If the dark matter (DM) were composed of axions, then structure formation in the Universe would be suppressed below the axion Jeans scale. Using an analytic model for the halo mass function of a mixed DM model with axions and cold dark matter, combined with the abundance-matching technique, we construct the UV-luminosity function. Axions suppress high-$z$ galaxy formation and the UV-luminosity function is truncated at a faintest limiting magnitude. From the UV-luminosity function, we predict the reionization history of the universe and find that axion DM causes reionization to occur at lower redshift. We search for evidence of axions using the Hubble Ultra Deep Field UV-luminosity function in the redshift range $z=6$-$10$, and the optical depth to reionization, $tau$, as measured from cosmic microwave background polarization. All probes we consider consistently exclude $m_alesssim 10^{-23}text{ eV}$ from contributing more than half of the DM, with our strongest constraint ruling this model out at more than $8sigma$ significance. In conservative models of reionization a dominant component of DM with $m_a=10^{-22}text{ eV}$ is in $3sigma$ tension with the measured value of $tau$, putting pressure on an axion solution to the cusp-core problem. Tension is reduced to $2sigma$ for the axion contributing only half of the DM. A future measurement of the UV-luminosity function in the range $z=10$-$13$ by JWST would provide further evidence for or against $m_a=10^{-22}text{ eV}$. Probing still higher masses of $m_a=10^{-21}text{ eV}$ will be possible using future measurements of the kinetic Sunyaev-Zeldovich effect by Advanced ACTPol to constrain the time and duration of reionization.
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