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Wonwoong Lee

Publications and source records attributed to Wonwoong Lee.

8 recordsLinked to original sources
arXiv2026

Divisor moments of polynomials in Fourier coefficients of modular forms

We study higher moments of the divisor function evaluated at polynomial expressions in the Fourier coefficients of a non-CM newform. The logarithmic exponent appearing in our estimates depends only on the number of irreducible factors of the polynomial and remains unchanged under a Sato--Tate restriction. The proof combines an effective Chebotarev theorem, or an effective Chebotarev--Sato--Tate theorem, with the arithmetic of joint cycle types and a mean value estimation for multivariable multiplicative functions with Frobenian support.

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arXiv2026

Rectangular representations and $λ$-independence of algebraic monodromy groups

Let $\mathfrak g$ be a complex semisimple Lie algebra. We define what it means for a finite dimensional representation of $\mathfrak g$ to be rectangular and completely classify faithful rectangular representations. As an application, we obtain new $λ$-independence results on the algebraic monodromy groups of compatible systems of $λ$-adic Galois representations of number fields.

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arXiv2026

Divisor problems for restricted Fourier coefficients of modular forms

Let $d(n)$ be the number of divisors of $n$. We investigate the average value of $d(a_f(p))^r$ for $r$ a positive integer and $a_f(p)$ the $p$-th Fourier coefficient of a cuspidal eigenform $f$ having integral Fourier coefficients, where $p$ is a prime subject to a constraint on the angle associated with the normalized Fourier coefficient.

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arXiv2025

On the real zeros of depth 1 quasimodular forms

We discuss the critical points of modular forms, or more generally the zeros of quasimodular forms of depth $1$ for $\mathrm{PSL}_2(\mathbb Z)$. In particular, we consider the derivatives of the unique weight $k$ modular forms $f_k$ with the maximal number of consecutive zero Fourier coefficients following the constant $1$. Our main results state that (1) every zero of a depth $1$ quasimodular form near the derivative of the Eisenstein series in the standard fundamental domain lies on the geodesic segment $\{z \in \mathbb H: \Re(z)=1/2\}$, and (2) more than half of zeros of $f_k$ in the standard fundamental domain lie on the geodesic segment $\{z \in \mathbb H: \Re(z)=1/2\}$ for large enough $k$ with $k\equiv 0 \pmod{12}$.

math.NT↗
arXiv2025

Monodromy and irreducibility of type $A_1$ automorphic Galois representations

Let $K$ be a totally real field and $π$ be a regular algebraic polarized cuspidal automorphic representation of $\mathrm{GL}_n(\mathbb A_K)$. Let $\{ρ_{π,λ}:\mathrm{Gal}_K\to\mathrm{GL}_n(\overline E_λ)\}_λ$ be the compatible system of Galois representations attached to $π$ and denote by $\mathbf G_λ$ the algebraic monodromy group of $ρ_{π,λ}$. Suppose there exists $λ_0$ such that (a) $ρ_{π,λ_0}$ is irreducible; (b) $\mathbf G_{λ_0}$ is connected and of type $A_1$; and (c) the tautological representation of $\mathbf G_{λ_0}$ is of a certain type. We prove that $\bullet$ $\mathbf G_{λ,\mathbb C}\subset\mathrm{GL}_{n, \mathbb C}$ is independent of $λ$; $\bullet$ $ρ_{π,λ}$ is irreducible for all $λ$, and residually irreducible for almost all $λ$. Moreover, if $K=\mathbb Q$ or $n$ is odd, we prove that the same conclusions hold without the assumption that $π$ is polarized. We also prove that if $K=\mathbb Q$, then the compatible system $\{ρ_{π,λ}\}_λ$ is constructed from certain two-dimensional modular compatible systems up to twist.

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arXiv2022

Non-holomorphic Eisenstein series for certain Fuchsian groups and class numbers

We study certain types of Fuchsian groups of the first kind denoted by $R(N)$, which coincide with the Fricke groups or the arithmetic Hecke triangle groups of low levels. We find all elliptic points and cusps of $R(p)$ for a prime $p$, and prove that there is a one-to-one correspondence between the set of equivalence classes of elliptic points of $R(p)$ and the imaginary quadratic class group. We also find the explicit formula of the Fourier expansion of the non-holomorphic Eisenstein series for $R(N)$ and study their analytic properties. These non-holomorphic Eisenstein series together with cusp forms provide a basis for the space of polyharmonic Maass forms for $R(N)$.

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arXiv2022

On the common zeros of quasi-modular forms for $Γ_0^+(N)$ of level $N=1,2,3$

In this paper, we study common zeros of the iterated derivatives of the Eisenstein series for $Γ_0^+(N)$ of level $N=1,2$ and $3$, which are quasi-modular forms. More precisely, we investigate the common zeros of quasi-modular forms, and prove that all the zeros of the iterated derivatives of the Eisenstein series $\frac{d^m E_k^{(N)}(τ)}{dτ^m}$ of weight $k=2,4,6$ for $Γ_0^+(N)$ of level $N=2,3$ are simple by generalizaing the results of Meher \cite{MEH} and Gun and Oesterlé \cite{SJ20} for SL$_2(\mathbb{Z})$.

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