  (Abridged) We study the host galaxy regions in close proximity to Type Ia
supernovae (SNe Ia) to analyze relations between the properties of SN Ia events
and environments most similar to where their progenitors formed. We focus on
local H\alpha\ emission as an indicator of young environments. The Nearby
Supernova Factory has obtained flux-calibrated spectral timeseries for SNe Ia
using integral field spectroscopy, allowing the simultaneous measurement of the
SN and its immediate vicinity. For 89 SNe Ia we measure H\alpha\ emission
tracing ongoing star formation within a 1 kpc radius around each SN. This
constitutes the first direct study of the local environment for a large sample
of SNe Ia also having accurate luminosity, color and stretch measurements. We
find that SNe Ia with local H\alpha\ emission are redder by 0.036+/-0.017 mag,
and that the previously-noted correlation between stretch and host mass is
entirely driven by the SNe Ia coming from passive regions. Most importantly,
the mean standardized brightness for SNe Ia with local H\alpha\ emission is
0.094+/-0.031 mag fainter than for those without. This offset arises from a
bimodal structure in the Hubble residuals, that also explains the
previously-known host-mass bias. We combine this bimodality with the cosmic
star-formation rate to predict changes with redshift in the mean SN Ia
brightness and the host-mass bias. This change is confirmed using high-redshift
SNe Ia from the literature. These environmental dependences point to remaining
systematic errors in SNe Ia standardization. The observed brightness offset is
predicted to cause a significant bias in measurements of the dark energy
equation of state. Recognition of these effects offers new opportunities to
improve SNe Ia as cosmological probes - e.g. SNe Ia having local H\alpha\
emission are more homogeneous, having a brightness dispersion of 0.105+/-0.012
mag.
