Relating invariant linear form and local epsilon factors via global methods

Abstract

We use the recent proof of Jacquet's conjecture due to Harris and Kudla [HK] and the Burger-Sarnak principle (see [BS]) to give a proof of the relationship between the existence of trilinear forms on representations of ${\mathrm{GL}}_2(k_u)$ for a non-Archimedean local field $k_u$ and local epsilon factors which was earlier proved only in the odd residue characteristic by this author in [P1, Theorem 1.4]. The method used is very flexible and gives a global proof of a theorem of Saito and Tunnell about characters of ${\mathrm{GL}}_2$ using a theorem of Waldspurger [W, Theorem 2] about period integrals for ${\mathrm{GL}}_2$ and also an extension of the theorem of Saito and Tunnell by this author in [P3, Theorem 1.2] which was earlier proved only in odd residue characteristic. In the appendix to this article, H. Saito gives a local proof of Lemma 4 which plays an important role in the article