On periods of cusp forms and algebraic cycles for \(U(3)\) (Q690058)

Let \(G\) be a quasi-split unitary group \(U(3)\) over a number field. Let \(\pi\) be a cuspidal representation of \(G\). There is an obvious notion of periods of \(\pi\) over any unitary group \(U(2)\) imbedded in \(G\). It is proved in the paper that the following two conditions are equivalent. (1) \(\pi\) has a nonzero theta-lift to the quasi-split \(U(2)\). (2) There is a character \(\chi\) such that \(\pi \otimes \chi\) has a nonzero period over some \(U(2)\). Examples are given where \(\pi\) has a nonzero theta-lift to an anisotropic \(U(2)\) and does not satisfy (2). This means that there are cuspidal \(\pi\) such that \(L(s,\pi)\) has a pole at \(s=1\), but all periods over all \(U(2)\) are zero. A consequence is the existence of algebraic cycles on the Shimura variety corresponding to \(G\) not spanned by modular curves.