  Relating the quantized Hall response of correlated insulators to many-body
topological invariants is a key challenge in topological quantum matter. Here,
we use Streda's formula to derive an expression for the many-body Chern number
in terms of the single-particle interacting Green's function and its derivative
with respect to a magnetic field. In this approach, we find that this many-body
topological invariant can be decomposed in terms of two contributions, $N_3[G]
+ \Delta N_3[G]$, where $N_3[G]$ is known as the Ishikawa-Matsuyama invariant
and where the second term involves derivatives of Green's function and the self
energy with respect to the magnetic perturbation. As a by-product, the
invariant $N_3[G]$ is shown to stem from the derivative of Luttinger's theorem
with respect to the probe magnetic field. These results reveal under which
conditions the quantized Hall conductivity of correlated topological insulators
is solely dictated by the invariant $N_3[G]$, providing new insight on the
origin of fractionalization in strongly-correlated topological phases.
