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The observed delay-time distribution (DTD) of Type-Ia supernovae (SNe Ia) isa valuable probe of SN Ia progenitors and physics, and of the role of SNe Ia incosmic metal enrichment. The SN Ia rate in galaxy clusters as a function ofcluster redshift is an almost-direct measure of the DTD, but current estimateshave been limited out to a mean redshift z=1.1, corresponding to time delays,after cluster star-formation, of over 3.2 Gyr. We analyze data from a HubbleSpace Telescope monitoring project of 12 galaxy clusters at z=1.13-1.75, wherewe discover 29 SNe, and present their multi-band light curves. Based on the SNphotometry and the apparent host galaxies, we assess cluster membership and SNtype, finding 11 cases that are likely SNe Ia in cluster galaxies and 4 morecases which are possible but not certain cluster SNe Ia. We conduct simulationsto estimate the SN detection efficiency, the experiment's completeness, and thephotometric errors, and perform photometry of the cluster galaxies to derivethe cluster stellar masses. Separating the cluster sample into high-z and low-zbins, we obtain rest-frame SN Ia rates per unit formed stellar mass of $2.2^{+2.6}_{-1.3}\times 10^{-13}{\rm yr}^{-1}{\rm M}_\odot^{-1}$ at a meanredshift z=1.25, and $3.5^{+6.6}_{-2.8} \times 10^{-13}{\rm yr}^{-1}{\rmM}_\odot^{-1}$ at z=1.58. Combining our results with previous cluster SN Iarates, we fit the DTD, now down to delays of 1.5 Gyr, with a power-lawdependence, $t^\alpha$, with $\alpha=-1.30^{+0.23}_{-0.16}$. We confirmprevious indications for a Hubble-time-integrated SN Ia production efficiencythat is several times higher in galaxy clusters than in the field, perhapscaused by a peculiar stellar initial mass function in clusters, or by a higherincidence of binaries that will evolve into SNe Ia.

The rate of Type-Ia supernovae in galaxy clusters and the delay-time distribution out to redshift 1.75