  The stellar initial mass function (IMF) is thought to be bottom-heavy in the
cores of the most massive galaxies, with an excess of low mass stars compared
to the Milky Way. However, studies of the kinematics of quiescent galaxies at
2<z<5 find M/L ratios that are better explained with lighter IMFs. Light IMFs
have also been proposed for the unexpected populations of luminous galaxies
that JWST has uncovered at z>7, to reduce tensions with galaxy formation
models. Here we explore IMFs that are simultaneously bottom-heavy, with a steep
slope at low stellar masses, and top-heavy, with a shallow slope at high
masses. We derive a form of the IMF for massive galaxies that is consistent
with measurements in the local universe and yet produces relatively low M/L
ratios at high redshift. This `concordance' IMF has slopes
$\gamma_1=2.40\pm0.09$, $\gamma_2=2.00\pm0.14$, and $\gamma_3=1.85\pm0.11$ in
the regimes 0.1-0.5 Msun, 0.5-1 Msun, and >1 Msun respectively. The IMF
parameter $\alpha$, the mass excess compared to a Milky Way IMF, ranges from
$\log(\alpha)\approx+0.3$ for present-day galaxies to $\log(\alpha)\approx-0.1$
for their star forming progenitors. The concordance IMF applies only to the
central regions of the most massive galaxies, with velocity dispersions ~300
km/s, and their progenitors. However, it can be generalized using a
previously-measured relation between $\alpha$ and $\sigma$. We arrive at the
following modification to the Kroupa (2001) IMF for galaxies with
$\sigma\gtrsim 160$ km/s: $\gamma_1\approx1.3+4.3\log\sigma_{160}$;
$\gamma_2\approx2.3-1.2\log\sigma_{160}$; and
$\gamma_3\approx2.3-1.7\log\sigma_{160}$, with $\sigma_{160}=\sigma/160$ km/s.
If galaxies grow primarily inside-out, so that velocity dispersions are
relatively stable, these relations should also hold at high redshift.
