Near-infrared Census of RR Lyrae variables in the Messier 3 globular
  cluster and the Period--Luminosity Relations
  We present new near-infrared ($JHK_s$) time-series observations of RR Lyrae
variables in the Messier 3 (NGC 5272) globular cluster using the WIRCam
instrument at the 3.6-m Canada France Hawaii Telescope. Our observations cover
a sky area of $\sim 21'\times 21'$ around the cluster center and provide an
average of twenty epochs of homogeneous $JHK_s$-band photometry. New
homogeneous photometry is used to estimate robust mean magnitudes for 175
fundamental-mode (RRab), 47 overtone-mode (RRc), and 11 mixed-mode (RRd)
variables. Our sample of 233 RR Lyrae variables is the largest thus far
obtained in a single cluster with time-resolved, multi-band near-infrared
photometry. Near-infrared to optical amplitude ratios for RR Lyrae in Messier 3
exhibit a systematic increase moving from RRc to short-period ($P < 0.6$~days)
and long-period ($P \gtrsim 0.6$~days) RRab variables. We derive $JHK_s$-band
Period--Luminosity relations for RRab, RRc, and the combined sample of
variables. Absolute calibrations based on the theoretically predicted
Period--Luminosity--Metallicity relations for RR Lyrae stars yield a distance
modulus, $\mu = 15.041 \pm 0.017~(\textrm{statistical}) \pm
0.036~(\textrm{systematic})$~mag, to Messier 3. When anchored to trigonometric
parallaxes for nearby RR Lyrae stars from the {\it Hubble Space Telescope} and
the {\it Gaia} mission, our distance estimates are consistent with those
resulting from the theoretical calibrations, albeit with relatively larger
systematic uncertainties.
