EXPLICIT DOUBLING INTEGRALS FOR S̃p2(Q2) USING “GOOD TEST VECTORS”

In a previous paper (see http:/www.math.ohio-state.edu/ ̃czorn/ works.html), we computed examples of the doubling integral for constituents of the unramified principal series of Sp2(F ) and S̃p2(F ) where F was a nondyadic field. These computations relied on certain “good test vectors” and “good theta test sections” motivated by the non-vanishing of theta lifts. In this paper, we aim to prove a partial analog for S̃p2(Q2). However, due to several complexities, we compute the doubling integral only for certain irreducible principal series representations induced from characters with ramified quadratic twists. We develop some 2-adic analogs for the machinery in the paper mentioned above; however, these tend to be more delicate and have more restrictive hypotheses than the non-dyadic case. Ultimately, this paper and the one mentioned above develop several computations intended to be used for future research into the non-vanishing of theta lifts.