 Automorphic form

An automorphic form is a function with arithmetic symmetries, together with analytic regularity and growth conditions. In a classical example, a modular form of integer weight $k$ on the modular group is a holomorphic function on the upper half-plane satisfyingand a holomorphy condition at the cusp. The transformation rule expresses compatibility with a group action, rather than ordinary periodicity alone: translation gives $f(z+1)=f(z)$, while inversion gives $f(-1/z)=z^{k}f(z)$.

The broader theory also permits nonholomorphic forms and groups beyond ${\mathrm{SL}}_2$. In the adelic formulation for ${\mathrm{GL}}_2$, automorphic forms are smooth, arithmetically invariant functions satisfying moderate growth and finiteness conditions under compact-group translations and central differential operators. The classical modular forms embed in this larger space. This connects number theory with representation theory and harmonic analysis.^[ref]