Arithmeticity for periods of automorphic forms
Abstract
A cuspidal automorphic representation \pi of a group G is said to to be distinguished with respect to a subgroup H if the integral of f along H is nonzero for a cusp form f in the space of \pi. Such period integrals are related to (non)vanishing of interesting L-values and also to Langlands functoriality. This article discusses a general principle, labelled arithmeticity, which roughly states that "\pi is H-distinguished if and only if any Galois conjugate of \pi is H-distinguished." We study this principle via several examples; starting with GL(2) and leading up to more complicated situations where the ambient group is a higher GL(n) or a classical group.
Cite
@article{arxiv.1207.4641,
title = {Arithmeticity for periods of automorphic forms},
author = {Wee Teck Gan and A. Raghuram},
journal= {arXiv preprint arXiv:1207.4641},
year = {2012}
}
Comments
32 pages. The final version is to appear in the proceedings of the International Colloquium on Automorphic Representations and L-functions, held in TIFR, Mumbai, January 2012