  We present the discovery of a new Jovian-sized planet, TOI-3757 b, the lowest
density planet orbiting an M dwarf (M0V). It orbits a solar-metallicity M dwarf
discovered using TESS photometry and confirmed with precise radial velocities
(RV) from HPF and NEID. With a planetary radius of $12.0^{+0.4}_{-0.5}$
$R_{\oplus}$ and mass of $85.3^{+8.8}_{-8.7}$ $M_{\oplus}$, not only does this
object add to the small sample of gas giants ($\sim 10$) around M dwarfs, but
also, its low density ($\rho =$ $0.27^{+0.05}_{-0.04}$ $\textrm{g~cm}^{-3}$)
provides an opportunity to test theories of planet formation. We present two
hypotheses to explain its low density; first, we posit that the low metallicity
of its stellar host ($\sim$ 0.3 dex lower than the median metallicity of M
dwarfs hosting gas giants) could have played a role in the delayed formation of
a solid core massive enough to initiate runaway accretion. Second, using the
eccentricity estimate of $0.14 \pm 0.06$ we determine it is also plausible for
tidal heating to at least partially be responsible for inflating the radius of
TOI-3757b b. The low density and large scale height of TOI-3757 b makes it an
excellent target for transmission spectroscopy studies of atmospheric escape
and composition (TSM $\sim$ 190). We use HPF to perform transmission
spectroscopy of TOI-3757 b using the helium 10830 \AA~ line. Doing this, we
place an upper limit of 6.9 \% (with 90\% confidence) on the maximum depth of
the absorption from the metastable transition of He at $\sim$ 10830 \AA, which
can help constraint the atmospheric mass loss rate in this energy limited
regime.
