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A local-global question in automorphic forms
- Part of
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- Journal:
- Compositio Mathematica / Volume 149 / Issue 6 / June 2013
- Published online by Cambridge University Press:
- 26 April 2013, pp. 959-995
- Print publication:
- June 2013
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- Article
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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.
