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Binary quadratic forms and Eichler orders

Montserrat Alsina — 2005

Journal de Théorie des Nombres de Bordeaux

For any Eichler order $𝒪 (D, N)$ of level $N$ in an indefinite quaternion algebra of discriminant $D$ there is a Fuchsian group $Γ (D, N) \subseteq SL (2, ℝ)$ 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...

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