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

TL;DR

The Iwaniec-Sarnak conjecture for $L^4$-norms of the Hecke-Maass cusp forms is proved and the number of inert nodal domains meeting any compact vertical segment on the imaginary axis is found.

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 $φ$ with the eigenvalue $λ_φ=\frac{1}{4}+t_φ^2$, for $a>0$, a sufficiently large $h>0$ and for any $0<ε_1<ε/10^7$ ($ε>0$) , for almost all $1\le k<t_φ^{1-ε}$, we are able to find $β_k=\{X_k+yi:a<y<a+h\}$ with $-\frac{1}{2}+\frac{k-1}{t_φ^{1-ε}}\le X_k\le-\frac{1}{2}+\frac{k}{t_φ^{1-ε}}$ such that the number of sign changes of $φ$ along the segment $β_k$ is $\gg_ε t_φ^{1-ε_1}$ as $t_φ\to\infty$. Also, we obtain the similar result for horizontal lines. On the other hand, we conditionally prove that for a sufficiently large segment $β$ on $\textrm{Re}(z)=0$ and $\textrm{Im}(z)>0$, the number of sign changes of $φ$ along $β$ is $\gg_ε t_φ^{1-ε}$ and consequently, the number of inert nodal domains meeting any compact vertical segment on the imaginary axis is $\gg_ε t_φ^{1-ε}$ as $t_φ\to\infty$.

Theorems & Definitions (26)

• Theorem 1
• Remark 2
• Corollary 3
• Theorem 4
• Theorem 5
• Lemma 2.1
• Lemma 2.2
• Proposition 2.3
• Proposition 2.4
• Theorem 2.5