×

Binary quadratic forms and Eichler orders. (English) Zbl 1079.11022

Let \(\mathcal O\) be an Eichler order in an indefinite quaternion algebra \(H\). So there is a standard embedding \(H\hookrightarrow\text{M}_2(\mathbb R)\), and each element \(\left(\begin{smallmatrix} a & b\\ c & d\end{smallmatrix}\right)\) of \(H\) can be associated with a quadratic form \(cx^2+(d-a)xy-by^2\). Therefore, the order \(\mathcal O\) gives rise to a class \(\mathcal H(\mathcal O)\) of binary quadratic forms. With respect to the Fuchsian group \(\Gamma(\mathcal O) :=\{ \alpha\in\mathcal O\mid \text{nr}(\alpha)>0\}\subset\text{SL}_2(\mathbb R)\), the author classifies primitive forms in \(\mathcal H(\mathcal O)\), generalizing the classical theory over \(\text{SL}_2(\mathbb R)\). Furthermore, the \(\Gamma(\mathcal O)\)-reduced forms are descibed in terms of fundamental domains.

MSC:

11E16 General binary quadratic forms
11G18 Arithmetic aspects of modular and Shimura varieties
11R52 Quaternion and other division algebras: arithmetic, zeta functions
PDF BibTeX XML Cite
Full Text: DOI Numdam EuDML Link

References:

[1] M. Alsina, A. Arenas, P. Bayer (eds.), Corbes de Shimura i aplicacions. STNB, Barcelona, 2001.
[2] M. Alsina, P. Bayer, Quaternion orders, quadratic forms and Shimura curves. CRM Monograph Series, vol. 22, American Mathematical Society, Providence, RI, 2004. · Zbl 1073.11040
[3] M. Alsina, Dominios fundamentales modulares. Rev. R. Acad. Cienc. Exact. Fis. Nat. 94 (2000), no. 3, 309-322. · Zbl 1024.11024
[4] M. Eichler, Zur Zahlentheorie der Quaternionen-Algebren. J. reine angew. Math. 195 (1955), 127-151. · Zbl 0068.03303
[5] 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. · Zbl 0531.14014
[6] G. Shimura, Construction of class fields and zeta functions of algebraic curves. Annals of Math. 85 (1967), 58-159. · Zbl 0204.07201
[7] M.F. Vigneras, Arithmétique des algèbres de quaternions. Lecture Notes in Math., no. 800, Springer, 1980. · Zbl 0422.12008
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.