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

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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 
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     author = {L. V. Kuz'min},
     title = {On $\ell$-adic logarithms of {Gauss} sums},
     journal = {Izvestiya. Mathematics },
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     number = {3},
     year = {2014},
     language = {en},
     url = {http://geodesic.mathdoc.fr/item/IM2_2014_78_3_a5/}
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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/