Arithmetic in Quaternion Algebras and Quaternionic Modular Forms

Wiebe, Jordan

Arithmetic in Quaternion Algebras and Quaternionic Modular Forms

dc.contributor.advisor	Martin, Kimball
dc.contributor.author	Wiebe, Jordan
dc.contributor.committeeMember	Schmidt, Ralf
dc.contributor.committeeMember	Roche, Alan
dc.contributor.committeeMember	Pitale, Ameya
dc.contributor.committeeMember	Cheng, Qi
dc.date.accessioned	2019-05-09T15:28:45Z
dc.date.available	2019-05-09T15:28:45Z
dc.date.issued	2019-05-10
dc.date.manuscript	2019-05-09
dc.description.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.	en_US
dc.identifier.uri	https://hdl.handle.net/11244/319651
dc.language	en_US	en_US
dc.rights	Attribution-NoDerivatives 4.0 International	*
dc.rights.uri	http://creativecommons.org/licenses/by-nd/4.0/	*
dc.subject	Number theory	en_US
dc.subject	Quaternion algebras	en_US
dc.subject	Orders in quaternion algebras	en_US
dc.subject	Modular forms	en_US
dc.thesis.degree	Ph.D.	en_US
dc.title	Arithmetic in Quaternion Algebras and Quaternionic Modular Forms	en_US
ou.group	College of Arts and Sciences::Department of Mathematics	en_US
shareok.orcid	0000-0002-2990-8616	en_US