Finite modular Coleman-Weinberg inflation

We propose a modular symmetric inflationary model based on a Coleman--Weinberg potential generated by integrating out heavy vector-like quarks that couple to the complex modulus field $\tau$ through modular forms. In this framework, the imaginary part of modulus $\tau$ plays the role of the inflaton, while the real part is identified with a heavy axion. We show that the model successfully explains the current cosmological observations. We further discuss reheating through modulus-dependent gauge kinetic functions and the cosmology of the axion. The axion oscillation dominates over the Universe after the reheating via inflaton decay, and then it decays before Big Bang Nucleosynthesis in the viable parameter region. The quantum fluctuation of the axion can be of order $\mathcal{O}(1)\% $ of that of the inflaton, which would induce isocurvature perturbations that may be detectable in future observations.