Date of Award
Open Access Dissertation
Matthew G. Boylan
In 1973, Shimura introduced a family of maps between modular forms of half-integral weight and modular forms of even integral weight. We will give explicit formulas for the images of two different classes of modular forms under these maps. In contrast to Shimura's difficult analytic construction, our formulas will fall out of relatively simple combinatorial derivations. Using the Shimura correspondence, we will prove congruences for the eigenvalues of a family of eigenforms introduced by Garvan. Using deep results of Eichler and Shimura, we state these congruences in terms of the number of points on associated elliptic curves, and we provide a table of these congruences for reference. Finally, we find a family of integral weight modular forms analogous to Garvan's half-integral weight family.
Brown, K. A.(2013). Shimura Images of A Family of Half-Integral Weight Modular Forms. (Doctoral dissertation). Retrieved from https://scholarcommons.sc.edu/etd/2436