Symmetric Spectral Reciprocity for $\mathrm{GL}(2)$ and Uniform Subconvexity

Abstract

We construct a new analytic regularization of the Petersson norm identity for Eisenstein series on $\mathrm{GL}_2$ over a number field $F$, and derive from it an explicit symmetric spectral reciprocity formula for twisted fourth moments of $\mathrm{GL}_2$ $L$-functions, reflecting the intrinsic rank decomposition $4=2+2$. Independently, we identify a square-level phenomenon arising from amplification, whereby the dual spectral family acquires square-level conductors. This additional arithmetic rigidity permits a refined analysis within the relative trace formula and leads to refined hybrid subconvexity bounds for twisted $L$-functions. As a consequence, we obtain new uniform subconvexity bounds for $\mathrm{GL}_2/F$; in particular, \begin{align*} L(1/2,π)\ll C(π)^{\frac14-\frac{1}{120}+\varepsilon} \end{align*} for every unitary cuspidal representation $π$ of $\mathrm{GL}_2/F$. We also obtain refined bounds for certain Artin $L$-functions and applications to class group arithmetic.