We present the first results with the ROentgen Survey with an Imaging Telescope Array (eROSITA) on board the Russian Spektrum-Roentgen-Gamma mission, and we combine the new X-ray data with observations with the Transiting Exoplanet Survey Satellite (TESS). We used theSUPERBLINKproper motion catalog of nearby M dwarfs as input sample to search for eROSITA and TESS data. We extractedGaiaDR2 data for the full M dwarf catalog, which comprises ~9000 stars, and we calculated the stellar parameters from empirical relations with optical/IR colors. Then we cross-matched this catalog with the eROSITA Final Equatorial Depth Survey (eFEDS) and the first eROSITA all-sky survey (eRASS1). After a meticulous source identification in which we associated the closestGaiasource with the eROSITA X-ray detections, our sample of M dwarfs is defined by 687 stars with SpT = K5..M7 (673 from eRASS1 and 14 from eFEDS). While for eRASSl we used the data from the source catalog provided by the eROSITA_DE consortium, for the much smaller eFEDS sample, we performed the data extraction, and we analyzed the X-ray spectra and light curves. This unprecedented data base for X-ray emitting M dwarfs allowed us to place a quantitative constraint on the mass dependence of the X-ray luminosity, and to determine the change in the activity level with respect to pre-main-sequence stars. TESS observations are available for 489 of 687 X-ray detected M dwarfs. By applying standard period search methods, we were able to determine the rotation period for 180 X-ray detected M dwarfs. This is about one-forth of the X-ray sample. With the joint eROSITA and TESS sample, and combining it with our compilation of historical X-ray and rotation data for M dwarfs, we examined the mass dependence of the saturated regime of the rotation-activity relation. A first comparison of eROSITA hardness ratios and spectra shows that 65% of the X-ray detected M dwarfs have coronal temperatures of ~0.5 keV. We performed a statistical investigation of the long-term X-ray variability of M dwarfs by comparing the eROSITA measurements to those obtained ~30 yr earlier during the ROSAT all-sky survey (RASS). Evidence for X-ray flares is found in various parts of our analysis: directly from an inspection of the eFEDS light curves, in the relation between RASS and eRASSl X-ray luminosities, and in a subset of stars that displays hotter X-ray emission than the bulk of the sample according to the hardness ratios. Finally, we point out the need to obtain X-ray spectroscopy for more M dwarfs to study the coronal temperature-luminosity relation, which is not well constrained by our eFEDS results.
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