The newly established luminosity functions of high-z galaxies at $4 lesssim z lesssim 10$ can provide a stringent check on dark matter models that aim to explain the core properties of dwarf galaxies. The cores of dwarf spheroidal galaxies are understood to be too large to be accounted for by free streaming of warm dark matter without overly suppressing the formation of such galaxies. Here we demonstrate with cosmological simulations that wave dark matter, $psi$DM, appropriate for light bosons such as axions, does not suffer this problem, given a boson mass of $m_{psi} ge 1.2 times 10^{-22}{,rm eV}$ ($2sigma$). In this case, the halo mass function is suppressed below $sim 10^{10}{,M_odot}$ at a level that is consistent with the high-z luminosity functions, while simultaneously generating the kpc-scale cores in dwarf galaxies arising from the solitonic ground state in $psi$DM. We demonstrate that the reionization history in this scenario is consistent with the Thomson optical depth recently reported by Planck, assuming a reasonable ionizing photon production rate. We predict that the luminosity function should turn over slowly around an intrinsic UV luminosity of $M_{rm UV} gtrsim -16$ at $z gtrsim 4$. We also show that for galaxies magnified $mathord{>}10times$ in the Hubble Frontier Fields, $psi$DM predicts an order of magnitude fewer detections than cold dark matter at $z gtrsim 10$ down to $M_{rm UV} sim -15$, allowing us to distinguish between these very different interpretations for the observed coldness of dark matter.
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