Perfect unary forms over real quadratic fields
Journal de Théorie des Nombres de Bordeaux, Tome 25 (2013) no. 3, pp. 759-775.

Soit F=(d) un corps quadratique réel avec anneau d’entiers 𝒪. Dans cet article, nous analysons le nombre h d de GL 1 (𝒪)-orbites de classes d’homothétie des formes parfaites unaires sur F en fonction de d. Nous calculons h d exactement pour d200000, sans carré. En reliant les formes parfaites aux fractions continues, nous donnons des bornes sur h d et répondons à certaines questions de Watanabe, Yano et Hayashi.

Let F=(d) be a real quadratic field with ring of integers 𝒪. In this paper we analyze the number h d of GL 1 (𝒪)-orbits of homothety classes of perfect unary forms over F as a function of d. We compute h d exactly for square-free d200000. By relating perfect forms to continued fractions, we give bounds on h d and address some questions raised by Watanabe, Yano, and Hayashi.

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DOI : https://doi.org/10.5802/jtnb.854
Classification : 11E12
Mots clés : quadratic forms, perfect forms, continued fractions, real quadratic fields
@article{JTNB_2013__25_3_759_0,
     author = {Dan  Yasaki},
     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 = {https://jtnb.centre-mersenne.org/articles/10.5802/jtnb.854/}
}
Dan  Yasaki. 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. https://jtnb.centre-mersenne.org/articles/10.5802/jtnb.854/

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