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Yueke Hu

Publications and source records attributed to Yueke Hu.

17 recordsLinked to original sources
arXiv2026

The cubic moment of $L$-functions for specified local component families

We prove Lindelöf-on-average upper bounds on the cubic moment of central values of $L$-functions over certain families of $\operatorname{PGL}_2/\mathbb{Q}$ automorphic representations $π$ given by specifying the local representation $π_p$ of $π$ at finitely many primes. Such bounds were previously known in the case that $π_p$ belongs to the principal series or is a ramified quadratic twist of the Steinberg representation; here we handle the supercuspidal case. Crucially, we use new Petersson/Bruggeman-Kuznetsov forumulas for supercuspidal local component families recently developed by the authors. As corollaries, we derive Weyl-strength subconvex bounds for central values of $\operatorname{PGL}_2$ $L$-functions in the square-full aspect, and in the depth aspect, or in a hybrid of these two situations. A special case of our results is the Weyl-subconvex bound for all cusp forms of level $p^2$. Previously, such a bound was only known for forms that are twists from level $p$, which cover roughly half of the level $p^2$ forms.

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arXiv2026

A generalized PGL(2) Petersson/Bruggeman-Kuznetsov formula for analytic applications

We develop generalized Petersson/Bruggeman-Kuznetsov (PBK) formulas for specified local components at non-archimedean places. In fact, we introduce two hypotheses on non-archimedean test function pairs $f \leftrightarrow π(f)$, called geometric and spectral hypotheses, under which one obtains `nice' PBK formulas by the adelic relative trace function approach. Then, given a supercuspidal representation $σ$ of ${\rm PGL}_2(\mathbb{Q}_p)$, we study extensively the case that $π(f)$ is a projection onto the line of the newform if $π$ is isomorphc to $σ$ or its unramified quadratic twist, and $π(f) = 0$ otherwise. As a first application, we prove an optimal large sieve inequality for families of automorphic representations that arise in our framework.

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arXiv2025

The subconvexity bound for standard L-function in level aspect

In this paper we prove a new subconvexity result for the standard L-function of a unitary cuspidal automorphic representation $π$ of $\text{GL}_n$, where the finite set of places $S$ with large conductors is allowed to vary, provided that the local parameters at every place in $S$ satisfy certain uniform growth condition.

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arXiv2023

The subconvexity problem for Rankin-Selberg and triple product L-functions

In this paper we study the subconvexity problem for the Rankin-Selberg L-function and triple product L-function, allowing joint ramifications and conductor dropping range. We first extend the method of Michel-Venkatesh to reduce the bounds for L-functions to local conjectures on test vectors, then verify these local conjectures under certain conditions, giving new subconvex bounds as long as the representations are not completely related.

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arXiv2023

The Petersson/Kuznetsov trace formula with prescribed local ramifications

In this paper we derive refined Petersson/Kuznetsov trace formulae with prescribed local ramifications. The spectral side of these formulae picks out newforms whose associated local components come from specific sub-families of representations of given level, and are much shorter compared with the classical versions. We use them to study the first moment and the subconvexity bound of certain Rankin-Selberg L-function in a hybrid setting, obtaining Weyl bound in a wider range compared to previous works.

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arXiv2020

Sup-norms of eigenfunctions in the level aspect for compact arithmetic surfaces, II: newforms and subconvexity

We improve upon the local bound in the depth aspect for sup-norms of newforms on $D^\times$ where $D$ is an indefinite quaternion division algebra over $\mathbb{Q}$. Our sup-norm bound implies a depth-aspect subconvexity bound for $L(1/2, f \times θ_χ)$, where $f$ is a (varying) newform on $D^\times$ of level $p^n$, and $θ_χ$ is an (essentially fixed) automorphic form on $\mathrm{GL}_2$ obtained as the theta lift of a Hecke character $χ$ on a quadratic field. For the proof, we augment the amplification method with a novel filtration argument and a recent counting result proved by the second-named author to reduce to showing strong quantitative decay of matrix coefficients of local newvectors along compact subsets, which we establish via $p$-adic stationary phase analysis. Furthermore, we prove a general upper bound in the level aspect for sup-norms of automorphic forms belonging to \emph{any} family whose associated matrix coefficients have such a decay property.

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arXiv2020

New test vector for Waldspurger's period integral, relative trace formula, and hybrid subconvexity bounds

In this paper we give quantitative local test vectors for Waldspurger's period integral (i.e., a toric period on $\text{GL}_2$) in new cases with joint ramifications. The construction involves minimal vectors, rather than newforms and their variants. This paper gives a uniform treatment for the matrix algebra and division algebra cases under mild assumptions, and establishes an explicit relation between the size of the local integral and the finite conductor $C(π\timesπ_{χ^{-1}})$. As an application, we combine the test vector results with the relative trace formula, and prove a hybrid type subconvexity bound which can be as strong as the Weyl bound in proper range.

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arXiv2019

Waldspurger's period integral for newforms

In this paper we discuss Waldspurger's local period integral for newforms in new cases. The main ingredient is the work \cite{HN18} on Waldspurger's period integral using the minimal vectors, and the explicit relation between the newforms and the minimal vectors. We use a representation theoretical trick to simplify computations for newforms. As an example, we compute the local integral coming from a special arithmetic setting which was used to study 3-part full BSD conjecture in \cite{HSY}.

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arXiv2018

An explicit Gross-Zagier formula related to the Sylvester Conjecture

Let $p\equiv 4,7\mod 9$ be a rational prime number such that $3\mod p$ is not a cubic residue. In this paper we prove the 3-part of the product of the full BSD conjectures for $E_p$ and $E_{3p^3}$ is true using an explicit Gross-Zagier formula, where $E_p: x^3+y^3=p$ and $E_{3p^2}: x^3+y^3=3p^2$ are the elliptic curves related to the Sylvester conjecture and cube sum problems.

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arXiv2018

Sup norm on $\text{PGL}_n$ in depth aspect

In this short paper we give the sub-local upper bound for the sup norm of an automorphic form on $\text{PGL}_n$, whose associated automorphic representation has finite conductor $C(π)=p^c$ with $c\rightarrow \infty$, and its local component at the place of ramification is a minimal vector belonging to an irreducible representation with generic induction datum.

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arXiv2018

Some analytic aspects of automorphic forms on GL(2) of minimal type

Let $π$ be a cuspidal automorphic representation of $PGL_2(\mathbb{A}_\mathbb{Q})$ of arithmetic conductor $C$ and archimedean parameter $T$, and let $ϕ$ be an $L^2$-normalized automorphic form in the space of $π$. The sup-norm problem asks for bounds on $\| ϕ\|_\infty$ in terms of $C$ and $T$. The quantum unique ergodicity (QUE) problem concerns the limiting behavior of the $L^2$-mass $|ϕ|^2 (g) \, d g$ of $ϕ$. All previous work on these problems in the conductor-aspect has focused on the case that $ϕ$ is a newform. In this work, we study these problems for a class of automorphic forms that are not newforms. Precisely, we assume that for each prime divisor $p$ of $C$, the local component $π_p$ is supercuspidal (and satisfies some additional technical hypotheses), and consider automorphic forms $ϕ$ for which the local components $ϕ_p \in π_p$ are "minimal" vectors. Such vectors may be understood as non-archimedean analogues of lowest weight vectors in holomorphic discrete series representations of $PGL_2(\mathbb{R})$. For automorphic forms as above, we prove a sup-norm bound that is sharper than what is known in the newform case. In particular, if $π_\infty$ is a holomorphic discrete series of lowest weight $k$, we obtain the optimal bound $C^{1/8 -ε} k^{1/4 - ε} \ll_ε |ϕ|_\infty \ll_ε C^{1/8 + ε} k^{1/4+ε}$. We prove also that these forms give analytic test vectors for the QUE period, thereby demonstrating the equivalence between the strong QUE and the subconvexity problems for this class of vectors. This finding contrasts the known failure of this equivalence for newforms of powerful level.

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arXiv2016

Test vectors for Waldspurger's period integral and application to the mass equidistribution on nonsplit torus

In this paper we provide local test vector for Waldspurger's period integral, when the level of the representation $π_v$ is sufficiently large compared to the level of the character $Ω_v$ over quadratic extension, while allowing joint ramifications. The test vectors we shall use are variants of classical newforms, and the size of the resulting local integral is asymptotically the inverse of convexity bound for $L(Π\otimesΩ,1/2)$. Such test vectors are used to recover Gross-Prasad type test vectors. We also get vanishing result for local integral when using other test vectors. This phenomenon is used to prove the mass equidistribution of cuspidal newforms on nonsplit torus in depth aspect.

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arXiv2014

The subconvexity bound for triple product L-function in level aspect

In this paper we generalized Venkatesh and Woodbury's work on the subconvexity bound of triple product L-function in level aspect, allowing joint ramifications, higher ramifications, general unitary central characters and general special values of local epsilon factors. In particular we derived a nice general formula for the local integrals whenever one of the representations has sufficiently higher level than the other two.

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arXiv2013

Cuspidal part of an Eisenstein series restricted to an index 2 subfield

Let $\mathbb{E}$ be a quadratic extension of a number field $\mathbb{F}$. Let $E(g, s)$ be an Eisenstein series on $GL_2(\mathbb{E})$, and let $F$ be a cuspidal automorphic form on $GL_2(\mathbb{F})$. We will consider in this paper the following automorphic integral: $$\int_{Z_{A}GL_{2}(\mathbb{F})\backslash GL_{2}(\mathbb{A}_{\mathbb{F}})} F(g)E(g,s) dg.$$ This is in some sense the complementary case to the well-known Rankin-Selberg integral and the triple product formula. We will approach this integral by Waldspurger's formula. We will discuss when the integral is automatically zero, and otherwise the L-function it represents. We will calculate local integrals at some ramified places, where the level of the ramification can be arbitrarily large.

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