arXiv subjects

Chun-Hsien Hsu

Publications and source records attributed to Chun-Hsien Hsu.

9 recordsLinked to original sources

arXiv2026

On triple product $L$-functions and the fiber bundle method

We introduce multi-variable zeta integrals which unfold to Euler products representing the triple product $L$-function times a product of $L$-functions with known analytic properties. We then formulate a generalization of the Poisson summation conjecture and show how it implies the analytic properties of triple product $L$-functions. Finally, we propose a strategy, the fiber bundle method, to reduce this generalized conjecture to a simpler case of the Poisson summation conjecture along with certain local compatibility statements.

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arXiv2026

Plancherel and Poisson summation formulae for a family of affine $Ψ$-bundles

We prove a Plancherel formula for a family of affine $Ψ$-bundles. As an application, we construct Fourier transforms and asymptotic Schwartz spaces for the family. We then prove the corresponding Poisson summation formula under suitable assumptions. The choice of affine $Ψ$-bundles we consider is motivated by applications to triple product $L$-functions explored in another paper of the authors and Leslie.

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arXiv2026

Schwartz spaces on L-monoids: non-Archimedean

We complete the Braverman-Kazhdan-Ngô program over non-Archimedean local fields assuming local Langlands conjecture for tempered representations and a natural assumption on the $γ$-factors of non-supercuspidal discrete series. In particular, the program is complete unconditionally for general linear groups over non-Archimedean local fields.

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arXiv2026

Asymptotics of Schwartz functions

Let $G$ be a split, simply connected, almost simple algebraic group, and let $P$ be a maximal parabolic subgroup of $G$. Braverman and Kazhdan in \cite{BKnormalized} defined a Schwartz space on the affine closure $X_P$ of $P^{\mathrm{der}}\backslash G$. An alternate, more analytically tractable definition was given in \cite{Getz:Hsu:Leslie}, following several earlier works. When $G$ is a classical group or $G_2$, we show the two definitions coincide and prove several previously conjectured properties of the Schwartz space that will be useful in applications. Along the way, we give an alternative construction of the ring of differential operators on $X_P$ using the Fourier theory. We also establish the Poisson summation formulae in these cases.

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arXiv2026

Weyl algebras on Braverman-Kazhdan spaces

Let $G$ be a split, simply connected, almost simple algebraic group over a field of characteristic zero, and let $P$ be a maximal parabolic subgroup of $G$. We study the ring of differential operators on $P^{\mathrm{der}}\backslash G$, showing that it shares several key structural properties with classical Weyl algebras. We also develop a corresponding theory of $D$-modules.

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arXiv2026

Geometrization of summation formulae for quadrics

We geometrize the Poisson summation formula for the zero locus of a split quadratic form in an even number of variables over number fields. We do so by making explicit the relationship between Schwartz spaces on quadrics defined in two different ways: via Braverman-Kazhdan spaces and via theta lifts.

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arXiv2024

The Fourier transform for triples of quadratic spaces

Let $V_1,V_2,V_3$ be a triple of even dimensional vector spaces over a number field $F$ equipped with nondegenerate quadratic forms $\mathcal{Q}_1,\mathcal{Q}_2,\mathcal{Q}_3$, respectively. Let $Y \subset \prod_{i=1}^3 V_i$ be the closed subscheme consisting of $(v_1,v_2,v_3)$ such that $\mathcal{Q}_1(v_1)=\mathcal{Q}_2(v_2)=\mathcal{Q}_3(v_3)$. One has a Poisson summation formula for this scheme under suitable assumptions on the functions involved, but the relevant Fourier transform was previously only defined as a correspondence. In the current paper we employ a novel global-to-local argument to prove that this Fourier transform is well-defined on the Schwartz space of $Y(\mathbb{A}_F).$ To execute the global-to-local argument, we introduce boundary terms and thereby extend the Poisson summation formula to a broader class of test functions. This is the first time a summation formula with boundary terms has been proven for a spherical variety that is not a Braverman-Kazhdan space.

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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 $3n-2.$

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arXiv2022

Harmonic analysis on certain spherical varieties

Braverman and Kazhdan proposed a conjecture, later refined by Ngô and broadened to the framework of spherical varieties by Sakellaridis, that asserts that affine spherical varieties admit Schwartz spaces, Fourier transforms, and Poisson summation formulae. The first author in joint work with B.~Liu and later the first two authors proved these conjectures for certain spherical varieties $Y$ built out of triples of quadratic spaces. However, in these works the Fourier transform was only proven to exist. In the present paper we give, for the first time, an explicit formula for the Fourier transform on $Y.$ We also prove that it is unitary in the nonarchimedean case. As preparation for this result, we give explicit formulae for Fourier transforms on Braverman-Kazhdan spaces attached to maximal parabolic subgroups of split, simple, simply connected groups. These Fourier transforms are of independent interest, for example, from the point of view of analytic number theory.

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