Soit un corps quadratique réel avec anneau d’entiers . Dans cet article, nous analysons le nombre de -orbites de classes d’homothétie des formes parfaites unaires sur en fonction de . Nous calculons exactement pour , sans carré. En reliant les formes parfaites aux fractions continues, nous donnons des bornes sur et répondons à certaines questions de Watanabe, Yano et Hayashi.
Let be a real quadratic field with ring of integers . In this paper we analyze the number of -orbits of homothety classes of perfect unary forms over as a function of . We compute exactly for square-free . By relating perfect forms to continued fractions, we give bounds on and address some questions raised by Watanabe, Yano, and Hayashi.
Mots-clés : quadratic forms, perfect forms, continued fractions, real quadratic fields
@article{JTNB_2013__25_3_759_0,
author = {Yasaki, Dan},
title = {Perfect unary forms over real quadratic fields},
journal = {Journal de th\'eorie des nombres de Bordeaux},
pages = {759--775},
publisher = {Soci\'et\'e Arithm\'etique de Bordeaux},
volume = {25},
number = {3},
year = {2013},
doi = {10.5802/jtnb.854},
zbl = {06291373},
mrnumber = {3179682},
language = {en},
url = {http://www.numdam.org/articles/10.5802/jtnb.854/}
}
TY - JOUR AU - Yasaki, Dan TI - Perfect unary forms over real quadratic fields JO - Journal de théorie des nombres de Bordeaux PY - 2013 SP - 759 EP - 775 VL - 25 IS - 3 PB - Société Arithmétique de Bordeaux UR - http://www.numdam.org/articles/10.5802/jtnb.854/ DO - 10.5802/jtnb.854 LA - en ID - JTNB_2013__25_3_759_0 ER -
Yasaki, Dan. Perfect unary forms over real quadratic fields. Journal de théorie des nombres de Bordeaux, Tome 25 (2013) no. 3, pp. 759-775. doi : 10.5802/jtnb.854. http://www.numdam.org/articles/10.5802/jtnb.854/
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