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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]

Matthew Emerton is a mathematician working in number theory whose work connects modular forms, representation theory and arithmetic geometry. His authored account of classical modular forms as automorphic forms gives a precise translation between a weight-$k$ holomorphic function on the upper half-plane and an arithmetically invariant function on an adelic group.^[ref]

The translation converts a transformation rule into invariance and differentiation conditions. In particular, being holomorphic becomes annihilation by a suitable differential operator, while the weight becomes a character under rotations. This places classical modular forms in the same representation-theoretic language as more general smooth automorphic forms. His author page provides the note and identifies his research area as number theory.^[ref]

A smooth automorphic form is an automorphic form whose archimedean variables are infinitely differentiable, without necessarily being holomorphic. Conventions differ over whether compact-group finiteness is included or one takes a larger smooth representation space. In the classical finite-vector convention on $G={\mathrm{SL}}_2(\mathbb{R})$, one asks for left invariance under an arithmetic subgroup $\Gamma$, moderate growth, finite-dimensional span under right translations by $K=\mathrm{SO}(2)$, and finiteness under the centre of the universal enveloping algebra.^[ref]

The constant function on $\Gamma \backslash G$ is an example: it is smooth, bounded, $K$-invariant and annihilated by positive-order invariant differentiation. More interesting examples come from Maass forms, whose Laplacian eigenfunction condition replaces holomorphy. Smoothness is therefore a regularity requirement, not a claim that the Cauchy-Riemann equations hold.