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Zhining Wei

Publications and source records attributed to Zhining Wei.

12 recordsLinked to original sources
arXiv2026

First Moment of derivatives of $L$-functions in a nonlinear family

In this paper, we prove an asymptotic formula for the moment of the first order derivative of modular $L$-functions at the center of the critical strip, weighted by generalized divisor functions formed by primitive quadratic characters. Such moments were previously studied by Munshi, which naturally arise in the study of elliptic fibration. To the best of our knowledge, such asymptotic formulae have only been proven in the setting of higher-order derivatives, or under the specialization to dihedral forms.

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arXiv2025

Second Moment of Central Values of Half-Integral Weight Modular Forms and Subconvexity

We let $f$ be a half-integral weight modular form of weight $\kappa>4$ on $\Gamma_0(4)$ that is an eigenfunction of all Hecke operators $T_n$, so that $T_nf = \Lambda_f(n)n^{\frac{\kappa-1}{2}}f$. Let $\|f\|$ denote the Petersson norm of $f$. We study a weighted second moment of the central value of the $L$-function associated to $f$ over an orthogonal basis $H_\kappa(4)$ of $S_{\kappa}(\Gamma_0(4))$. This corresponds to studying the following sum: $$\sum_{f\in H_\kappa(4)}\frac{\Lambda_f(n)\vert L(1/2,f)\vert^2}{\|f\|^2}.$$ Using the relative trace formula, we obtain an asymptotic formula for the second moment. We then use the method of amplification to get the subconvexity bound $$L(1/2,f)\ll_{\varepsilon} (\kappa^2)^{\frac{1}{4}-\frac{1}{40}+\varepsilon}.$$ This is the first subconvexity result for half-integral weight modular forms in the weight aspect. We also apply our second moment result to get a quantitative simultaneous non-vanishing result for central values of $L$-functions.

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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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arXiv2025

Effective open image theorem and a Linnik type problem for elliptic curves

We study an effective open image theorem for families of elliptic curves and products of elliptic curves ordered by conductor. Unconditionally, we prove that for $100\%$ of pairs of elliptic curves, the largest prime $\ell$, for which the associated mod $\ell$ Galois representation fails to be surjective, is small. Additionally, for semistable families, our bound on $\ell$ is comparable to the result obtained under the Generalized Riemann Hypothesis. We reduce the problem to a Linnik type problem for modular forms and apply the zero density estimates. This method, together with an analysis of the local Galois representations of elliptic curves, allows us to show similar results for single elliptic curves.

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arXiv2025

A note on a classical relative trace formula

In this note, we derive a relative trace formula (RTF) using classical methods. We obtain a closed formula for the second moment of the central values of holomorphic cusp forms, a result originally established in Kuznetsov's preprint.

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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

Some remarks on strong multiplicity one for paramodular forms

We establish several refined strong multiplicity one results for paramodular cusp forms by using automorphic and Galois-theoretic methods. We also give an application to distinguishing eigenforms by the twisted central values of the spinor $L$-functions, which is based on a result in Radziwi{\l}{\l} and Yang 2023 (arXiv:2304.09171).

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arXiv2023

On Möbius functions from automorphic forms and a generalized Sarnak's conjecture

In this paper, we consider Möbius functions associated with two types of $L$-functions: Rankin-Selberg $L$-functions of symmetric powers of distinct holomorphic cusp forms and $L$-functions of Maass cusp forms. We show that these Möbius functions are weakly orthogonal to bounded sequences. As a direct corollary, a generalized Sarnak's conjecture holds for these two types of Möbius functions.

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arXiv2023

Generalizations of the Erd\H{o}s-Kac Theorem and the Prime Number Theorem

In this paper, we study the linear independence between the distribution of the number of prime factors of integers and that of the largest prime factors of integers. Respectively, under a restriction on the largest prime factors of integers, we will refine the Erd\H{o}s-Kac Theorem and Loyd's recent result on Bergelson and Richter's dynamical generalizations of the Prime Number Theorem. At the end, we will show that the analogue of these results holds with respect to the Erd\H{o}s-Pomerance Theorem as well.

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arXiv2022

On distinguishing Siegel cusp forms of degree two

In this work, we establish several results on distinguishing Siegel cusp forms of degree two. In particular, a Hecke eigenform of level one can be determined by its second Hecke eigenvalue under a certain assumption. Moreover, we can distinguish two Hecke eigenforms of level one by using $L$-functions.

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