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We present a summary of our project that studies galaxies hosting type Iasupernova (SN Ia) at different redshifts. We present Gran Telescopio deCanarias (GTC) optical spectroscopy of six SN Ia host galaxies at redshift$z\sim 0.4-0.5$. They are joined to a set of SN Ia host galaxies atintermediate-high redshift, which include galaxies from surveys SDSS andCOSMOS. The final sample, after a selection of galaxy spectra in terms ofsignal-to-noise and other characteristics, consists of 680 galaxies withredshift in the range $0.04 < z < 1$. We perform an inverse stellar populationsynthesis with the code {\sc fado} to estimate the star formation andenrichment histories of this set of galaxies, simultaneously obtaining theirmean stellar age and metallicity and stellar mass. After analysing thecorrelations among these characteristics, we look for possible dependencies ofthe Hubble diagram residuals and supernova features (luminosity, color andstrength parameter) on these stellar parameters. We find that the Hubbleresiduals show a clear dependence on the stellar metallicity weighted by masswith a slope of -0.061\,mag\,dex$^{-1}$, when represented in logarithmic scale,$\log{ \langle Z_{M}/Z_{\odot}\rangle }$. This result supports our previousfindings obtained from gas oxygen abundances for local and SDSS-surveygalaxies. Comparing with other works from the literature that also use thestellar metallicity, we find a similar value, but with more precision and abetter significance (2.08 vs $\sim$ 1.1), due to the higher number of objectsand wider range of redshift of our sample.

Stellar Populations in type Ia supernova host galaxies at intermediate-high redshift: Star formation and metallicity enrichment histories