  There are few constructions of square-integrable automorphic functions on
metaplectic groups. Such functions may be obtained by the residues of certain
Eisenstein series on covers of groups, "theta functions," but the Fourier
coefficients of these residues are not well-understood, even for low degree
covers of $GL_2$. Patterson and Chinta-Friedberg-Hoffstein proposed conjectured
relations for the Fourier coefficients of the $GL_2$ quartic and sextic theta
functions (resp.), each obtained from a conjectured equality of non-Eulerian
Dirichlet series. In this article we propose a new framework for constructing
specific $L^2$ metaplectic functions and for understanding these conjectures:
descent integrals. We study descent integrals which begin with theta functions
on covers of larger rank classical groups and use them to construct certain
$L^2$ metaplectic functions on covers related to $GL_2$. We then establish
information about the Fourier coefficients of these metaplectic automorphic
functions, properties which are consistent with the conjectures of Patterson
and Chinta-Friedberg-Hoffstein. In particular, we prove that Fourier
coefficients of the descent functions are arithmetic for infinitely many primes
$p$. We also show that they generate a representation with non-zero projection
to the space of theta. We conjecture that the descents may be used to realize
the quartic and sextic theta functions. Moreover, this framework suggests that
each of the conjectures of Patterson and Chinta-Friedberg-Hoffstein is the
first in a series of relations between certain Fourier coefficients of two
automorphic forms on different covering groups.

Number Theory