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Liyang Yang

Publications and source records attributed to Liyang Yang.

At least 19 recordsLinked to original sources
arXiv2026

Fourier Spectral Reciprocity and Canonical Hecke $L$-Functions

We prove a Fourier-type toric spectral reciprocity formula over number fields, relating Tate integrals over a quadratic extension to a dual family of Hecke periods over the base field. The formula keeps the spectral parameter free and allows test functions adapted to arithmetic applications. We apply this reciprocity formula to canonical Hecke $L$-functions over CM fields and obtain explicit first moment formulas, including both central values and central derivatives. These formulas are uniform in the weight and the twisting conductor, and require no Heegner-type splitting hypothesis. Together with subconvexity bounds, they yield quantitative nonvanishing results for canonical Hecke $L$-functions and their derivatives. In weight one, this gives a rank-one analogue of the arithmetic applications of Masri--Yang, yielding quantitative Mordell--Weil rank-one results for quadratic twists of the associated CM abelian varieties.

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arXiv2026

Weyl Subconvexity for $\mathrm{GL}_2$ with Simple Supercuspidal Ramification

We establish Weyl-type subconvexity bounds in the level aspect for cuspidal automorphic representations of $\mathrm{GL}_2/F$ with simple supercuspidal ramification at a prime ideal $\mathfrak{q}$. More precisely, for the family $\mathcal{F}_t^\zeta(\mathfrak{q}^3;\omega)$ consisting of representations of conductor $\mathfrak{q}^3$, central character $\omega$, and prescribed simple supercuspidal local component, we prove the fourth moment estimate \begin{align*} \sum_{\substack{\pi \in \mathcal{F}_{t}^{\zeta}(\mathfrak{q}^3;\omega) \\ C_v(\pi) \leq \mathbf{C}_v,\ v \mid \infty}} |L(1/2,\pi)|^4 \ll_{F,\varepsilon} \mathbf C_\infty^{1+\varepsilon} N_F(\mathfrak{q})^{2+\varepsilon}. \end{align*} As a consequence, we deduce the Weyl-type bound \begin{align*} L(1/2,\pi) \ll_{F,\varepsilon} C_{\infty}(\pi)^{1/4+\varepsilon} C_{\mathrm{fin}}(\pi)^{1/6+\varepsilon}. \end{align*} In particular, this bound applies to a genuinely non-self-dual family of odd conductor exponent, beyond the reach of the cubic moment method.

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arXiv2026

Symmetric Spectral Reciprocity for $\mathrm{GL}(2)$ and Uniform Subconvexity

We construct a new analytic regularization of the Petersson norm identity for Eisenstein series on $\mathrm{GL}_2$ over a number field $F$, and derive from it an explicit symmetric spectral reciprocity formula for twisted fourth moments of $\mathrm{GL}_2$ $L$-functions, reflecting the intrinsic rank decomposition $4=2+2$. Independently, we identify a square-level phenomenon arising from amplification, whereby the dual spectral family acquires square-level conductors. This additional arithmetic rigidity permits a refined analysis within the relative trace formula and leads to refined hybrid subconvexity bounds for twisted $L$-functions. As a consequence, we obtain new uniform subconvexity bounds for $\mathrm{GL}_2/F$; in particular, \begin{align*} L(1/2,\pi)\ll C(\pi)^{\frac14-\frac{1}{120}+\varepsilon} \end{align*} for every unitary cuspidal representation $\pi$ of $\mathrm{GL}_2/F$. We also obtain refined bounds for certain Artin $L$-functions and applications to class group arithmetic.

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arXiv2026

Rankin--Selberg Subconvexity via Spectral Reciprocity

We establish explicit subconvex bounds for central values of Rankin--Selberg $L$-functions $L(1/2,\pi\times\pi')$ associated with pairs of unitary cuspidal automorphic representations of $\mathrm{GL}_2$ over a number field. Building on the spectral reciprocity framework of Michel and Venkatesh, we develop a refined, fully explicit form of spectral reciprocity that allows for precise control of conductors and local test vectors. As a consequence, we obtain an explicit subconvex bound, which, even over $F=\mathbb{Q}$, improves all previously known results. We further apply these bounds to several arithmetic problems. These include effective equidistribution of CM suborbits on quaternionic Shimura varieties, quantitative equidistribution of totally geodesic submanifolds, a uniform quantitative form of dihedral quantum unique ergodicity over number fields, and an application to distinguishing cuspidal automorphic representations.

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arXiv2025

Spectral Reciprocity: A Fourier--Analytic Approach

We develop a Fourier--analytic framework for establishing spectral reciprocity formulas linking $\mathrm{GL}_3$ and $\mathrm{GL}_2$ automorphic spectra over number fields. The method applies uniformly to cuspidal and non-cuspidal $\mathrm{GL}_3$ representations and treats Motohashi-type and Blomer--Khan-type reciprocities in a parallel manner, revealing intrinsic connections between them and extending each to new settings. We also obtain explicit weight transforms in the analytic newvector and spherical cases. Applications include first-moment estimates for $\mathrm{GL}_3\times\mathrm{GL}_2$ $L$-functions over number fields, an explicit twisted fourth moment for $\mathrm{GL}_2$ $L$-functions over totally real fields, a sharp upper bound for the fifth moment, subconvexity for triple product $L$-functions, and new simultaneous nonvanishing results.

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arXiv2025

Low-lying zeros of Hilbert modular $L$-functions weighted by powers of central $L$-values

Let $\mathcal{F}(\textbf{k},\mathfrak{q})$ be the set of primitive Hilbert modular forms of weight $\textbf{k}$ and prime level $\mathfrak{q}$, with trivial central character. We study the one-level density of low-lying zeros of $L(s,\pi)$ weighted by powers of central $L$-values $L(1/2,\pi)^r$, where $\pi$ runs through $\mathcal{F}(\textbf{k},\mathfrak{q})$. For $r=1,2,3$, we show that the resulting distributions $W_r$ match with predictions from Random Matrix Theory. For general $r \geq 1$, we also formulate a conjectural formula for $W_r$ based on the ``recipe'' method.

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arXiv2024

Relative Trace Formula And Simultaneous Nonvanishing for GL_3 x GL_2 and GL_3 x GL_1 L-functions

Fix a Dirichlet character $\chi$ and a cuspidal GL$(2)$ eigenform $\phi$ with relatively prime conductors. Then we show that there are infinitely many cusp forms $\pi$ on GL$(3)$ such that $L(1/2, \pi \times \chi)$ and $L(1/2, \pi \times \phi)$ are simultaneously non-zero. We achieve this by use of Jacquet's Relative Trace Formula. We derive an expression of the average over the GL$(3)$ cuspidal spectrum as a sum of a non-zero main term and two subsidiary terms which are forced to be zero for large enough level by use of a suitable test function. This modest article is dedicated to the memory of Harish Chandra, on the occasion of his hundredth birthday.

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arXiv2024

Relative Trace Formula and Uniform non-vanishing of Central $L$-values of Hilbert Modular Forms

Let $\mathcal{F}(\mathbf{k},\mathfrak{q})$ be the set of normalized Hilbert newforms of weight $\mathbf{k}$ and prime level $\mathfrak{q}$. In this paper, utilizing regularized relative trace formulas, we establish a positive proportion of $\#\{\pi\in\mathcal{F}(\mathbf{k},\mathfrak{q}):L(1/2,\pi)\neq 0\}$ as $\#\mathcal{F}(\mathbf{k},\mathfrak{q})\to+\infty$. Moreover, our result matches the strength of the best known results in both the level and weight aspects.

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arXiv2023

Bessel Periods on $U(2,1) \times U(1,1)$, Relative Trace Formula and Non-Vanishing of Central $L$-values

In this paper we calculate the asymptotics of the second moment of the Bessel periods associated to certain holomorphic cuspidal representations $(\pi, \pi')$ of $U(2,1) \times U(1,1)$ of regular infinity type (averaged over $\pi$). Using these, we obtain quantitative non-vanishing results for the Rankin-Selberg central $L$-values $L(1/2, \pi \times \pi')$, which are of degree twelve over $\mathbb{Q}$, with concomitant difficulty in applying standard methods, especially since we are in a `conductor dropping' situation. We use the relative trace formula, and the orbital integrals are evaluated rather than compared with others. Besides their intrinsic interest, non-vanishing of these critical values also lead, by known results, to deducing certain associated Selmer groups have rank zero.

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arXiv2023

Relative Trace Formula, Subconvexity and Quantitative Nonvanishing of Rankin-Selberg $L$-functions for $\mathrm{GL}(n+1)\times\mathrm{GL}(n)$

Let $\pi'$ be a fixed unitary cuspidal representation of $\mathrm{GL}(n)/\mathbb{Q}.$ We establish a subconvex bound in the $t$-aspect $$ L(1/2+it,\pi\times\pi')\ll_{\pi,\pi',\varepsilon}(1+|t|)^{\frac{n(n+1)}{4}-\frac{1}{4\cdot (4n^2+2n-1)}+\varepsilon}, $$ for any unitary pure isobaric automorphic representation $\pi$ of $\mathrm{GL}(n+1)/\mathbb{Q}.$ Moreover, the bound improves in the standard $L$-function case $$ L(1/2+it, \pi')\ll_{\pi',\varepsilon}(1+|t|)^{\frac{n}{4}-\frac{1}{4(n+1)(4n-1)}+\varepsilon}. $$ We also prove an explicit lower bound for nonvanishing of central $L$-values $$ \sum_{\pi\in\mathcal{A}_0}\textbf{1}_{L(1/2,\pi\times\pi')\neq 0}\gg_{\varepsilon}|\mathcal{A}_0|^{\frac{1}{n(n+1)(4n^2+2n-1)}-\varepsilon}, $$ for a suitable finite family $\mathcal{A}_0$ of unitary cuspidal representations of $\mathrm{GL}(n+1)/\mathbb{Q}.$ More generally, we address the spectral side subconvexity in the case of uniform parameter growth, and a quantitative form of simultaneous nonvanishing of central $L$-values for $\mathrm{GL}(n+1)\times\mathrm{GL}(n)$ (over $\mathbb{Q}$) in both level and eigenvalue aspects. Among other ingredients, our proofs employ a new relative trace formula in conjunction with P. Nelson's construction of archimedean test functions in \cite{Nel21} and volume estimates in \cite{Nel20}.

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arXiv2023

Average of Central L-values for GL(2)$\times$GL(1), Hybrid Subconvexity, and Simultaneous Nonvanishing

We employ a regularized relative trace formula to establish a second moment estimate for twisted $L$-functions across all aspects over a number field. Our results yield hybrid subconvex bounds for both Hecke $L$-functions and twisted $L$-functions, comparable to the Weyl bound in suitable ranges. Moreover, we present an application of our results to address the simultaneous nonvanishing problem.

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arXiv2023

Relative Trace Formula and Twisted $L$-functions: the Burgess Bound

Let $F$ be a number field, $π$ either a unitary cuspidal automorphic representation of $\mathrm{GL}(2)/F$ or a unitary Eisenstein series, and $χ$ a unitary Hecke character of analytic conductor $C(χ).$ We develop a regularized relative trace formula to prove a refined hybrid subconvex bound for $L(1/2,π\timesχ).$ In particular, we obtain the Burgess subconvex bound \begin{align*} L(1/2,π\timesχ)\ll_{π,F,\varepsilon}C(χ)^{\frac{1}{2}-\frac{1}{8}+\varepsilon}, \end{align*} where the implied constant depends on $π,$ $F$ and $\varepsilon.$

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arXiv2023

Non-vanishing of twists of $\text{GL}_4(\mathbb{A}_{\mathbb{Q}})$ $L$-functions

Let $π$ be a unitary cuspidal automorphic representation of $\text{GL}_{4}(\mathbb{A}_{\mathbb{Q}})$. Let $f \geq 1$ be given. We show that there exists infinitely many primitive even (resp. odd) Dirichlet characters $χ$ with conductor co-prime to $f$ such that $L(s, π\otimes χ)$ is non-vanishing at the central point. Our result has applications for the construction of $p$-adic $L$-functions for $\text{GSp}_{4}$ following Loeffler-Pilloni-Skinner-Zerbes, the Bloch-Kato conjecture and the Birch-Swinnerton-Dyer conjecture for abelian surfaces following Loeffler-Zerbes, strong multiplicity one results for paramodular cuspidal representations of $\text{GSp}_{4}(\mathbb{A}_{\mathbb{Q}})$ and the rationality of the central values of $\text{GSp}_{4}(\mathbb{A}_{\mathbb{Q}})$ $L$-functions in the remaining non-regular weight case.

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arXiv2023

Relative Trace Formula and $L$-functions for $\mathrm{GL}(n+1)\times \mathrm{GL}(n)$

We establish a relative trace formula on $\mathrm{GL}(n+1)$ weighted by cusp forms on $\mathrm{GL}(n)$ over number fields. The spectral side is a weighted average of Rankin-Selberg $L$-functions for $\mathrm{GL}(n+1)\times\mathrm{GL}(n)$ over the \textit{full spectrum}, and the geometric side consists of Rankin-Selberg $L$-functions for $\mathrm{GL}(n)\times\mathrm{GL}(n),$ and certain explicit holomorphic functions. The formula yields new results towards central $L$-values for $\mathrm{GL}(n+1)\times\mathrm{GL}(n)$ (over number fields): the second moment evaluation, and simultaneous nonvnaishing in the level aspect.

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arXiv2020

On Exceptional Maass Forms

We prove certain relations between Satake parameters of cuspidal representations of $\GL_2(\mathbb{A}_{\mathbb{Q}})$ at finite and archimedean places. Consequently, we show that the Ramanujan-Petersson conjecture at a fixed prime $p\nmid N$ for \textit{non-exceptional} Maass forms of level $N$ implies the conjecture at $p$ for \textit{all} Maass forms of level $N$ and the Selberg's $1/4$-eigenvalue conjecture simultaneously. As an application, we improve Kim and Sarnak's $7/64$-bound towards the Satake parameters at all $p\nmid N$ for exceptional Maass forms.

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arXiv2020

A Coarse Jacquet-Zagier Trace Formula for GL($n$) with Applications

In this paper we establish a coarse Jacquet-Zagier trace identity for GL$(n).$ We prove the absolute convergence when $\Re(s)>1$ and $0<\Re(s)<1;$ and obtain holomorphic continuation under almost all character twist. Moreover, as an application, we prove that holomorphy of certain adjoint $L$-functions for GL$(n)$ implies Dedekind conjecture of degree $n$. Some nonvanishing results are also discussed.

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arXiv2020

A constraint for twist equivalence of cusp forms on GL$(n)$

This Note answers, and generalizes, a question of Kaisa Matomäki. We show that give two cuspidal automorphic representations $π_1$ and $π_2$ of $GL_n$ over a number field $F$ of respective conductors $N_1,$ $N_2,$ every character $χ$ such that $π_1\otimesχ\simeqπ_2$ of conductor $Q,$ satisfies the bound: $Q^n\mid N_1N_2.$ If at every finite place $v,$ $π_{1,v}$ is a discrete series whenever it is ramified, then $Q^n$ divides the least common multiple $[N_1, N_2].$

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