J-PLUS: Beyond Spectroscopy III. Stellar Parameters and Elemental-abundance Ratios for Five Million Stars from DR3

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.