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Irreducibility and cuspidality

Dinakar Ramakrishnan
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Abstract

Irreducible representations are the building blocks of general, semisimple Galois representations ρ, and cuspidal representations are the building blocks of automorphic forms πof the general linear group. It is expected that when an object of the former type is associated to one of the latter type, usually in terms of an identity of L-functions, the irreducibility of the former should imply the cuspidality of the latter, and vice-versa. It is not a simple matter - at all - to prove this expectation in either direction, and nothing much is known in dimensions >2. The main result of this article shows for n < 6, in particular, that the cuspidality of a regular algebraic πis implied by the irreducibility of ρ.

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Dinakar Ramakrishnan. 2006-09-23. Irreducibility and cuspidality. https://arxiv.org/abs/math/0609460

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