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).
This paper studies the Prasad conjecture for the special orthogonal group \mathrm{SO}_{3,3}.
Then we use the local theta correspondence between \mathrm{Sp}_{4} and \mathrm{O}(V) to study the \mathrm{Sp}_{4}-distinction problems over a quadratic field extension E/F and \dim V=4 or 6.
Thus we can verify the Prasad conjecture for a square-integrable representation of \mathrm{Sp}_{4}(E).
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