Nonvanishing of Gl ( 2 ) automorphic L functions at 1 / 2

Let k be a number field with ring of adeles A, let B be a quaternion algebra defined over k, and letG = B×. Let π be an infinite dimensional irreducible cuspidal automorphic representation of G(A). Then the vanishing or nonvanishing of L(1/2, π) has been conjectured or shown to be equivalent to conditions of considerable interest in number theory or automorphic representation theory. For example, if k = Q, B = M2×2 and π corresponds to an elliptic curveE defined over Q, then Birch and Swinnerton-Dyer conjectured that the order of vanishing of L(s, π) at 1/2 is the rank of the torsion free part of E(Q). To take another example, if the central character of π is trivial, then Waldspurger showed in [W1] and [W2] that the nonvanishing of L(1/2, π) is equivalent to the nonvanishing of the theta lift of π to Mp(2,A), the metaplectic cover of Sl(2,A). In this paper, again when the central character of π is trivial, we show how another condition is related to the nonvanishing of L(1/2, π). We also consider the implications of our results for modular forms. Our first main result relates the nonvanishing of L(1/2, π) to the existence of another irreducible cuspidal automorphic representation σ ofG(A) along with an embedding of π in σ ⊗ σ∨. For a precise account we need some notation. If σ is an infinite dimensional irreducible cuspidal automorphic representation of G(A), define the trilinear form