Modular Inflation at Higher Level $N$
  We introduce the framework of modular inflation with level structure,
generalizing the level one theory considered previously to higher levels. We
analyze the modular structure of CMB observables in this framework and show
that the nontrivial geometry of the target space suffices to ensure the almost
holomorphic modularity of the relevant parameters. We further introduce a
concrete class of models based on hauptmodul functions that provide generators
of the corresponding inflationary potentials at level $N>1$. The phenomenology
of this class of models provides targets for ground-based CMB experiments in
the immediate future. In the framework of our models we also discuss the status
of the quantum gravity conjectures that have been formulated in the context of
the swampland.
