$L^4$-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 $\varphi$ with the eigenvalue ${\lambda }_{\varphi }=\frac{1}{4}+t_{\varphi }^2$, for $a>0$, a sufficiently large $h>0$ and for any $0<{\epsilon }_1<\epsilon /10^7$ ($\epsilon >0$) , for almost all $1\le k<t_{\varphi }^{1-\epsilon }$, we are able to find ${\beta }_k=\{X_k+yi:a<y<a+h\}$ with $-\frac{1}{2}+\frac{k-1}{t_{\varphi }^{1-\epsilon }}\le X_k\le -\frac{1}{2}+\frac{k}{t_{\varphi }^{1-\epsilon }}$ such that the number of sign changes of $\varphi$ along the segment ${\beta }_k$ is ${\gg }_{\epsilon }{t}_{\varphi }^{1-{\epsilon }_1}$ as $t_{\varphi }\to \infty$. Also, we obtain the similar result for horizontal lines. On the other hand, we conditionally prove that for a sufficiently large segment $\beta$ on $\text{Re}(z)=0$ and $\text{Im}(z)>0$, the number of sign changes of $\varphi$ along $\beta$ is ${\gg }_{\epsilon }{t}_{\varphi }^{1-\epsilon }$ and consequently, the number of inert nodal domains meeting any compact vertical segment on the imaginary axis is ${\gg }_{\epsilon }{t}_{\varphi }^{1-\epsilon }$ as $t_{\varphi }\to \infty$.