The subconvexity problem for symmetric square L-functions in level aspect
Abstract
In this paper, we address the subconvexity problem in level aspect for symmetric square L-functions for cuspidal automorphic representation of GL2(ℚ) with a prescribed local ramification at prime p. To be more precise, let π be a tempered cuspidal automorphic representation of conductor q(π)=p2 with a non-quadratic central character of conductor p. We prove that if the corresponding local representation πp belongs to a suitable class of representations 𝒮, then L(1/2, Sym2π)≪ε, π∞ q(Sym2π)1/4−1/168+o(1), where implied constant depends polynomially on the spectral parameters of π∞. This is the first instance of level-aspect subconvex bound for L-functions of a GL3(ℚ) automorphic representation. Our approach is based on the delta-symbol method. Apart from some standard analytic number theoretic tools, Katz's theory of hypergeometric sums, and Deligne's proof of Weil-conjectures play an important role in the proof.
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BibTeX
@misc{mitra2026subconvexity,
title = {{The subconvexity problem for symmetric square $L$-functions in level aspect}},
author = {Pratim Mitra},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2609.04155},
eprint = {2609.04155},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2609.04155},
}
RIS
TY - UNPB
TI - The subconvexity problem for symmetric square L-functions in level aspect
AU - Pratim Mitra
PY - 2026
DA - 2026/09/03
PB - arXiv
DO - 10.48550/arxiv.2609.04155
UR - https://arxiv.org/abs/2609.04155
AB - In this paper, we address the subconvexity problem in level aspect for symmetric square L-functions for cuspidal automorphic representation of GL_2(ℚ) with a prescribed local ramification at prime p. To be more precise, let π be a tempered cuspidal automorphic representation of conductor q(π)=p^2 with a non-quadratic central character of conductor p. We prove that if the corresponding local representation π_p belongs to a suitable class of representations 𝒮, then L(1/2, Sym^2π)≪_ε, π_∞ q(Sym^2π)^1/4−1/168+o(1), where implied constant depends polynomially on the spectral parameters of π_∞. This is the first instance of level-aspect subconvex bound for L-functions of a GL_3(ℚ) automorphic representation. Our approach is based on the delta-symbol method. Apart from some standard analytic number theoretic tools, Katz's theory of hypergeometric sums, and Deligne's proof of Weil-conjectures play an important role in the proof.
ER -