Geodesic
On $\ell$-adic logarithms of Gauss sums
Izvestiya. Mathematics , Tome 78 (2014) no. 3, pp. 531-553.

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

For the Gauss sum $S(\chi)$ corresponding to a character $\chi$ of order $\ell^md$ of the multiplicative group of a finite field $\mathbb F_q$ of characteristic $p$, we obtain an approximate formula for the $\ell$-adic logarithm of $S(\chi)$. We construct a special basis in the group of logarithms and define modulo $\ell^m$ the coefficients of $\log_\ell(S(\chi))$ relative to this basis (modulo $\ell^{m-1}$ if $\ell=2$). These coefficients are defined in terms of power residues of some cyclotomic numbers at the places over $p$.
Keywords: cyclotomic units, Iwasawa theory, reciprocity laws.
Mots-clés : Gauss sums 
@article{IM2_2014_78_3_a5,
     author = {L. V. Kuz'min},
     title = {On $\ell$-adic logarithms of {Gauss} sums},
     journal = {Izvestiya. Mathematics },
     pages = {531--553},
     publisher = {mathdoc},
     volume = {78},
     number = {3},
     year = {2014},
     language = {en},
     url = {http://geodesic.mathdoc.fr/item/IM2_2014_78_3_a5/}
}
TY  - JOUR
AU  - L. V. Kuz'min
TI  - On $\ell$-adic logarithms of Gauss sums
JO  - Izvestiya. Mathematics 
PY  - 2014
SP  - 531
EP  - 553
VL  - 78
IS  - 3
PB  - mathdoc
UR  - http://geodesic.mathdoc.fr/item/IM2_2014_78_3_a5/
LA  - en
ID  - IM2_2014_78_3_a5
ER  - 
%0 Journal Article
%A L. V. Kuz'min
%T On $\ell$-adic logarithms of Gauss sums
%J Izvestiya. Mathematics 
%D 2014
%P 531-553
%V 78
%N 3
%I mathdoc
%U http://geodesic.mathdoc.fr/item/IM2_2014_78_3_a5/
%G en
%F IM2_2014_78_3_a5
L. V. Kuz'min. On $\ell$-adic logarithms of Gauss sums. Izvestiya. Mathematics , Tome 78 (2014) no. 3, pp. 531-553. http://geodesic.mathdoc.fr/item/IM2_2014_78_3_a5/

[1] S. Lang, “Cyclotomic fields I and II”, Grad. Texts in Math., 121, Springer-Verlag, New York, 1990 | DOI | MR | Zbl

[2] B. H. Gross, N. Koblitz, “Gauss sums and the $p$-adic $\Gamma$-function”, Ann. of Math. (2), 109:3 (1979), 569–581 | DOI | MR | Zbl

[3] H. Miki, “On the $l$-adic expansion of certain Gauss sums and its applications”, Galois representations and arithmetic algebraic geometry (Kyoto 1985/Tokyo 1986), Adv. Stud. Pure Math., 12, North-Holland, Amsterdam, 1987, 87–118 | MR | Zbl

[4] L. V. Kuz'min, “Some explicit calculations in local and global cyclotomic fields”, Proc. Steklov Inst. Math., 208 (1995), 179–197 | MR | Zbl

[5] W. Sinnott, “On the Stickelberger ideal and the circular units of a cyclotomic field”, Ann. of Math. (2), 108:1 (1978), 107–134 | DOI | MR | Zbl