Dark matter stability at fixed points in a modular $A_4$ symmetry

We propose a new mechanism of stabilization for dark matter under fixed points of modular $A_4$ symmetry, thanks to recovering a residual symmetry for each of fixed points at $\tau=\omega,\ i, i\times\infty$. We find that a model at $\tau =i$ successfully reproduces the neutrino oscillation data without conflict of the DM stability.