Generalised Form of a Conjecture of Jacquet and a Local Consequence

Following the work of Harris and Kudla we prove a more general form of a conjecture of Jacquet relating the non-vanishing of a certain period integral to non-vanishing of the central critical value of a certain L-function. As a consequence we deduce certain local results about the existence of GL2(k)-invariant linear forms on irreducible, admissible representations of GL2(K) for K a commutative semi-simple cubic algebra over a non-archimedean local field k in terms of certain local epsilon factors which were proved only in certain cases by the first author in his earlier work in [16]. This has been achieved by globalising a locally distinguished representation to a globally distinguished representation, a result of independent interest.