Automorphic inflation

A framework of inflation is formulated based on symmetry groups and their associated automorphic functions. In this setting the inflaton multiplet takes values in a curved target space constructed from a continuous group $G$ and a discrete subgroup $\Gamma$. The dynamics of inflationary models is essentially determined by the choice of the pair $(G,\Gamma)$ and a function $\Phi$ on the group $G$ relative to $\Gamma$. Automorphic inflation provides a natural structure in which the shift symmetry of large field inflation arises as one of generators of $\Gamma$. The model of $j-$inflation is discussed as an example of modular inflation associated to the special linear group.