  We present results from a high-contrast adaptive optics imaging search for
giant planets and brown dwarfs (>1 MJup) around 122 newly identified nearby
(<40 pc) young M dwarfs. Half of our targets are younger than 135 Myr and 90%
are younger than the Hyades (620 Myr). Our H- and K-band coronagraphic
observations with Keck/NIRC2 and Subaru/HiCIAO achieve typical contrasts of
12-14 mag and 9-13 mag at 1", respectively, which corresponds to limiting
planet masses of 0.5-10 MJup at 5-33 AU for 85% of our sample. We discovered
four young brown dwarf companions: 1RXS J235133.3+312720 B (32 $\pm$ 6 MJup;
L0$^{+2}_{-1}$; 120 $\pm$ 20 AU), GJ 3629 B (64$^{+30}_{-23}$ MJup; M7.5 $\pm$
0.5; 6.5 $\pm$ 0.5 AU), 1RXS J034231.8+121622 B (35 $\pm$ 8 MJup; L0 $\pm$ 1;
19.8 $\pm$ 0.9 AU), and 2MASS J15594729+4403595 B (43 $\pm$ 9 MJup; M8.0 $\pm$
0.5; 190 $\pm$ 20 AU). Over 150 candidate planets were identified; we obtained
follow-up imaging for 56% of these but all are consistent with background
stars. Our null detection of planets enables strong statistical constraints on
the occurrence rate of long-period giant planets around single M dwarfs. We
infer an upper limit (at the 95% confidence level) of 10.3% and 16.0% for 1-13
MJup planets between 10-100 AU for hot-start and cold-start (Fortney)
evolutionary models, respectively. Fewer than 6.0% (9.9%) of M dwarfs harbor
massive gas giants in the 5-13 MJup range like those orbiting HR 8799 and
$\beta$ Pictoris between 10-100 AU for a hot-start (cold-start) formation
scenario. Although the first directly imaged planets were found around massive
stars, there is currently no statistical evidence for a trend of giant planet
frequency with stellar host mass at large separations as predicted by the disk
instability model of giant planet formation.
