$L^4$-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 $1\le k < t_{\phi}^{1-\epsilon}$, we are able to find $\beta_k = \{X_k +yi : a < y < a+h\}$ with $-\frac{1}{2} + \frac{k-1}{t_{\phi}^{1-\epsilon}} \le X_k \le -\frac{1}{2} +\frac{k}{t_{\phi}^{1-\epsilon}}$ such that the number of sign changes of $\phi$ along the segment $\beta_k$ is $\gg_{\epsilon} t_{\phi}^{1-\epsilon_1}$ as $t_\phi\to \infty$. Also, we obtain the similar result for horizontal lines. On the other hand, we conditionally prove that for a sufficiently large segment $\beta$ on $\mathrm{Re}(z) = 0$ and $\mathrm{Im}(z) > 0$, 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 $t_{\phi}\to\infty$.