The main objects of study in this work are the function field analogues of Shimura curves arising from indefinite quaternion algebras over Q . We study these curves and their Jacobians mostly using rigid-analytic techniques. Our main results are: (1) A description of the action of the Hecke operators in terms of the rigid-analytic uniformization. (2) An analytic construction of elliptic curves associated to harmonic Hecke eigenforms via Tate periods. (3) The calculation of the degree of the modular parametrization in terms of a pairing on harmonic cochains. (4) Applications to computer calculations.
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