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-[$α$/Fe] (''thick disc'') stars and low-[$α$/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 [$α$/Fe] could affect spectroscopic age estimates for red giants that rely on the observed C/N ratio as a diagnostic of stellar mass.