Subconvex bounds for compact toric integrals

We generalize our method for subconvex bounds for $$\mathrm{G}\mathrm{L}$_2 \times $\mathrm{G}\mathrm{L}$_1$ to the setting of the Waldspurger’s formula for compact torical integrals. We address the two major difficulties: one is the lack of split places with small norm, the other is the test vector problem. The final bound is valid with arbitrary high probability and is better than the known bounds for a non-empty interval.