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An automorphic form is a well-behaved function from a topological group $G$ to the complex numbers (or complex vector space) which is invariant under the action of a discrete subgroup $\Gamma \subset G$ of the topological group. Automorphic forms are a generalization of the idea of periodic functions in Euclidean space to general topological groups.
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votes
Is there a characterization of the cuspidal automorphic representations that arise from elli...
I'll assume that you're talking about elliptic curves over $\mathbf{Q}$ - much of what I'm going to say should generalize to totally real fields, replacing modular forms by Hilbert modular forms. I th …