D4 modular forms

In this paper, we study modular forms on two simply connected groups of type D4 over Q. One group, G_s, is a globally split group of type D₄, viewed as the group of isotopies of the split rational octonions. The other, G_c, is the isotopy group of the rational (nonsplit) octonions. We study automorphic forms on G_s in analogy to the work of Gross, Gan, and Savin on G₂; namely we study automorphic forms whose component at infinity corresponds to a quaternionic discrete series representation. We study automorphic forms on G_c using Gross's formalism of "algebraic modular forms." Finally, we follow work of Gan, Savin, Gross, Rallis, and others, to study an exceptional theta correspondence connecting modular forms on G_c and G_s. This can be thought of as an octonionic generalization of the Jacquet-Langlands correspondence.