Generalised form of a conjecture of Jacquet, and a local consequence

Abstract

Following the work of Harris and Kudla, we prove a 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 a theorem relating the existence of GL₂ (K) -invariant linear forms on irreducible, admissible representations of GL₂(K) for a commutative semi-simple cubic algebra K over a non-archimedean local field k in terms of local epsilon factors which was proved only in some cases by the first author in his earlier work in [D. Prasad, Invariant forms for representations of GL₂ over a local field, Amer. J. Math. 114 (1992), no. 6, 1317–1363.]. This has been achieved by globalising a locally distinguished supercuspidal representation to a globally distinguished representation, a result of independent interest which is proved by an application of the relative trace formula.