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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.

15 votes

Where stands functoriality in 2009?

The work of Ngo should allow for the treatment of all endoscopic cases of functoriality; this is a kind of technical condition, but includes transfer from classical groups to GL(n) and base change for …
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12 votes
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Sato-Tate measure for GL(3) Automorphic forms

(2017-11-26 edit by j.c.: earlier versions of this answer consisted of David Hansen's screenshot of the following, with the text "Here is a screenshot of a semi-answer which froze my computer when I h …
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8 votes

Langlands in dimension 2: the Yoshida conjecture

There's no need to require irreducibility. If $A=E_1 \times E_2$ is a product of elliptic curves the conjecture is true, by modularity of elliptic curves combined with Yoshida's lifting from pairs of …
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5 votes

Overview of automorphic representations for $SL(2)/{\mathbf{Q}}$?

The multiplicity of any cusp form in the spectrum is one; this follows from the analysis in LL and the results proven in Ramakrishnan's paper "Modularity of the Rankin-Selberg L-series and multiplici …
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2 votes

Terminology occuring in automorphic representation and relationship between them

For global automorphic representations, square-integrable is weaker than cuspidal. According to Moeglin-Waldspurger, the square-integrable automorphic representations of $GL_n(\mathbb{A})$ are genera …
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