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Bounds for Average toward the Resonance Barrier for GL(3) × GL(2) Automorphic Forms

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Abstract

Let f be a fixed Maass form for SL3 (ℤ) with Fourier coefficients Af(m, n). Let g be a Maass cusp form for SL2 (ℤ) with Laplace eigenvalue \({1 \over 4} + {k^2}\) and Fourier coefficient λg(n), or a holomorphic cusp form of even weight k. Denote by SX(f × g, α, β) a smoothly weighted sum of Af(1, n)λg(n)e(αnβ) for X < n < 2X, where α ≠ 0 and β > 0 are fixed real numbers. The subject matter of the present paper is to prove non-trivial bounds for a sum of SX(f × g, α, β) over g as k tends to ∞ with X. These bounds for average provide insight for the corresponding resonance barriers toward the Hypothesis S as proposed by Iwaniec, Luo, and Sarnak.

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Authors and Affiliations

  1. San Diego State University—Imperial Valley, Calexico, CA, 92231, USA

    Huan Qin

  2. Department of Mathematics, The University of Iowa, Iowa City, Iowa, 52242-1419, USA

    Yang Bo Ye

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  1. Huan Qin
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  2. Yang Bo Ye
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Correspondence to Yang Bo Ye.

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Qin, H., Ye, Y.B. Bounds for Average toward the Resonance Barrier for GL(3) × GL(2) Automorphic Forms. Acta. Math. Sin.-English Ser. 39, 1667–1683 (2023). https://doi.org/10.1007/s10114-023-1022-4

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  • DOI: https://doi.org/10.1007/s10114-023-1022-4

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