Automorphic form

Summary

An automorphic form is a function on a Lie group $G$ that is invariant under a discrete subgroup $\Gamma$. It must also satisfy strong finiteness and growth conditions. Modular forms and Maass wave forms give the basic examples, for the group $\mathrm{SL}_2(\R)$ and $\Gamma = \mathrm{SL}_2(\Z)$ or a congruence subgroup. Automorphic forms carry deep arithmetic information, and they are one side of the correspondences predicted by the Langlands program.