  Let $g$ be a Hecke-Maass cusp form on the modular surface ${\rm
SL}_2(\mathbb{Z})\backslash\mathbb{H}$, namely an $L^2$-normalised nonconstant
Laplacian eigenfunction on ${\rm SL}_2(\mathbb{Z})\backslash\mathbb{H}$ 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. Via interpolation, 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.
Furthermore, these methods yield for compact arithmetic surfaces the best
$L^4$-norm bound to date.
  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 ${\rm
GL}_3\times{\rm GL}_2\leftrightsquigarrow{\rm GL}_4\times{\rm GL}_1$ spectral
reciprocity, which relates a ${\rm GL}_2$ moment of ${\rm GL}_3\times{\rm
GL}_2$ Rankin-Selberg $L$-functions to a ${\rm GL}_1$ moment of ${\rm
GL}_4\times{\rm GL}_1$ 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 ${\rm GL}_4\times{\rm
GL}_2\leftrightsquigarrow{\rm GL}_4\times{\rm GL}_2$ spectral reciprocity,
which is a cuspidal analogue of a formula of Kuznetsov for the fourth moment of
central values of Hecke $L$-functions.
