Binary quadratic forms and Eichler orders
Montserrat Alsina1
1Dept. Matemàtica Aplicada III EUPM Av. Bases de Manresa 61-73, Manresa-08240, Catalunya, Spain
Journal de théorie des nombres de Bordeaux, Volume 17 (2005) no. 1, pp. 13-23

For any Eichler order 𝒪(D,N) of level N in an indefinite quaternion algebra of discriminant D there is a Fuchsian group Γ(D,N)⊆SL(2,ℝ) and a Shimura curve X(D,N). We associate to 𝒪(D,N) a set ℋ(𝒪(D,N)) of binary quadratic forms which have semi-integer quadratic coefficients, and we develop a classification theory, with respect to Γ(D,N), for primitive forms contained in ℋ(𝒪(D,N)). In particular, the classification theory of primitive integral binary quadratic forms by SL(2,ℤ) is recovered. Explicit fundamental domains for Γ(D,N) allow the characterization of the Γ(D,N)-reduced forms.

Pour tout ordre d’Eichler 𝒪(D,N) de niveau N dans une algèbre de quaternions indéfinie de discriminant D, il existe un groupe Fuchsien Γ(D,N)⊆SL(2,ℝ) et une courbe de Shimura X(D,N). Nous associons à 𝒪(D,N) un ensemble ℋ(𝒪(D,N)) de formes quadratiques binaires ayant des coefficients semi-entiers quadratiques et developpons une classification des formes quadratiques primitives de ℋ(𝒪(D,N)) pour rapport à Γ(D,N). En particulier nous retrouvons la classification des formes quadratiques primitives et entières de SL(2,ℤ). Un domaine fondamental explicite pour Γ(D,N) permet de caractériser les Γ(D,N) formes réduites.

Published online:
DOI: 10.5802/jtnb.473

Montserrat Alsina  1

1 Dept. Matemàtica Aplicada III EUPM Av. Bases de Manresa 61-73, Manresa-08240, Catalunya, Spain 
Montserrat Alsina. Binary quadratic forms and Eichler orders. Journal de théorie des nombres de Bordeaux, Volume 17 (2005) no. 1, pp. 13-23. doi: 10.5802/jtnb.473
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