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Flare Rates, Rotation Periods. and Spectroscopic Activity Indicators of a Volume-Complete Sample of Mid-to-Late M dwarfs within 15 Parsecs

We present a study of flare rates, rotation periods, and spectroscopic activity indicators of 125 single stars within 15 parsecs and with masses between 0.1$-$0.3 $M_\odot$ observed during the first year of the TESS mission, with the goal of elucidating the relationship between these various magnetically connected phenomena. We gathered multi-epoch high resolution spectra of each target and we measured equivalent widths of the activity indicators Helium I D$_3$, $Hα$, and the Calcium infrared triplet line at 8542.09 angstroms. We present 18 new rotation periods from MEarth photometry and 19 new rotation periods from TESS photometry. We present a catalog of 1392 flares. After correcting for sensitivity, we find the slope of the flare frequency distribution for all stars to have a standard value of $α$ = 1.98 $\pm$ 0.02. We determine R$_{31.5}$, the rate of flares per day with energies above E = 3.16$\times$10$^{31}$ ergs in the TESS bandpass. We find that below a critical value of $Hα$ EW = -0.71 angstroms, log R$_{31.5}$ increases linearly with increasing $Hα$ emission; above this value, log R$_{31.5}$ declines rapidly. The stars divide into two groups: 26% have $Hα$ in emission, high flare rates with typical values of log R$_{31.5}$ = -1.30 $\pm$ 0.08, and have Rossby numbers $<$ 0.50. The remaining 74% show little to no $Hα$ in emission and exhibit log R$_{31.5}$ $<$ -3.86, with the majority of these stars not showing a single flare during the TESS observations.