The Hubble Tension, The $M$ Crisis of Late Time $H(z)$ Deformation
  Models and the Reconstruction of Quintessence Lagrangians
  We present a detailed and pedagogical analysis of recent cosmological data,
including CMB, BAO, SnIa and the recent local measurement of $H_0$. We thus
obtain constraints on the parameters of standard dark energy parameterizations,
including $\Lambda CDM$, and $H(z)$ deformation models such as $wCDM$ (constant
equation of state $w$ of dark energy), and the CPL model (corresponding to the
evolving dark energy equation-of-state parameter $w(z) = w_0 + w_a
\frac{z}{1+z}$). The fitted parameters include the dark matter density
$\Omega_{0m}$, the SnIa absolute magnitude $M$, the Hubble constant $H_0$ and
the dark energy parameters (e.g., $w$ for $wCDM$). All models considered lead
to a best-fit value of $M$ that is inconsistent with the locally determined
value obtained by Cepheid calibrators ($M$ tension). We then use the best-fit
dark energy parameters to reconstruct the quintessence Lagrangian that would be
able to reproduce these best-fit parameterizations. Due to the derived late
phantom behavior of the best-fit dark energy equation-of-state parameter
$w(z)$, the reconstructed quintessence models have a negative kinetic term and
are therefore plagued with instabilities.
