  We present a novel integral representation for a quotient of global
automorphic L-functions, the symmetric square over the exterior square. The
pole of this integral characterizes a period of a residual representation of an
Eisenstein series. As such, the integral itself constitutes a period, of an
arithmetic nature. The construction involves the study of local and global
aspects of a new model for double covers of general linear groups, the
metaplectic Shalika model. In particular, we prove uniqueness results over
p-adic and Archimedean fields, and a new Casselman-Shalika type formula.
