Arithmetic in Quaternion Algebras and Quaternionic Modular Forms

Abstract

This dissertation has two parts. In the first part, we revisit the correspondence between spaces of modular forms and orders in quaternion algebras addressed first by Eichler and completed by Hijikata, Pizer, and Shemanske, using an arbitrary definite quaternion algebra with arbitrary level. We present explicit bases for orders of arbitrary level N>1 in definite rational quaternion algebras. These orders have applications to computations of spaces of elliptic and quaternionic modular forms.

In the second part, we investigate the behavior of quaternionic modular forms. In particular, we calculate quaternionic modular forms of weight 2, and illustrate a use of the orders constructed in the first part. We use these forms to explore the behavior of spaces of quaternionic cusp forms of weight 2 and level N, and make a number of conjectures concerning the behavior of zeros of such quaternionic modular forms. In particular, we use dimension formulas and the action of involutions on our space to predict certain zeros of quaternionic modular forms (which we call trivial zeros), and conjecture that the ratio of the number of zerofree forms of level < N to the number of forms with no trivial zeros tends to 1 as N goes to infinity. Finally, we analyze asymptotics of the growth rate of trivial zeros, and provide a histogram of the distribution of nontrivial zeros with respect to the degrees of factors associated to them. We also provide data on a variety of quaternionic modular forms in Appendix A.