  We present a catalog of stellar parameters (effective temperature $T_{\rm
eff}$, surface gravity $\log g$, age, and metallicity [Fe/H]) and
elemental-abundance ratios ([C/Fe], [Mg/Fe], and [$\alpha$/Fe]) for some five
million stars (4.5 million dwarfs and 0.5 million giants stars) in the Milky
Way, based on stellar colors from the Javalambre Photometric Local Universe
Survey (J-PLUS) DR3 and \textit{Gaia} EDR3. These estimates are obtained
through the construction of a large spectroscopic training set with parameters
and abundances adjusted to uniform scales, and trained with a Kernel Principal
Component Analysis. Owing to the seven narrow/medium-band filters employed by
J-PLUS, we obtain precisions in the abundance estimates that are as good or
better than derived from medium-resolution spectroscopy for stars covering a
wide range of the parameter space: 0.10-0.20 dex for [Fe/H] and [C/Fe], and
0.05 dex for [Mg/Fe] and [$\alpha$/Fe]. Moreover, systematic errors due to the
influence of molecular carbon bands on previous photometric-metallicity
estimates (which only included two narrow/medium-band blue filters) have now
been removed, resulting in photometric-metallicity estimates down to [Fe/H]
$\sim -4.0$, with typical uncertainties of 0.25 dex and 0.40 dex for dwarfs and
giants, respectively. This large photometric sample should prove useful for the
exploration of the assembly and chemical-evolution history of our Galaxy.
