  Topological pairing of composite fermions has led to remarkable ideas, such
as excitations obeying non-Abelian braid statistics and topological quantum
computation. We construct a $p$-wave paired Bardeen-Cooper-Schrieffer (BCS)
wave function for composite fermions in the torus geometry, which is a
convenient geometry for formulating momentum space pairing as well as for
revealing the underlying composite-fermion Fermi sea. Following the standard
BCS approach, we minimize the Coulomb interaction energy at half filling in the
lowest and the second Landau levels, which correspond to filling factors
$\nu=1/2$ and $\nu=5/2$ in GaAs quantum wells, by optimizing two variational
parameters that are analogous to the gap and the Debye cut-off energy of the
BCS theory. Our results show no evidence for pairing at $\nu=1/2$ but a clear
evidence for pairing at $\nu=5/2$. To a good approximation, the highest overlap
between the exact Coulomb ground state at $\nu=5/2$ and the BCS state is
obtained for parameters that minimize the energy of the latter, thereby
providing support for the physics of composite-fermion pairing as the mechanism
for the $5/2$ fractional quantum Hall effect. We discuss the issue of modular
covariance of the composite-fermion BCS wave function, and calculate its Hall
viscosity and pair correlation function. By similar methods, we look for but do
not find an instability to $s$-wave pairing for a spin-singlet
composite-fermion Fermi sea at half-filled lowest Landau level in a system
where the Zeeman splitting has been set to zero.
