Symmetric Spectral Reciprocity for GL(2) and Uniform Subconvexity
Abstract
We establish an explicit symmetric spectral reciprocity formula for twisted fourth moments of GL2 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 L(1/2,π)≪F,ε C(π)1/4−1/120+ε for every unitary cuspidal automorphic representation π of GL2/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.
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BibTeX
@misc{yang2026symmetric,
title = {{Symmetric Spectral Reciprocity for $\mathrmGL(2)$ and Uniform Subconvexity}},
author = {Liyang Yang},
howpublished = {arXiv},
year = {2026},
month = nov,
doi = {10.48550/arxiv.2607.04476},
eprint = {2607.04476},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2607.04476},
}
RIS
TY - UNPB
TI - Symmetric Spectral Reciprocity for GL(2) and Uniform Subconvexity
AU - Liyang Yang
PY - 2026
DA - 2026/11/02
PB - arXiv
DO - 10.48550/arxiv.2607.04476
UR - https://arxiv.org/abs/2607.04476
AB - We establish an explicit symmetric spectral reciprocity formula for twisted fourth moments of 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 L(1/2,π)≪_F,ε C(π)^1/4−1/120+ε for every unitary cuspidal automorphic representation π of 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.
ER -