Year: 2026
Type: preprint
Abstract: We establish an explicit symmetric spectral reciprocity formula for twisted fourth moments of $\mathrm{GL}_{2}$ $L$-functions over an arbitrary number field. A Rankin--Selberg kernel gives a new analytic regularization of the formal Petersson identity for products of Eisenstein series, based on the rank decomposition $4=2+2$. The formula preserves the nonnegativity of the amplified fourth-moment spectral side at general levels. We further identify a new \emph{square-level concentration phenomenon}: after amplification, the dominant off-diagonal contribution comes from representations whose conductor is the square of a product of two distinct primes. Using square-level local test functions in the relative trace formula, we prove new hybrid subconvexity bounds for twisted $L$-functions. Combining these bounds with symmetric spectral reciprocity and amplification, we obtain the uniform estimate \begin{align*} L(1/2,π)\ll_{F,\varepsilon} C(π)^{\frac{1}{4}-\frac{1}{120}+\varepsilon} \end{align*} for every unitary cuspidal automorphic representation $π$ of $\mathrm{GL}_{2}/F$ with arbitrary central character. We also obtain stronger bounds for fixed central characters, a prime-level fifth-moment estimate, and applications to Rankin--Selberg and ring class $L$-functions.
Source: arXiv (Cornell University)
Author: Yang, Liyang
Language: English
Subfield: Algebra and Number Theory
Field: Mathematics
Domain: Physical Sciences
Open Access status: green
OpenAlex ID: W7167619909