Theta correspondence and the Prasad conjecture for SL(2)

TL;DR

This paper presents a new proof for local period problems of SL(2) over quadratic extensions using theta correspondence and verifies the Prasad conjecture for SL(2), extending methods to Sp(4).

Contribution

It introduces a novel approach linking base change and theta lifts to address the Prasad conjecture for SL(2) and Sp(4).

Findings

Verified the Prasad conjecture for SL(2).

Provided a new proof for local period problems of SL(2).

Extended the strategy to Sp(4).

Abstract

We use relations between the base change representations and theta lifts, to give a new proof to the local period problems of SL(2) over a nonarchimedean quadratic field extension E/F. Then we will verify the Prasad conjecture for SL(2). With a similar strategy, we obtain a certain result for the Prasad conjecture for Sp(4).