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Soon-Yi Kang

Publications and source records attributed to Soon-Yi Kang.

At least 19 recordsLinked to original sources
arXiv2026

Moments of partition statistics, Bell polynomials and Eisenstein-type series

We develop a systematic method to express generating functions for moments of combinatorial statistics in terms of partition traces. We employ an algebraic approach based on the complete Bell polynomials and their inversion formula, alongside an analytic approach via Fa\`a di Bruno's formula. Our approach can be applied to a wide class of combinatorial statistics, such as the largest part of an integer partition, the partition crank and rank, and the unimodal sequence rank.

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arXiv2026

Hecke Equivariance of Divisor Lifting with respect to Sesquiharmonic Maass Forms

We investigate the properties of Hecke operator for sesquiharmonic Maass forms. We begin by proving Hecke equivariance of the divisor lifting with respect to sesquiharmonic Mass functions, which maps an integral weight meromorphic modular form to the holomorphic part of the Fourier expansion of a weight 2 sesquiharmonic Maass form. Using this Hecke equivariance, we show that the sesquiharmonic Maass functions, whose images under the hyperbolic Laplace operator are the Faber polynomials $J_n$ of the $j$-function, form a Hecke system analogous to $J_n$. By combining the Hecke equivariance of the divisor lifting with that of the Borcherds isomorphism, we extend Matsusaka's finding on the twisted traces of sesquiharmonic Maass functions.

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arXiv2025

On the Alexander polynomials of modular knots

Closed geodesics associated with indefinite binary quadratic forms, or equivalently with real quadratic irrationals, have long been studied as geometric $\mathrm{SL}_2(\mathbb{Z})$-invariants. Building on the Birman-Williams approach to Lorenz knots and following the notion of modular knots introduced by Ghys, this article investigates the topological $\mathrm{SL}_2(\mathbb{Z})$-invariants arising from modular knots. Our main focus is the Alexander polynomial of modular knots. Using the Burau representation, we highlight two contrasting features of this family. On the one hand, for each fixed degree, only finitely many Alexander polynomials of modular knots occur. On the other hand, any integer appears as a coefficient of the Alexander polynomial of some modular knot, and coefficients of the same sign can occur in runs of arbitrarily long length.

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arXiv2025

Hecke equivariance of the divisor map

We study the multiplicative Hecke operators acting on the space of meromorphic modular forms, and show that the divisor map to divisors on $X_0(N)$ is a Hecke equivariant map. As applications, we investigate the divisor sum formula of Bruinier-Kohnen-Ono and more general Rohrlich-type divisor sums for polyharmonic Maass forms, discussing several implications for the Hecke action and its relation to the self-adjointness of the Hecke operators.

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arXiv2024

A unified approach to Rohrlich-type divisor sums

We propose a systematic method for analyzing Rohrlich-type divisor sums for arbitrary congruence subgroups $Γ_0(N)$. Our main theorem unifies various results from the literature, and its significance is illustrated through the following five applications: (1) the valence formula, (2) a natural generalization of classical Rohrlich's formula to level $N$, (3) an explicit version of the theorem by Bringmann-Kane-Löbrich-Ono-Rolen, (4) an extension of the generalized Rohrlich formula proposed by Bringmann-Kane, and (5) an alternative proof of the decomposition formula for twisted traces of CM values of weight 0 Eisenstein series.

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arXiv2024

Hauptmoduln and even-order mock theta functions modulo 2

The Fourier coefficients $c_1(n)$ of the elliptic modular $j$-function are always even for $n \not\equiv 7 \pmod{8}$. In contrast, for $n \equiv 7 \pmod{8}$, it is conjectured that ``half" of the coefficients take odd values. In this article, we first observe in detail when $c_1(8n-1)$ is odd and show that the coefficients share the same parity as the coefficients $c_{μ_2}(n)$ of the 2nd order mock theta function $μ_2(q)$. Furthermore, we prove that this phenomenon also holds among several hauptmoduln and between hauptmoduln and even-order mock theta functions.

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arXiv2023

Alder-type partition inequality at the general level

A Known Alder-type partition inequality of level $a$, which involves the second Rogers-Ramanujan identity when the level $a$ is 2, states that the number of partitions of $n$ into parts differing by at least $d$ with the smallest part being at least $a$ is greater than or equal to that of partitions of $n$ into parts congruent to $\pm a \pmod{d+3}$, excluding the part $d+3-a$. In this paper, we prove that for all values of $d$ with a finite number of exceptions, an arbitrary level $a$ Alder-type partition inequality holds without requiring the exclusion of the part $d+3-a$ in the latter partition.

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arXiv2023

Hecke equivariance of generalized Borcherds products of type $O(2,1)$

Recently, a weak converse theorem for Borcherds' lifting operator of type $O(2,1)$ for $\G_0(N)$ is proved and the logarithmic derivative of a modular form for $\G_0(N)$ is explicitly described in terms of the values of Niebur-Poincaré series at its divisors in the complex upper half-plane. In this paper, we prove that the generalized Borcherds' lifting operator of type $O(2,1)$ is Hecke equivariant under the extension of Guerzhoy's multiplicative Hecke operator on the integral weight meromorphic modular forms and the Hecke operator on half-integral weight vector-valued harmonic weak Maass forms. Additionally, we show that the logarithmic differential operator is also Hecke equivariant under the multiplicative Hecke operator and the Hecke operator on integral weight meromorphic modular forms. As applications of Hecke equivariance of the two operators, we obtain relations for twisted traces of singular moduli modulo prime powers and congruences for twisted class numbers modulo primes, including those associated to genus $1$ modular curves.

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arXiv2021

On values of weakly holomorphic modular functions at divisors of meromorphic modular forms

We show that the values of a certain family of weakly holomorphic modular functions at points in the divisors of any meromorphic modular form with algebraic Fourier coefficients are algebraic. We use this to extend the classical result of Schneider by proving that zeros or poles of any non-zero meromorphic modular form with algebraic Fourier coefficients are either transcendental or imaginary quadratic irrational.

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arXiv2020

Divisibility Properties of the Fourier Coefficients of (Mock) Modular Functions and Ramanujan

We survey divisibility properties of the Fourier coefficients of modular functions inspired by Ramanujan. Then using recent results of the generalized Hecke operator on harmonic Maass functions and known divisibility of Fourier coefficients of modular functions, we establish congruence relations of the Fourier coefficients of certain modular functions and mock modular functions of various levles.

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arXiv2020

Hecke System of Harmonic Maass Functions and Applications to Modular Curves of Higher Genera

In Monstrous moonshine, genus 0 property and the notion of replicability are strongly connected. With regards to recent developments of moonshine, we investigate a higher genus generalization of replicability for a general automorphic form. Specifically, we extend the definitions of replicates and a Hecke operator to harmonic Maass functions on modular curves of higher genera to obtain number theoretic generalizations of important results in Monstrous moonshine. Furthermore, we show the utility of the extended notions in yielding uniform proofs for numerous arithmetic properties of Fourier coefficients of modular functions of arbitrary level, which have been proved only for special cases of curves of genus zero or small prime levels.

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arXiv2016

Quantum mock modular forms arising from eta-theta functions

In 2013, Lemke Oliver classified all eta-quotients which are theta functions. In this paper, we unify the eta-theta functions by constructing mock modular forms from the eta-theta functions with even characters, such that the shadows of these mock modular forms are given by the eta-theta functions with odd characters. In addition, we prove that our mock modular forms are quantum modular forms. As corollaries, we establish simple finite hypergeometric expressions which may be used to evaluate Eichler integrals of the odd eta-theta functions, as well as some curious algebraic identities.

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arXiv2012

Weak Maass-Poincare series and weight 3/2 mock modular forms

The primary goal of this paper is to construct the basis of the space of weight 3/2 mock modular forms which is an extension of the Borcherd-Zagier basis of weight 3/2 weakly holomorphic modular forms. The shadows of the members of this basis form the Borcherds- Zagier basis of the space of weight 1/2 weakly holomorphic modular forms. For the purpose, we use a weak Maass-Poincaré Series. The secondary goal is to provide a full computation of the Fourier coefficients for the weak Maass-Poincaré Series in most general form as a weak Maass-Poincaré Series has played a key role in the recent advances in the theory of weak Maass forms.

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arXiv2012

Cycle integrals of a sesqui-harmonic Maass form of weight zero

Borcherds-Zagier bases of the spaces of weakly holomorphic modular forms of weights 1/2 and 3/2 share the Fourier coefficients which are traces of singular moduli. Recently, Duke, Imamoglu, and Tóth have constructed a basis of the space of weight 1/2 mock modular forms, each member in which has Zagier's generating series of traces of singular moduli as its shadow. They also showed that Fourier coefficients of their mock modular forms are sums of cycle integrals of the $j$-function which are real quadratic analogues of singular moduli. In this paper, we prove the Fourier coefficients of a basis of the space of weight 3/2 mock modular forms are sums of cycle integrals of a sesqui-harmonic Maass form of weight zero whose image under hyperbolic Laplacian is the $j$-function. Furthermore, we express these sums as regularized inner products of weakly holomorphic modular forms of weight 1/2.

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arXiv2009

Arithmetic Properties of Traces of Singular Moduli on Congruence Subgroups

After Zagier proved that the traces of singular moduli $j(z)$ are Fourier coefficients of a weakly holomorphic modular form, various properties of the traces of the singular values of modular functions mostly on the full modular group $PSL_2(\mathbb{Z})$ have been investigated such as their exact formulas, limiting distribution, duality, and congruences. The purpose of this paper is to generalize these arithmetic properties of traces of singular values of a weakly holomorphic modular function on the full modular group to those on a congruence subgroup $Γ_0(N)$.

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arXiv2008

Mock Jacobi forms in basic hypergeometric series

We show that some $q$-series such as universal mock theta functions are linear sums of theta quotients and mock Jacobi forms of weight 1/2, which become holomorphic parts of real analytic modular forms when they are restricted to torsion points and multiplied by suitable powers of $q$. And we prove that certain linear sums of $q$-series are weakly holomorphic modular forms of weight 1/2 due to annihilation of mock Jacobi forms or completion by mock Jacobi forms. As an application, we obtain a relation between the rank and crank of a partition.

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arXiv2007

Partitions weighted by the parity of the crank

A partition statistic ` crank' gives combinatorial interpretations for Ramanujan's famous partition congruences. In this paper, we establish an asymptotic formula, Ramanujan type congruences, and q-series identities that the number of partitions with even crank $M_e(n)$ minus the number of partitions with odd crank $M_o(n)$ satisfies. For example, we show that $M_e(5n+4)-M_o(5n+4)\equiv 0 \pmod 5.$ We also determine the exact values of $M_e(n)-M_o(n)$ in case of partitions into distinct parts, which are at most two and zero for infinitely many $n$.

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