UC San Diego

In this dissertation, we study a class of non-holomorphic, cohomological automorphic functions on the split, simply connected, spin group G=Spin(4,4). Following ideas of Gross-Wallach, Gan-Gross-Savin, M. Weissman, and A. Pollack, we term these automorphic functions quaternionic modular forms on G, and analyze a theory of scalar-valued Fourier coefficients associated to them. Our results build parallels between the theory of quaternionic modular forms on G, and the arithmetically rich theory of genus two Siegel modular forms. The main result states that a level one quaternionic modular form on G is determined by its primitive Fourier coefficients. As an input to this result, we develop a theory of Fourier-Jacobi expansions for quaternionic modular forms, in which the non-degenerate coefficients are themselves genus two Siegel modular forms. Our primary application strengthens earlier joint work of the author with J. Johnson-Leung, I. Negrini, M. Roy, and A. Pollack. More precisely, in this dissertation we show that the level one quaternionic modular forms on SO(4,4) that arise as theta lifts from Sp(4) admit an elementary Fourier coefficient theoretic characterization, which is akin to a characterization of the classical Saito-Kurokawa subspace proven by D. Zagier.