About

Wenzhi Luo is a professor in the Department of Mathematics at The Ohio State University. His areas of expertise include Number Theory, with a specific focus on the Analytic and Arithmetic Theory of Automorphic Forms. He earned his Ph.D. from Rutgers University in 1993. His research involves the study of automorphic forms and their arithmetic properties, contributing to the broader understanding of number theory. As a faculty member, he is involved in teaching and mentoring within the department, supporting the academic growth of students and advancing mathematical research in his field.

Research topics

• Mathematics
• Pure mathematics
• Mathematical analysis
• Combinatorics
• Arithmetic

Selected publications

• Separated Pairs of Submodules in Hilbert $C^*$-modules

  arXiv (Cornell University) · 2024-05-08

  preprintOpen access

  We introduce the notion of the separated pair of closed submodules in the setting of Hilbert $C^*$-modules. We demonstrate that even in the case of Hilbert spaces this concept has several nice characterizations enriching the theory of separated pairs of subspaces in Hilbert spaces. Let $\mathscr H$ and $\mathscr K$ be orthogonally complemented closed submodules of a Hilbert $C^*$-module $\mathscr E$. We establish that $ (\mathscr H,\mathscr K)$ is a separated pair in $\mathscr{E}$ if and only if there are idempotents $Π_1$ and $Π_2$ such that $Π_1Π_2=Π_2Π_1=0$ and $\mathscr R(Π_1)=\mathscr H$ and $\mathscr R(Π_2)=\mathscr K$. We show that $\mathscr R(Π_1+λΠ_2)$ is closed for each $λ\in \mathbb{C}$ if and only if $\mathscr R(Π_1+Π_2)$ is closed. We use the localization of Hilbert $C^*$-modules to define the angle between closed submodules. We prove that if $(\mathscr H^\perp,\mathscr K^\perp)$ is concordant, then $(\mathscr H^{\perp\perp},\mathscr K^{\perp\perp})$ is a separated pair if the cosine of this angle is less than one. We also present some surprising examples to illustrate our results.

  PublisherOA PDFDOI

• Moments of the central <i>L</i>-values of the Asai lifts

  Canadian Mathematical Bulletin · 2024-03-04

  article1st authorCorresponding

  Abstract We study some analytic properties of the Asai lifts associated with cuspidal Hilbert modular forms, and prove sharp bounds for the second moment of their central L -values.

  PublisherDOI

• Non-existence of Siegel zeros for cuspidal functorial products on 𝐺𝐿(2)×𝐺𝐿(3)

  Proceedings of the American Mathematical Society · 2022-11-16 · 2 citations

  article1st authorCorresponding

  Let <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="pi 1"> <mml:semantics> <mml:msub> <mml:mi> π </mml:mi> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mn>1</mml:mn> </mml:mrow> </mml:msub> <mml:annotation encoding="application/x-tex">\pi _{1}</mml:annotation> </mml:semantics> </mml:math> </inline-formula> and <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="pi 2"> <mml:semantics> <mml:msub> <mml:mi> π </mml:mi> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mn>2</mml:mn> </mml:mrow> </mml:msub> <mml:annotation encoding="application/x-tex">\pi _{2}</mml:annotation> </mml:semantics> </mml:math> </inline-formula> be two cuspidal automorphic forms over a number field <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="bold upper F"> <mml:semantics> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mi mathvariant="bold">F</mml:mi> </mml:mrow> <mml:annotation encoding="application/x-tex">\mathbf {F}</mml:annotation> </mml:semantics> </mml:math> </inline-formula> on <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper G upper L left-parenthesis 2 right-parenthesis"> <mml:semantics> <mml:mrow> <mml:mi>G</mml:mi> <mml:mi>L</mml:mi> <mml:mo stretchy="false">(</mml:mo> <mml:mn>2</mml:mn> <mml:mo stretchy="false">)</mml:mo> </mml:mrow> <mml:annotation encoding="application/x-tex">GL(2)</mml:annotation> </mml:semantics> </mml:math> </inline-formula> and <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper G upper L left-parenthesis 3 right-parenthesis"> <mml:semantics> <mml:mrow> <mml:mi>G</mml:mi> <mml:mi>L</mml:mi> <mml:mo stretchy="false">(</mml:mo> <mml:mn>3</mml:mn> <mml:mo stretchy="false">)</mml:mo> </mml:mrow> <mml:annotation encoding="application/x-tex">GL(3)</mml:annotation> </mml:semantics> </mml:math> </inline-formula> respectively. In this work, we prove the non-existence of Siegel zeros for the automorphic <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper L"> <mml:semantics> <mml:mi>L</mml:mi> <mml:annotation encoding="application/x-tex">L</mml:annotation> </mml:semantics> </mml:math> </inline-formula> -function associated to a cuspidal functorial product <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="pi 1 times pi 2"> <mml:semantics> <mml:mrow> <mml:msub> <mml:mi> π </mml:mi> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mn>1</mml:mn> </mml:mrow> </mml:msub> <mml:mo> × </mml:mo> <mml:msub> <mml:mi> π </mml:mi> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mn>2</mml:mn> </mml:mrow> </mml:msub> </mml:mrow> <mml:annotation encoding="application/x-tex">\pi _{1} \times \pi _{2}</mml:annotation> </mml:semantics> </mml:math> </inline-formula> .

  PublisherDOI

• Sums of k-th powers and the Whittaker–Fourier coefficients of automorphic forms

  The Ramanujan Journal · 2021-02-09 · 2 citations

  article1st authorCorresponding
  PublisherDOI

• On the Hecke eigenvalues of Maass forms

  American Journal of Mathematics · 2019-01-01 · 3 citations

  articleOpen access1st authorCorresponding

  Let denote a primitive Hecke-Maass cusp form for o (N) with the Laplacian eigenvalue = 1/4 + t 2 . In this work we show that there exists a prime p such that p N, | p | = | p | = 1, and p (N(1 + |t |)) c , where p , p are the Satake parameters of at p, and c is an absolute constant with 0 < c < 1. In fact, c can be taken as 0.27332. In addition, we prove that the natural density of such primes p (p N and | p | = | p | = 1) is at least 34/35.

  PublisherOA PDFDOI

• On simultaneous nonvanishing of the central 𝐿-values

  Proceedings of the American Mathematical Society · 2016-11-28 · 4 citations

  articleOpen access1st authorCorresponding

  In this note we derive a new quantitative result on the simultaneous nonvanishing of the central <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper L"> <mml:semantics> <mml:mi>L</mml:mi> <mml:annotation encoding="application/x-tex">L</mml:annotation> </mml:semantics> </mml:math> </inline-formula> -values twisted by quadratic characters, for pairs of holomorphic cuspidal Hecke eigenforms with large weight <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="2 k"> <mml:semantics> <mml:mrow> <mml:mn>2</mml:mn> <mml:mi>k</mml:mi> </mml:mrow> <mml:annotation encoding="application/x-tex">2k</mml:annotation> </mml:semantics> </mml:math> </inline-formula> .

  PublisherOA PDFDOI

• Nonvanishing of the central L-values with large weight

  Advances in Mathematics · 2015-08-24 · 21 citations

  article1st authorCorresponding
  PublisherDOI

• On the Hecke Eigenvalues of Maass Forms

  arXiv (Cornell University) · 2014-05-20

  preprintOpen access1st authorCorresponding

  Let $ϕ$ denote a primitive Hecke-Maass cusp form for $Γ_o(N)$ with the Laplacian eigenvalue $λ_ϕ=1/4+t_ϕ^2$. In this work we show that there exists a prime $p$ such that $p\nmid N$, $|α_{p}|=|β_{p}| = 1$, and $p\ll(N(1+|t_ϕ|))^c$, where $α_{p},\;β_{p}$ are the Satake parameters of $ϕ$ at $p$, and $c$ is an absolute constant with $0

  PublisherOA PDFDOI

• L 4-Norms of the Dihedral Maass Forms

  International Mathematics Research Notices · 2013-01-17 · 15 citations

  article1st authorCorresponding

  In this paper, we prove the optimal bound for L4-norms of the dihedral Maass forms associated to Hecke’s grossencharacters of a fixed real quadratic field, as their Laplacian eigenvalues tend to infinity.

  PublisherDOI

• Central values of the symmetric square $L$-functions

  Proceedings of the American Mathematical Society · 2012-01-31 · 8 citations

  articleOpen access1st authorCorresponding

  We establish a sharp bound for the square mean of the central values of the symmetric square $L$-functions associated to holomorphic cusp forms of level $1$, as the weight $k$ varies in the short interval $[K,\; K + K^{1/2 + \epsilon }]$.

  PublisherOA PDFDOI

Recent grants

• Equidistribution of Hecke Eigenforms and Related Problems

  NSF · $195k · 2009–2013

• Collaborative Research: Arithmetic and Equidistribution on Homogeneous Spaces

  NSF · $374k · 2006–2011

• Analytic Aspects of L-functions and Related Problems

  NSF · $196k · 2012–2017

Frequent coauthors

• Peter Sarnak

  Institute for Advanced Study

  12 shared

• Henryk Iwaniec

  4 shared

• Zeév Rudnick

  Tel Aviv University

  4 shared

• R. Eskandari

  2 shared

• Zhou Fan

  2 shared

• Dinakar Ramakrishnan

  2 shared

• Mohammad Sal Moslehian

  2 shared

• Zhang Hai

  2 shared

Awards & honors

• Graduate Teaching Awards