$L^4$-norms and sign changes of Maass forms
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arXiv
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| Format: | Preprint |
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2023
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| _version_ | 1866911448614043648 |
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| author | Ki, Haseo |
| author_facet | Ki, Haseo |
| contents | 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 $t_ϕ\to\infty$. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2302_02625 |
| institution | arXiv |
| publishDate | 2023 |
| record_format | arxiv |
| spellingShingle | $L^4$-norms and sign changes of Maass forms Ki, Haseo Number Theory Primary: 11F12, 11F30. Secondary: 35P20 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 $t_ϕ\to\infty$. |
| title | $L^4$-norms and sign changes of Maass forms |
| topic | Number Theory Primary: 11F12, 11F30. Secondary: 35P20 |
| url | https://arxiv.org/abs/2302.02625 |