Binary quadratic forms and Eichler orders
Journal de théorie des nombres de Bordeaux, Volume 17 (2005) no. 1, p. 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,) and a Shimura curve X(D,N). We associate to 𝒪(D,N) a set (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 (𝒪(D,N)). In particular, the classification theory of primitive integral binary quadratic forms by SL(2,) 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,) et une courbe de Shimura X(D,N). Nous associons à 𝒪(D,N) un ensemble (𝒪(D,N)) de formes quadratiques binaires ayant des coefficients semi-entiers quadratiques et developpons une classification des formes quadratiques primitives de (𝒪(D,N)) pour rapport à Γ(D,N). En particulier nous retrouvons la classification des formes quadratiques primitives et entières de SL(2,). Un domaine fondamental explicite pour Γ(D,N) permet de caractériser les Γ(D,N) formes réduites.

     author = {Alsina, Montserrat},
     title = {Binary quadratic forms and Eichler orders},
     journal = {Journal de th\'eorie des nombres de Bordeaux},
     publisher = {Universit\'e Bordeaux 1},
     volume = {17},
     number = {1},
     year = {2005},
     pages = {13-23},
     doi = {10.5802/jtnb.473},
     mrnumber = {2152207},
     zbl = {1079.11022},
     language = {en},
     url = {}
Alsina, Montserrat. 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.

[AAB01] M. Alsina, A. Arenas, P. Bayer (eds.), Corbes de Shimura i aplicacions. STNB, Barcelona, 2001.

[AB04] M. Alsina, P. Bayer, Quaternion orders, quadratic forms and Shimura curves. CRM Monograph Series, vol. 22, American Mathematical Society, Providence, RI, 2004. | MR 2038122 | Zbl 1073.11040

[Als00] M. Alsina, Dominios fundamentales modulares. Rev. R. Acad. Cienc. Exact. Fis. Nat. 94 (2000), no. 3, 309–322. | MR 1843547 | Zbl 1024.11024

[Eic55] M. Eichler, Zur Zahlentheorie der Quaternionen-Algebren. J. reine angew. Math. 195 (1955), 127–151. | MR 80767 | Zbl 0068.03303

[Ogg83] A. P. Ogg, Real points on Shimura curves. Arithmetic and geometry, Vol. I, Progr. Math., vol. 35, Birkhäuser Boston, Boston, MA, 1983, pp. 277–307. | MR 717598 | Zbl 0531.14014

[Shi67] G. Shimura, Construction of class fields and zeta functions of algebraic curves. Annals of Math. 85 (1967), 58–159. | MR 204426 | Zbl 0204.07201

[Vig80] M.F. Vigneras, Arithmétique des algèbres de quaternions. Lecture Notes in Math., no. 800, Springer, 1980. | MR 580949 | Zbl 0422.12008