Rankin--Selberg Subconvexity via Spectral Reciprocity

Abstract

We establish explicit subconvex bounds for central values of Rankin--Selberg $L$-functions $L(1/2,π\timesπ')$ associated with pairs of unitary cuspidal automorphic representations of $\mathrm{GL}_2$ over a number field. Building on the spectral reciprocity framework of Michel and Venkatesh, we develop a refined, fully explicit form of spectral reciprocity that allows for precise control of conductors and local test vectors. As a consequence, we obtain an explicit subconvex bound, which, even over $F=\mathbb{Q}$, improves all previously known results. We further apply these bounds to several arithmetic problems. These include effective equidistribution of CM suborbits on quaternionic Shimura varieties, quantitative equidistribution of totally geodesic submanifolds, a uniform quantitative form of dihedral quantum unique ergodicity over number fields, and an application to distinguishing cuspidal automorphic representations.