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A local-global question in automorphic forms

Part of: Lie groups Discontinuous groups and automorphic forms

Published online by Cambridge University Press:  26 April 2013

U. K. Anandavardhanan  and
Dipendra Prasad
U. K. Anandavardhanan
Affiliation:
Department of Mathematics, Indian Institute of Technology Bombay, Mumbai - 400 076, India email anand@math.iitb.ac.in
Dipendra Prasad
Affiliation:
Tata Institute of Fundamental Research, Homi Bhabha Road, Mumbai - 400 005, India email dprasad@math.tifr.res.in
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Abstract

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In this paper, we consider the $\mathrm{SL} (2)$ analogue of two well-known theorems about period integrals of automorphic forms on $\mathrm{GL} (2)$: one due to Harder–Langlands–Rapoport about non-vanishing of period integrals on ${\mathrm{GL} }_{2} ({ \mathbb{A} }_{F} )$ of cuspidal automorphic representations on ${\mathrm{GL} }_{2} ({ \mathbb{A} }_{E} )$ where $E$ is a quadratic extension of a number field $F$, and the other due to Waldspurger involving toric periods of automorphic forms on ${\mathrm{GL} }_{2} ({ \mathbb{A} }_{F} )$. In both these cases, now involving $\mathrm{SL} (2)$, we analyze period integrals on global$L$-packets; we prove that under certain conditions, a global automorphic $L$-packet which at each place of a number field has a distinguished representation, contains globally distinguished representations, and further, an automorphic representation which is locally distinguished is globally distinguished.

Keywords

period integralslocally distinguished representationsglobally distinguished representationsbase changeAsai liftAsai L-functioncentral L-valuesepsilon factorsfibers of functorial liftssimultaneous non-vanishing of L-functions

MSC classification

Secondary: 11F70: Representation-theoretic methods; automorphic representations over local and global fields 22E55: Representations of Lie and linear algebraic groups over global fields and adèle rings
Type
Research Article
Information
Compositio Mathematica , Volume 149 , Issue 6 , June 2013 , pp. 959 - 995
Copyright
© The Author(s) 2013 

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