CNO dredge-up in a sample of APOGEE/Kepler red giants: Tests of stellar
  models and Galactic evolutionary trends of N/O and C/N
  Surface abundances of C, N, and O in red giants are affected by processed
material mixed into the stars' convective envelopes. Using a sample of $\sim
5100$ stars with elemental abundances from APOGEE and asteroseismic masses from
{\it Kepler}, we test theoretical stellar models that predict this mixing, then
apply these models to derive birth C, N, and O abundances for these stars. Our
models with standard mixing can reproduce the observed trends to within
plausible uncertainties in the birth abundances. Some models with ''extra''
mixing processes fail, predicting trends with surface gravity or evolutionary
state that are not observed. Applying mixing corrections to the APOGEE
abundances removes the observed age-dependence of log(N/O) and log(C/N), but it
leaves trends of log(N/O) and log(C/N) with metallicity, as expected based on
nucleosynthesis models. The stellar N/O trend agrees well with Dopita et al.'s
calibration of gas phase log(N/O) with metallicity, and with gas phase trends
in the MaNGA integral field survey of nearby galaxies. We also find a
substantial separation in birth [N/Mg] ratios between high-[$\alpha$/Fe]
(''thick disc'') stars and low-[$\alpha$/Fe] (''thin disc'') stars. We find a
smaller but still clear separation for [C/Mg]. The trends of birth C and N
abundances with [Fe/H] and [$\alpha$/Fe] could affect spectroscopic age
estimates for red giants that rely on the observed C/N ratio as a diagnostic of
stellar mass.
