1 results
On fields of rationality for automorphic representations
- Part of
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- Journal:
- Compositio Mathematica / Volume 150 / Issue 12 / December 2014
- Published online by Cambridge University Press:
- 11 September 2014, pp. 2003-2053
- Print publication:
- December 2014
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- Article
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This paper proves two results on the field of rationality
$\mathbb{Q}({\it\pi})$ for an automorphic representation 
${\it\pi}$, which is the subfield of 
$\mathbb{C}$ fixed under the subgroup of 
$\text{Aut}(\mathbb{C})$ stabilizing the isomorphism class of the finite part of 
${\it\pi}$. For general linear groups and classical groups, our first main result is the finiteness of the set of discrete automorphic representations 
${\it\pi}$ such that 
${\it\pi}$ is unramified away from a fixed finite set of places, 
${\it\pi}_{\infty }$ has a fixed infinitesimal character, and 
$[\mathbb{Q}({\it\pi}):\mathbb{Q}]$ is bounded. The second main result is that for classical groups, 
$[\mathbb{Q}({\it\pi}):\mathbb{Q}]$ grows to infinity in a family of automorphic representations in level aspect whose infinite components are discrete series in a fixed 
$L$-packet under mild conditions.
