Geodesic
On coherent families of uniformizing elements in some towers of Abelian extensions of local number fields
Fundamentalʹnaâ i prikladnaâ matematika, Tome 14 (2008) no. 8, pp. 151-157 
L. V. Kuz'min. On coherent families of uniformizing elements in some towers of Abelian extensions of local number fields. Fundamentalʹnaâ i prikladnaâ matematika, Tome 14 (2008) no. 8, pp. 151-157. http://geodesic.mathdoc.fr/item/FPM_2008_14_8_a8/
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     title = {On coherent families of uniformizing elements in some towers of {Abelian} extensions of local number fields},
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     url = {http://geodesic.mathdoc.fr/item/FPM_2008_14_8_a8/}
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Voir la notice de l'article provenant de la source Math-Net.Ru

For a local number field $K$ with the ring of integers $\mathcal O_K$, the residue field $\mathbb F_q$, and uniformizing $\pi$, we consider the Lubin–Tate tower $K_\pi=\bigcup_{n\geq0}K_n$, where $K_n=K(\pi_n)$, $f(\pi_0)=0$, and $f(\pi_{n+1})=\pi_n$. Here $f(X)$ defines the endomorphism $[\pi]$ of the Lubin–Tate group. If $q\neq2$, then for any formal power series $g(X)\in\mathcal O_K[[X]]$ the following equality holds: $\sum_{n=0}^\infty\mathrm{Sp}_{K_n/K}g(\pi_n)=-g(0)$. One has a similar equality in the case $q=2$.

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[2] Kuzmin L. V., “Novoe dokazatelstvo odnoi teoremy dvoistvennosti o $l$-adicheskikh logarifmakh lokalnykh edinits”, Itogi nauki i tekhn. Ser. Sovr. mat. i eë pril., 45, VINITI, M., 1997, 72–81

[3] Kuzmin L. V., “Nekotorye zamechaniya o $l$-adicheskom regulyatore. III”, Izv. RAN. Ser. mat., 63:6 (1999), 29–82 | MR | Zbl

[4] Kuzmin L. V., “Ob odnom svoistve $l$-adicheskikh logarifmov edinits lokalnykh neabelevykh polei”, Izv. RAN. Ser. mat., 70:5 (2006), 97–122 | MR | Zbl