In this thesis, we study modular forms on definite and indefinite quaternion algebras. These spaces are a priori very different. On the definite side they are abstract spaces of functions defined on finite sets, whereas on the indefinite side they are sections of an appropriate sheaf on a Shimura curve. We construct an explicit, canonical, and Hecke equivariant isomorphism between these spaces with $Q_p$-coefficients, where $p$ is a prime dividing the level of the modular forms on the definite quaternion algebra. Our map takes the form: [ cL_k(U, Q_p)^{p-new} ra H^0(X',Omega^{tensor k/2}) ] see Theorem ref{EJL-isom} for details. There are natural $Z_p$ lattices $cM$ and $mathcal{N}$ on the left and right respectively. This isomorphism carries $cM$ to $mathcal{N}$, and for $p > max(k-2,3)$ restricts to an isomorphism $cM isom mathcal{N}$. The quotient $mathcal{N}/cM$ is a canonical and finitely generated $p$-torsion Hecke module. Our isomorphism is an explicit, and $Z_p$-integral refinement of the Jacquet-Langlands correspondence in our setting.PhDMathematicsUniversity of Michigan, Horace H. Rackham School of Graduate Studieshttp://deepblue.lib.umich.edu/bitstream/2027.42/133472/1/psuchand_1.pdf