Subconvex bounds for compact toric integrals

TL;DR

This paper extends subconvex bounds for toric integrals in the context of Waldspurger's formula, overcoming key difficulties and achieving bounds that outperform previous results with high probability.

Contribution

It introduces a generalized method for subconvex bounds in the setting of compact toric integrals, addressing split place scarcity and test vector issues.

Findings

Bound is valid with high probability

Achieves better bounds than previous results

Addresses key difficulties in the method

Abstract

We generalize our method for subconvex bounds for $\mathrm{GL}_2 \times \mathrm{GL}_1$ 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.