$L^p$-Norm Bounds for Automorphic Forms via Spectral Reciprocity

Peter Humphries, Rizwanur Khan

Published 2022-08-11Version 1

Let $g$ be a Hecke-Maass cusp form on the modular surface $\mathrm{SL}_2(\mathbb{Z}) \backslash \mathbb{H}$, namely an $L^2$-normalised nonconstant Laplacian eigenfunction on $\mathrm{SL}_2(\mathbb{Z})\backslash\mathbb{H}$ that is additionally a joint eigenfunction of every Hecke operator. We prove the $L^4$-norm bound $\|g\|_4\ll_{\varepsilon}\lambda_g^{3/304+\varepsilon}$, where $\lambda_g$ denotes the Laplacian eigenvalue of $g$, which improves upon Sogge's $L^4$-norm bound $\|g\|_4\ll\lambda_g^{1/16}$ for Laplacian eigenfunctions on a compact Riemann surface by more than a six-fold power-saving. Interpolating with the sup-norm bound $\|g\|_{\infty}\ll_{\varepsilon}\lambda_g^{5/24+\varepsilon}$ due to Iwaniec and Sarnak, this yields $L^p$-norm bounds for Hecke-Maass cusp forms that are power-saving improvements on Sogge's bounds for all $p > 2$. Our paper marks the first improvement of Sogge's result on the modular surface. Via the Watson-Ichino triple product formula, bounds for the $L^4$-norm of $g$ are reduced to bounds for certain mixed moments of $L$-functions. We bound these using two forms of spectral reciprocity. The first is a form of $\mathrm{GL}_3\times\mathrm{GL}_2\leftrightsquigarrow\mathrm{GL}_4\times\mathrm{GL}_1$ spectral reciprocity, which relates a $\mathrm{GL}_2$ moment of $\mathrm{GL}_3\times\mathrm{GL}_2$ Rankin-Selberg $L$-functions to a $\mathrm{GL}_1$ moment of $\mathrm{GL}_4\times\mathrm{GL}_1$ Rankin-Selberg $L$-functions; this can be seen as a cuspidal analogue of Motohashi's formula relating the fourth moment of the Riemann zeta function to the third moment of central values of Hecke $L$-functions. The second is a form of $\mathrm{GL}_4\times\mathrm{GL}_2\leftrightsquigarrow\mathrm{GL}_4\times\mathrm{GL}_2$ spectral reciprocity, which is a cuspidal analogue of a formula of Kuznetsov for the fourth moment of central values of Hecke $L$-functions.