In this paper we describe how to perform computations with Witt vectors of length in an efficient way and give a formula that allows us to compute the third coordinate of the Greenberg transform of a polynomial directly. We apply these results to obtain information on the third coordinate of the -invariant of the canonical lifting as a function on the -invariant of the ordinary elliptic curve in characteristic .
Dans cet article, nous décrivons comment effectuer des calculs avec les vecteurs de Witt de longueur d’une manière efficace et donnons une formule qui permet de calculer directement la troisième coordonnée de la transformée de Greenberg d’un polynôme. Nous appliquons ces résultats afin d’obtenir des renseignements sur la troisième coordonnée de l’invariant du relèvement canonique en fonction de l’invariant de la courbe elliptique ordinaire en caractéristique .
Mots-clés : Witt vectors, elliptic curves, canonical lifting, pseudo-canonical lifting, modular polynomial
@article{JTNB_2011__23_2_417_0, author = {Finotti, Lu{\'\i}s R.~A.}, title = {Computations with {Witt} vectors of length $3$}, journal = {Journal de th\'eorie des nombres de Bordeaux}, pages = {417--454}, publisher = {Soci\'et\'e Arithm\'etique de Bordeaux}, volume = {23}, number = {2}, year = {2011}, doi = {10.5802/jtnb.770}, zbl = {1269.13003}, mrnumber = {2817938}, language = {en}, url = {http://www.numdam.org/articles/10.5802/jtnb.770/} }
TY - JOUR AU - Finotti, Luís R. A. TI - Computations with Witt vectors of length $3$ JO - Journal de théorie des nombres de Bordeaux PY - 2011 SP - 417 EP - 454 VL - 23 IS - 2 PB - Société Arithmétique de Bordeaux UR - http://www.numdam.org/articles/10.5802/jtnb.770/ DO - 10.5802/jtnb.770 LA - en ID - JTNB_2011__23_2_417_0 ER -
%0 Journal Article %A Finotti, Luís R. A. %T Computations with Witt vectors of length $3$ %J Journal de théorie des nombres de Bordeaux %D 2011 %P 417-454 %V 23 %N 2 %I Société Arithmétique de Bordeaux %U http://www.numdam.org/articles/10.5802/jtnb.770/ %R 10.5802/jtnb.770 %G en %F JTNB_2011__23_2_417_0
Finotti, Luís R. A. Computations with Witt vectors of length $3$. Journal de théorie des nombres de Bordeaux, Volume 23 (2011) no. 2, pp. 417-454. doi : 10.5802/jtnb.770. http://www.numdam.org/articles/10.5802/jtnb.770/
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