Theta Series of Quaternion Algebras over Function Fields

Let K be the function field over a finite field of odd order, and let H be a definite quaternion algebra over K. If Λ is an order of level M in H, we define theta series for each ideal I of Λ using the reduced norm on H. Using harmonic analysis on the completed algebra H_∞ and the arithmetic of quaternion algebras, we establish a transformation law for these theta series. We also define analogs of the classical Hecke operators and show that in general, the Hecke operators map the theta series to a linear combination of theta series attached to different ideals, a generalization of the classical Eichler Commutation Relation.