Arithmeticity for periods of automorphic forms

An automorphic form f on a group G is said to to be distinguished with respect to a subgroup H if the integral of f along H is nonzero. Such period integrals are related to (non)vanishing of interesting L-values and also to Langlands functoriality. In this talk, I will discuss a general principle, labelled arithmeticity, which roughly states that "f is H-distinguished if and only if any Galois conjugate of f is H-distinguished." This principle will be explored via examples starting with GL(2) and leading up to more complicated situations where the ambient group is a classical group. This talk is a report on some recent joint work with Wee Teck Gan.