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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.

Sonja Žunar presented joint work on topological aspects of smooth automorphic forms. Her 2022 slides explain the use of locally convex spaces and the strict inductive limit topology in organizing spaces of such functions. This supplies a topology compatible with an increasing sequence of function spaces, rather than treating automorphic forms only as an unstructured collection of functions.^[ref]