Soft bounds for local triple products and the subconvexity-QUE implication for $\mathrm{GL}_2$

TL;DR

This paper provides a soft proof of uniform bounds for local factors in the triple product formula, enabling effective quantum unique ergodicity results derived from subconvexity bounds.

Contribution

It introduces a soft proof technique for local bounds in the triple product formula, facilitating broader applications to QUE from subconvexity.

Findings

Established uniform upper bounds for local factors in the triple product formula.

Connected subconvexity bounds to effective quantum unique ergodicity results.

Provided a general framework for deriving QUE from subconvexity using local bounds.

Abstract

We give a soft proof of a uniform upper bound for the local factors in the triple product formula, sufficient for deducing effective and general forms of quantum unique ergodicity (QUE) from subconvexity.