  K2-19 (EPIC201505350) is an interesting planetary system in which two
transiting planets with radii ~ 7 $R_{Earth}$ (inner planet b) and ~ 4
$R_{Earth}$ (outer planet c) have orbits that are nearly in a 3:2 mean-motion
resonance. Here, we present results of ground-based follow-up observations for
the K2-19 planetary system. We have performed high-dispersion spectroscopy and
high-contrast adaptive-optics imaging of the host star with the HDS and HiCIAO
on the Subaru 8.2m telescope. We find that the host star is relatively old (>8
Gyr) late G-type star ($T_{eff}$ ~ 5350 K, $M_s$ ~ 0.9 $M_{Sun}$, and $R_{s}$ ~
0.9 $R_{Sun}$). We do not find any contaminating faint objects near the host
star which could be responsible for (or dilute) the transit signals. We have
also conducted transit follow-up photometry for the inner planet with KeplerCam
on the FLWO 1.2m telescope, TRAPPISTCAM on the TRAPPIST 0.6m telescope, and
MuSCAT on the OAO 1.88m telescope. We confirm the presence of transit-timing
variations, as previously reported by Armstrong and coworkers. We model the
observed transit-timing variations of the inner planet using the synodic
chopping formulae given by Deck & Agol (2015). We find two statistically
indistinguishable solutions for which the period ratios ($P_{c}/P_{b}$) are
located slightly above and below the exact 3:2 commensurability. Despite the
degeneracy, we derive the orbital period of the inner planet $P_b$ ~ 7.921 days
and the mass of the outer planet $M_c$ ~ 20 $M_{Earth}$. Additional transit
photometry (especially for the outer planet) as well as precise radial-velocity
measurements would be helpful to break the degeneracy and to determine the mass
of the inner planet.
