Relating invariant linear form and local epsilon factors via global methods

We use the recent proof of Jacquet’s conjecture due to Harris and Kudla, and the Burger-Sarnak principle to give a proof about the relationship between the existence of trilinear forms on representations of GL2(ku) for a non-Archimedean local field ku and local epsilon factors which was earlier proved only in the odd residue characteristic by this author in [P1]. The same method gives a global proof of a theorem of Saito and Tunnell about characters of GL2 using a theorem of Waldspurger about period integrals for GL2 and also an extension of the theorem of Saito-Tunnell by this author in [P2] which was earlier proved only in odd residue characteristic. 1 Triple products Let π1, π2 and π3 be three irreducible admissible infinite dimensional representations of GL2(ku) for a non-Archimedean local field ku with the product of their central characters trivial. Let Du denote the unique quaternion division algebra over ku. For an irreducible admissible discrete series representation π of GL2(ku), let π denote the representation of D u associated to π by the Jacquet-Langlands correspondence, and let π = 0, if π is not a discrete series representation. The author in his thesis [P1] studied the space of trilinear forms l : π1 ⊗ π2 ⊗ π3 → C which are GL2(ku)-invariant. Let m(π1 ⊗ π2 ⊗ π3) denote the dimension of the space of such trilinear forms, and let m(π 1 ⊗ π ′ 2 ⊗ π ′ 3) denote the dimension of the space of D u-invariant linear forms on π ′ 1 ⊗ π ′ 2 ⊗ π ′ 3 (so m(π 1 ⊗ π ′ 2 ⊗ π ′ 3) is nonzero only if all the πi are discrete series representations). We let ǫ(π1 ⊗ π2 ⊗ π3) = ǫ( 1 2 , π1 ⊗ π2 ⊗ π3) denote the triple product epsilon factor defined by the Langlands-Shahidi method; under the condition that the