:
(i) When no prime of \(F^+\) above \(p\) splits in \(F/F^+\).
(ii) When \(F=K_n\), where \(K_n\) is the \(n\)-th cyclotomic layer of the cyclotomic \({\mathbb Z}_p\)-extension of an abelian field extension \(K\) of \(F\) whose degree is prime to \(p\).
(iii) When \(p\) is tamely ramified in \(F/{\mathbb Q}\), and \(F\) does not contain a primitive \(p\)-th root of unity.

Reviewer: R. Sujatha (Bombay)

MSC:

11R23	Iwasawa theory
11R42	Zeta functions and \(L\)-functions of number fields
13C20	Class groups

Keywords:

Iwasawa theory; class group; Fitting ideals

References:

[1]	Coates J., Ann. Math. 95 pp 99– (1972)
[2]	Coates, J. and Greenberg, R., Kummer theory for abelian varieties over local fields, Invent. math. 124 (1996), 129-174. · Zbl 0858.11032
[3]	Coates, J. and Sinnott, W., An analogue of Stickelberger’s theorem for the higher K-groups, Invent. math. 24 (1974), 149-161. · Zbl 0282.12006
[4]	Colmez, P., Residu en s 1des fonctions ze^ta p-adiques, Invent. math. 91 (1988), 371-398.
[5]	Cornacchia, P. and stv r, P. A., On the Coates-Sinnott conjecture, K-Theory 19 (2000), 195-209. · Zbl 1076.11515
[6]	Cornacchia P., J. Number Th. 73 pp 459– (1998)
[7]	Deligne, P. and Ribet, K., Values of abelian L-functions at negative integers over totally real fields, Invent. math. 59 (1980), 227-286. · Zbl 0434.12009
[8]	Ferrero B., Ann. Math. 109 pp 377– (1979)
[9]	Greenberg R., Proc. Nat. Acad. Sci. US 94 pp 11125– (1997)
[10]	Greenberg R., Springer Lect. Notes Math. 1716 pp 51– (1999)
[11]	Greither C., Math. Z. 229 pp 107– (1998)
[12]	Greither C., Math. Z. 233 pp 515– (2000)
[13]	Hachimori Y., J. Alg. Geom. 8 pp 581– (1999)
[14]	Iwasawa K., J. 33 pp 263– (1981)
[15]	Kato K., Springer Lect. Notes Math. 1553 pp 50– (1993)
[16]	Koike M., J. Math. Sci. Univ. Tokyo 6 pp 371– (1999)
[17]	Kolyvagin, V. A., Euler systems, The Grothendieck Festschrift Vol. II (1990), 435-483. · Zbl 0742.14017
[18]	Kurihara M., J. Europ. Math. Soc. 1 pp 35– (1999)
[19]	Kurihara, M., On the TateShafarevich groups over cyclotomic fields of an elliptic curve with supersingular reduction I, Invent. math. 149 (2002), 195-224. · Zbl 1033.11028
[20]	Mazur B., Duke Math. J. 54 (2) pp 711– (1987)
[21]	Mazur, B. and Wiles, A., Class fields of abelian extensions of Q, Invent. math. 76 (1984), 179-330. · Zbl 0545.12005
[22]	Northcott, D. G., Finite free resolutions, Cambridge Univ. Press, Cambridge-New York 1976. · Zbl 0328.13010
[23]	Ritter J., Mem. Amer. Math. Soc. pp 157– (2002)
[24]	Ritter J., Manuscr. Math. 109 pp 131– (2002)
[25]	Rubin K., Progr. Math. 89 pp 309– (1991)
[26]	Rubin, K., Euler systems, Ann. Math. Stud.147, Princeton Univ. Press, 2000. · Zbl 0977.11001
[27]	Schoof, R., The structure of the minus class groups of abelian number fields, Sem. de Theorie des Nombres Paris 1988-89, Birkh user, Boston (1990), 185-204.
[28]	Schoof R., Math. Comput. 67 (223) pp 1225– (1998)
[29]	Serre, J.P., Corps Locaux, Hermann, Paris1968 (troisieme edition).
[30]	Serre, J.P., Cohomologie galoisienne, Lect. Notes Math.5, Springer-Verlag, 1973 (quatrieme edition). · Zbl 0259.12011
[31]	Serre J.-P., Comptes Rendus Acad. Sc. Paris (A) 287 pp 183– (1978)
[32]	Sinnott W., Ann. Math. 108 pp 107– (1978)
[33]	Sinnott, W., On theStickelberger ideal and the circular units of an abelian field, Invent. math. 62 (1980), 181-234. · Zbl 0465.12001
[34]	Solomon D., Grenoble 40 pp 467– (1990)
[35]	Stevens, G., Stickelberger elements and modular parametrizations of elliptic curves, Invent. math. 98 (1989), 75-106. · Zbl 0697.14023
[36]	Sumida H., J. Math. Soc. Japan 49 pp 689– (1997)
[37]	Tate, J., Les conjectures de Stark sur les Fonctions L d’Artin en s 0, Progr. Math. 47, Birkh user, 1984. · Zbl 0545.12009
[38]	Washington L., Grad. Texts Math. pp 83– (1982)
[39]	Wiles A., Ann. Math. 131 pp 493– (1990)
[40]	Wiles A., Ann. Math. 131 pp 555– (1990)
[41]	Yamazaki, Y., On theStickelberger elements and ideal class groups (in Japanese), Master’s thesis, Tokyo Metropolitan Univ., 1998. Department of Mathematics, Tokyo Metropolitan University, Hachioji, Tokyo, 192-0397, Japan