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Jiaqi Hou

Publications and source records attributed to Jiaqi Hou.

6 recordsLinked to original sources
arXiv2026

$L^q$-norm bounds for arithmetic eigenfunctions via microlocal Kakeya-Nikodym estimate

Let $X$ be a compact arithmetic congruence hyperbolic surface, and let $ψ$ be an $L^2$-normalized Hecke-Maass form on $X$ with sufficiently large spectral parameter $λ$. We give a new proof to obtain a power saving for the global $L^6$-norm $\|ψ\|_{L^6(X)}\lesssim_\varepsilonλ^{\frac{5}{36}+\varepsilon}$ over the local bound $\|ψ\|_{L^6(X)}\lesssimλ^{\frac{1}{6}}$ of Sogge. Our method uses a microlocal decomposition for $ψ$ and reduces the $L^6$-norm problem to microlocal Kakeya-Nikodym estimates for $ψ$, and we establish improved microlocal Kakeya-Nikodym estimates via arithmetic amplification developed by Iwaniec and Sarnak.

math.NT↗
arXiv2025

Restrictions of Maass forms on $\mathrm{SL}(2,\mathbb{C})$ to hyperbolic surfaces and geodesic tubes

Let $ψ$ be an $L^2$-normalized Hecke-Maass form with a large spectral parameter $λ>0$ on a compact arithmetic congruence hyperbolic 3-manifold $X=Γ\backslash\mathrm{SL}(2,\mathbb{C})/\mathrm{SU}(2)$, and let $Y$ be a totally geodesic surface in $X$ with bounded diameter. The local $L^2$-bound for the restriction of $ψ$ to $Y$ is $\|ψ|_Y\|_{L^2(Y)}\ll λ^{1/4}$ by Burq, Gérard, and Tzvetkov. We apply the method of arithmetic amplification developed by Iwaniec and Sarnak to obtain a power saving over the local bound. The new feature in the proof is that we establish two different estimates for the integrals of $ψ|_Y$ against geodesic beams over $Y$ via two amplification arguments. Combining these estimates, we can improve the local bound for generalized Fourier coefficients of $ψ|_Y$ against eigenfunctions on $Y$ with spectral parameters near $λ$. We also apply the amplification method to obtain a power saving over the trivial bound $O(1)$ for $L^2$-norms of $ψ$ restricted to $λ^{-1/2}$-neighborhoods of unit-length geodesic segments. Consequently, by applying a result of Blair and Sogge, we obtain power savings over the local $L^p$-bounds of $ψ$ by Sogge for $2<p<4$ from our improved bound for the Kakeya-Nikodym norm.

math.NT↗
arXiv2025

Kakeya-Nikodym norms of Maass forms on $\rm{U}(2,1)$

Let $ψ$ be a Hecke-Maass form with a large spectral parameter on a compact arithmetic complex hyperbolic surface. We apply the amplification method to obtain a power saving over the trivial bound for the Kakeya-Nikodym norm of $ψ$. As a consequence, we obtain power savings over the local bound of Sogge for $\|ψ\|_p$ when $2<p<10/3$.

math.NT↗
arXiv2025

Weighted geodesic restrictions of arithmetic eigenfunctions

Let $X$ be an arithmetic hyperbolic surface, $ψ$ a Hecke-Maass form, $\ell$ a geodesic segment on $X$, and $μ$ a Borel measure supported on $\ell$ with dimension greater than 1/2. We obtain a power saving over the local bound of Eswarathasan and Pramanik for the $L^2$ norm of $ψ$ with respect to $μ$, which is a weighted generalization of Marshall's geodesic restriction bound and is proved by applying the method of arithmetic amplification. On a general 2-dimensional Riemannian manifold, we also obtain a Kakeya-Nikodym bound for the $L^2$ norm of any Laplace-Beltrami eigenfunction with respect to a Borel measure supported on a geodesic segment with dimension greater than 1/2.

math.NT↗
arXiv2024

A nonabelian circle method

We count integral quaternion zeros of $γ_1^2 \pm \dots \pm γ_n^2$, giving an asymptotic when $n\ge 9$, and a likely near-optimal bound when $n=8$. To do so, we introduce a new, nonabelian delta symbol method, which is of independent interest. Our asymptotic at height $X$ takes the form $cX^{4n-8} + O(X^{3n+\varepsilon})$ for suitable $c \in \mathbb{C}$ and any $\varepsilon>0.$ We construct special subvarieties implying that, in general, $3n+\varepsilon$ can be at best improved to $3n-2.$

math.NT↗