Geodesic
Some remarks on the $l$-adic regulator.~II
Izvestiya. Mathematics , Tome 35 (1990) no. 1, pp. 113-144.

Voir la notice de l'article provenant de la source Math-Net.Ru

Given an algebraic number field $k$ with unit group $U(k)$ and a prime number $l$, consider the bilinear form $S\colon(U(k)\otimes\mathbf Z_l)(U(k)\otimes\mathbf Z_l)\to\mathbf Q_l$, $S(x,y)=\operatorname{Sp}_{k/\mathbf Q}(\log x\cdot\log y)$ where $\log$ is the $l$-adic logarithm. For certain types of fields it is shown that the form $S$ is nondegenerate. We investigate the behavior of the rank of the kernel of $S$ on the family of intermediate fields in a $\mathbf Z_l$-extension $k_\infty/k$. Bibliography: 11 titles. 
@article{IM2_1990_35_1_a5,
     author = {L. V. Kuz'min},
     title = {Some remarks on the $l$-adic {regulator.~II}},
     journal = {Izvestiya. Mathematics },
     pages = {113--144},
     publisher = {mathdoc},
     volume = {35},
     number = {1},
     year = {1990},
     language = {en},
     url = {http://geodesic.mathdoc.fr/item/IM2_1990_35_1_a5/}
}
TY  - JOUR
AU  - L. V. Kuz'min
TI  - Some remarks on the $l$-adic regulator.~II
JO  - Izvestiya. Mathematics 
PY  - 1990
SP  - 113
EP  - 144
VL  - 35
IS  - 1
PB  - mathdoc
UR  - http://geodesic.mathdoc.fr/item/IM2_1990_35_1_a5/
LA  - en
ID  - IM2_1990_35_1_a5
ER  - 
%0 Journal Article
%A L. V. Kuz'min
%T Some remarks on the $l$-adic regulator.~II
%J Izvestiya. Mathematics 
%D 1990
%P 113-144
%V 35
%N 1
%I mathdoc
%U http://geodesic.mathdoc.fr/item/IM2_1990_35_1_a5/
%G en
%F IM2_1990_35_1_a5
L. V. Kuz'min. Some remarks on the $l$-adic regulator.~II. Izvestiya. Mathematics , Tome 35 (1990) no. 1, pp. 113-144. http://geodesic.mathdoc.fr/item/IM2_1990_35_1_a5/

[1] Artin E., Hasse H., “Die beiden Ergänzungssätze zum Reziprozitätsgesetz der $l^n$-ten Potenzreste im Körper der $l^n$-ten Einheitswurzeln”, Abh. Math. Semin. Hamburg, 6 (1928), 146–162 | DOI | Zbl

[2] Babaitsev V. A., “O nekotorykh voprosakh teorii $\Gamma$-rasshirenii polei algebraicheskikh chisel”, Izv. AN SSSR. Ser. matem., 40:3 (1976), 477–487 | MR

[3] Ferrer B., Washington L., “The Iwasawa invariant $\mu_p$ vanishes for abelian number fields”, Ann. of Math., 109 (1979), 377–395 | DOI | MR | Zbl

[4] Iwasawa K., “On some modules in the theory of cyclotomic fields”, Math. Soc. Japan, 20 (1964), 42–82 | MR

[5] Iwasawa K., Lectures on $p$-adic $L$-functions, Princeton Univ. Press., Princeton, 1972 | MR | Zbl

[6] Kuzmin L. V., “Modul Teita polei algebraicheskikh chisel”, Izv. AN SSSR. Ser. matem., 36:2 (1972), 267–327 | MR

[7] Kuzmin L. V., “Nekotorye teoremy dvoistvennosti dlya krugovykh $\Gamma$-rasshirenii polei algebraicheskikh chisel SM-tipa”, Izv. AN SSSR. Ser. matem., 43:3 (1979), 483–546 | MR

[8] Kuzmin L. V., “Nekotorye zamechaniya o $l$-adicheskoi teoreme Dirikhle i $l$-adicheskom regulyatore”, Izv. AN SSSR. Ser. matem., 45:6 (1981), 1203–1240 | MR

[9] Lang S., Cyclotomic fields, II, Grad. Texts Math., 69, Springer, N.Y., 1980 | MR | Zbl

[10] Shafarevich I. R., “Obschii zakon vzaimnosti”, Matem. sb., 26(68):1 (1950), 113–146 | Zbl

[11] Iwasawa K., “On the $\mu$-invariants of $\mathbf{Z}_l$-extensions”, Number theory, algebraic geometry and commutative algebra in honour Akizuki, Tokyo, 1973, 1–11 | MR | Zbl