Searcharxiv⌕ Search

arXiv subjects

Qinghua Pi

Publications and source records attributed to Qinghua Pi.

5 recordsLinked to original sources
arXiv2022

Bias of Root Numbers for Hilbert Newforms of Cubic Level

We give a general formula of the bias of root numbers for Hilbert modular newforms of cubic level. Explicit calculation is given when the base field is $\mathbb{Q}, \mathbb{Q}(\sqrt{2}), \mathbb{Q}(\sqrt{5})$ and the level is the cube of certain rational integers. This complements a previous result of the second author and extends the bias phenomenon to the number fields. Our method is based on Jacquet-Zagier's trace formula, and the explicit calculation works generally for all real quadratic fields of narrow class number one and for rational cubic levels.

math.NT↗
arXiv2021

Bias of Root Numbers for Modular Newforms of Cubic Level

Let $H^{\pm}_{2k} (N^3)$ denote the set of modular newforms of cubic level $N^3$, weight $2 k$, and root number $\pm 1$. For $N > 1$ squarefree and $k>1$, we use an analytic method to establish neat and explicit formulas for the difference $|H^{+}_{2k} (N^3)| - |H^{-}_{2k} (N^3)|$ as a multiple of the product of $φ(N)$ and the class number of $\mathbb{Q}(\sqrt{- N})$. In particular, the formulas exhibit a strict bias towards the root number $+1$. Our main tool is a root-number weighted simple Petersson formula for such newforms.

math.NT↗
arXiv2020

Simple Fourier Trace Formulas of Cubic Level and Applications

With the method of the relative trace formula and the classification of simple supercuspidal representations, we establish some Fourier trace formulas for automorphic forms on $PGL(2)$ of cubic level. As applications, we obtain a non-vanishing result for central $L$-values of holomorphic newforms and a weighted Weyl's law for Maass newforms.

math.NT↗
arXiv2018

Central Values of $GL(2)\times GL(3)$ Rankin-Selberg $L$-functions with Applications

Let $f$ be a normalized holomorphic cusp form for $SL_2(\mathbb{Z})$ of weight $k$ with $k\equiv0\bmod 4$. By the Kuznetsov trace formula for $GL_3(\mathbb R)$, we obtain the first moment of central values of $L(s,f\otimes ϕ)$, where $ϕ$ varies over Hecke-Maass cusp forms for $SL_3(\mathbb Z)$. As an application, we obtain a non-vanishing result for $L(1/2,f\otimesϕ)$ and show that such $f$ is determined by $\{L(1/2,f\otimesϕ)\}$ as $ϕ$ varies.

math.NT↗