= Matthew Emerton
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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>.<Classical Modular Forms as Automorphic Forms by Matthew Emerton>[Definition 5.1 and Proposition 5.5]{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>.https://www.math.uchicago.edu/~emerton/[Author homepage and Miscellaneous Notes]{ref}
\Image[https://smp.uq.edu.au/files/29705/AMSI%2520AUSTMS%2520Public%2520Lecture.png]
{title=Matthew Emerton, University of Queensland announcement for his July 2024 lecture}
{height=240}
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