$L^4$-norms and sign changes of Maass forms

Abstract

Unconditionally, we prove the Iwaniec–Sarnak conjecture for $L^4$-norms of the Hecke–Maass cusp forms. From this result, we can justify that for even Maass cusp form $\phi$ with the eigenvalue $\lambda_{\phi}=\frac{1}{4}+t_{\phi}^2$, for $a>0$, a sufficiently large $h>0$ and for any $0<\epsilon_1<\epsilon/10^7$ ($\epsilon>0$) , for almost all $1\le k0$, the number of sign changes of $\phi$ along $\beta$ is $\gg_{\epsilon} t_{\phi}^{1-\epsilon}$ and consequently, the number of inert nodal domains meeting any compact vertical segment on the imaginary axis is $\gg_{\epsilon} t_{\phi}^{1-\epsilon}$ as $t_{\phi}\to\infty$.