Galactic archaeology with asteroseismology and spectroscopy: Red giants observed by CoRoT and APOGEE

16 pages + references and appendix, 17 figures, accepted for publication in A&A. Data described in Appendix B will be published through the CDS upon publicationInternational audienceWith the advent of the space missions CoRoT and Kepler, it has become feasible to determine precise asteroseismic masses and ages for large samples of red-giant stars. In this paper, we present the CoRoGEE dataset -- obtained from CoRoT lightcurves for 606 red giant stars in two fields of the Galactic disc which have been co-observed for an ancillary project of APOGEE. We have used the Bayesian parameter estimation code PARAM to calculate distances, extinctions, masses, and ages for these stars in a homogeneous analysis, resulting in relative statistical uncertainties of $\sim2\%$ in distance, $\sim4\%$ in radius, $\sim9\%$ in mass and $\sim25\%$ in age. We also assess systematic age uncertainties due to different input physics and mass loss. We discuss the correlation between ages and chemical abundance patterns of field stars over a large radial range of the Milky Way's disc (5 kpc $<R_{\rm Gal}<$ 14 kpc), focussing on the [$\alpha$/Fe]-[Fe/H]-age plane in five radial bins of the Galactic disc. We find an overall agreement with the expectations of chemical-evolution models computed before the present data were available, especially for the outer regions. However, our data also indicate that a significant fraction of stars now observed near and beyond the Solar Neighbourhood migrated from inner regions. Mock CoRoGEE observations of a chemo-dynamical Milky Way disc model show that the number of high-metallicity stars in the outer disc is too high to be accounted for even by the strong radial mixing present in the model. The mock observations also reveal that the age distribution of the [$\alpha$/Fe]-enhanced sequence in the CoRoGEE inner-disc field is much broader than expected from a combination of radial mixing and observational errors. We suggest that a thick disc/bulge component that formed stars for more than 3 Gyr may account for these discrepancies