= 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> satisfying
$$
f\!\left(\frac{az+b}{cz+d}\right)=(cz+d)^k f(z),\qquad \begin{pmatrix}a&b\\c&d\end{pmatrix}\in\operatorname{SL}_2(\mathbb Z),
$$
and 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 $\operatorname{SL}_2$. In the adelic formulation for $\operatorname{GL}_2$, <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>.<Classical Modular Forms as Automorphic Forms by Matthew Emerton>[Definition 5.1 and Proposition 5.5]{ref}
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