arXiv subjects

Xiao-Jie Zhu

Publications and source records attributed to Xiao-Jie Zhu.

11 recordsLinked to original sources

arXiv2026

Dimension formulas for modular form spaces of rational weights, the classification of eta-quotient characters and an extension of Martin's theorem

We give an explicit formula for dimensions of spaces of rational-weight modular forms whose multiplier systems are induced by eta-quotients of fractional exponents. As the first application, we give series expressions of Fourier coefficients of the $n$th root of certain infinite $q$-products. As the second application, we extend Yves Martin's list of multiplicative holomorphic eta-quotients of integral weights by first extending the meaning of multiplicativity, then identifying one-dimensional spaces, and finally applying Wohlfahrt's extension of Hecke operators. A table containing $2277$ of such eta-quotients is presented. As a related result, we completely classify the multiplier systems induced by eta-quotients of integral exponents. For instance, there are totally $384$ such multiplier systems on $Γ_0(4)$ for any fixed weight. There are also some new results on $n$-fold covers of modular groups for $n\geq3$. Finally, we provide SageMath programs for verifying the theorems and generating the tables.

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arXiv2026

Correspondence among congruence families for generalized Frobenius partitions via modular permutations

In 2024, Garvan, Sellers and Smoot discovered a remarkable symmetry in the families of congruences for generalized Frobenius partitions $cψ_{2,0}$ and $cψ_{2,1}$. They also emphasized that the considerations for the general case of $cψ_{k,β}$ are important for future work. In this paper, for each $k$ we construct a vector-valued modular form for the generating functions of $cψ_{k,β}$, and determine an equivalence relation among all $β$. Within each equivalence class, we can identify modular transformations relating the congruences of one $cψ_{k,β}$ to that of another $cψ_{k,β'}$. Furthermore, correspondences between different equivalence classes can also be obtained through linear combinations of modular transformations. As an example, with the aid of these correspondences, we prove a family of congruences of $cϕ_{3}$, the Andrews' $3$-colored Frobenius partition.

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arXiv2025

Quadratic Gauss sums over $\mathbb{Z}^n/c\mathbb{Z}^n$: explicit formulas, a duality theorem, and applications to Weil representations and cubic hypersurfaces over $\mathbb{F}_p$

We provide explicit formulas for quadratic Gauss sums over $\mathbb{Z}^n/c\mathbb{Z}^n$, which generalize some of the existing formulas, e.g., Skoruppa and Zagier's (for $n=2$), and Iwaniec and Kowalski's (for arbitrary $n$). We then give four main applications. As the first application, we prove a duality theorem, which relates a sum over a subgroup of $\mathbb{Z}^n/c\mathbb{Z}^n$ to another sum over the orthogonal complement. This allows us to give explicit formulas for quadratic Gauss partial sums over certain subgroups. As the second application, we give an explicit formula for the coefficients of Weil representations of $\mathrm{Mp}_2(\mathbb{Z})$, which has the advantage, compared to Scheithauer's, Strömberg's, and Boylan and Skoruppa's formulas, that it involves neither local data nor limits of theta series. As the third application, we provide an explicit formula for the number of solutions of an arbitrary quadratic congruence in $n$ variables modulo a prime, as well as an efficient formula modulo a general integer. As the final application, we give a uniform formula for the number of solutions of the Diophantine equation $dxyz=Q(x,y,z)$ over $\mathbb{F}_p$, where $Q$ is an arbitrary integral quadratic form with some weak restrictions. This generalizes a result of Baragar regarding the Markoff equation. There is also a formula relating Hecke Gauss sums to sums over $\mathbb{Z}^n/c\mathbb{Z}^n$, an explicit formula for Hecke Gauss sums over quadratic fields, and an example of computing a Hecke Gauss sum over a cyclotomic field.

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arXiv2025

A formula for Fourier coefficients of certain eta-quotients and their expansions as Eisenstein series

We give a list of $113$ holomorphic eta-quotients of integral weight ($66$ of which are primitive) and provide a uniform closed formula for their Fourier coefficients $c(l)$ where $l\equiv1\bmod{m}$ with some fixed $m\mid24$. The proof involves Wohlfahrt's extension of Hecke operators and a dimension formula for spaces of modular forms of general multiplier system. We further provide the expansions of these eta-quotients as linear combinations of standard Eisenstein series.

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arXiv2025

Explicit formulae for linear characters of $Γ_0(N)$

We give explicit formulae for a class of complex linear unitary characters of the congruence subgroups $Γ_0(N)$ which involve a variant of Rademacher's $Ψ$ function. We then prove that these characters cover all characters of $Γ_0(N)$ precisely when $N=1,2,3,4,5,6,7,8,10,12,13$.

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arXiv2024

Transformation properties of Andrews-Beck $NT$ functions and generalized Appell-Lerch series

In 2021, Andrews mentioned that George Beck introduced a partition statistic $NT(r,m,n)$ which is related to Dyson's rank statistic. Motivated by Andrews's work, scholars have established a number of congruences and identities involving $NT(r,m,n)$. In this paper, we strengthen and extend a recent work of Mao on the transformation properties of the $NT$ function and provide an analogy of Hickerson and Mortenson's work on the rank function. As an application, we demonstrate how one can deduce from our results many identities involving $NT(r,m,n)$ and another crank-analog statistic $M_ω(r,m,n)$. As a related result, some new properties of generalized Appell-Lerch series are given.

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arXiv2023

An Eisenstein series of rational weight

We study an Eisenstein series of weight $20/9$, level $11$ whose multiplier system is the same as $η^{44/9}(τ)η^{-4/9}(11τ)$. We prove that it is a modular form and obtain its Fourier expansion.

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arXiv2023

Holomorphic Eisenstein series of rational weights and special values of Gamma function

We give all possible holomorphic Eisenstein series on $Γ_0(p)$, of rational weights greater than $2$, and with multiplier systems the same as certain rational-weight eta-quotients at all cusps. We prove they are modular forms and give their Fourier expansions. We establish four sorts of identities that equate such series to rational-weight eta-quotients. As an application, we give series expressions of special values of Euler Gamma function at any rational arguments. These expressions involve exponential sums of Dedekind sums.

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arXiv2023

Double coset operators and eta-quotients

We study a type of generalized double coset operators which may change the characters of modular forms. For any pair of characters $v_1$ and $v_2$, we describe explicitly those operators mapping modular forms of character $v_1$ to those of $v_2$. We give three applications, concerned with eta-quotients. For the first application, we give many pairs of eta-quotients of small weights and levels, such that there are operators maps one eta-quotient to another. We also find out these operators. For the second application, we apply the operators to eta-powers whose exponents are positive integers not greater than $24$. This results in recursive formulas of the coefficients of these functions, generalizing Newman's theorem. For the third application, we describe a criterion and an algorithm of whether and how an eta-power of arbitrary integral exponent can be expressed as a linear combination of certain eta-quotients.

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arXiv2022

Taylor expansions of Jacobi forms and linear relations among theta series

We study Taylor expansions of Jacobi forms of lattice index. As the main result, we give an embedding from certain space of such forms, whether scalar-valued or vector-valued, integral-weight or half-integral-weight, of any level, with any character, into a product of finitely many spaces of modular forms. As an application, we investigate linear relations among Jacobi theta series of lattice index. Many linear relations among the second powers of such theta series associated with the $D_4$ lattice and $A_3$ lattice are obtained, along with relations among the third powers of series associated with the $A_2$ lattice. We present the complete SageMath code for the $D_4$ lattice.

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arXiv2021

Finite quadratic modules and lattices

We give a new proof of the fact that any finite quadratic module can be decomposed into indecomposable ones. For any indecomposable finite quadratic module, we construct a lattice, and a positive definite lattice, both of which are of the least rank, whose discriminant module is the given one. The resulting lattices are given by their Gram matrices explicitly.

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