Research

Xiao-Jie Zhu, Explicit formulae for linear characters of \(\Gamma_0(N)\), Communications in Algebra 2025, published online.

Abstract:

We give explicit formulae for a class of complex linear unitary characters of the congruence subgroups \(\Gamma_0(N)\) which involve a variant of Rademacher's \(\Psi\) function. We then prove that these characters cover all characters of \(\Gamma_0(N)\) precisely when \(N=1,2,3,4,5,6,7,8,10,12,13\).

Keywords:

unitary character, congruence subgroup, Dedekind sum, Rademacher's \(\Psi\) function, modular group, Dedekind eta function

MSC:

Primary 11F06; Secondary 11F20, 22D10, 20F05, 20H10

DOI: https://doi.org/10.1080/00927872.2025.2488029

arXiv: https://arxiv.org/abs/2402.14796

ar5iv: https://ar5iv.org/abs/2402.14796

AMS MR: None

zbMath: None

Xiao-Jie Zhu, Finite quadratic modules and lattices, Communications in Algebra 52 (2024): 604-629.

Abstract:

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.

Keywords:

Discriminant modules; finite quadratic modules; lattices

MSC:

Primary: 11E12; Secondary: 16D70

DOI: https://doi.org/10.1080/00927872.2023.2245924

arXiv: https://arxiv.org/abs/2110.06783

ar5iv: https://ar5iv.org/abs/2110.06783

AMS MR: https://mathscinet.ams.org/mathscinet/article?mr=4703543

zbMath: https://zbmath.org/1546.11051

Hai-Gang Zhou, Xiao-Jie Zhu, Double coset operators and eta-quotients, Journal of Number Theory 249 (2023): 537-601.

Abstract:

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.

Keywords:

Hecke operators, Eta quotients, Dedekind eta function, Double coset operators, Partition function

MSC:

Primary 11F25, Secondary 11F20, 11F11, 11F37, 11F03, 11F30

DOI: https://doi.org/10.1016/j.jnt.2023.02.017

arXiv: https://arxiv.org/abs/2110.06768

ar5iv: https://ar5iv.org/abs/2110.06768

AMS MR: https://mathscinet.ams.org/mathscinet/article?mr=4578663

zbMath: https://zbmath.org/1536.11074

Xiao-Jie Zhu, Holomorphic Eisenstein series of rational weights and special values of Gamma function, Acta Arithmetica 210 (2023): 279-305.

Abstract:

We give all possible holomorphic Eisenstein series on \(\Gamma_0(p)\), 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.

Keywords:

Eisenstein series, modular form, Dedekind sum, Dedekind eta function, Euler Gamma function, rational weight

MSC:

Primary 11F30; Secondary 11F03, 11F11, 11F20, 33B15

DOI: https://doi.org/10.4064/aa221110-1-4

arXiv: https://arxiv.org/abs/2210.06253

ar5iv: https://ar5iv.org/abs/2210.06253

AMS MR: https://mathscinet.ams.org/mathscinet/article?mr=4678128

zbMath: https://zbmath.org/1550.11060

Xiao-Jie Zhu, Taylor expansions of Jacobi forms and linear relations among theta series, Dissertationes Mathematicae 590 (2023): 1-66.

Abstract:

We study Taylor expansions of Jacobi forms of lattice index. As the main result, we give an embedding from a 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 a complete SageMath code for the \(D_4\) lattice.

Keywords:

Jacobi form, lattice, theta series, modular form, Weil representation, quadratic form

MSC:

Primary 11F50; Secondary 11F37, 11F11, 11F27

DOI: https://doi.org/10.4064/dm880-12-2023

arXiv: https://arxiv.org/abs/2204.08262

ar5iv: https://ar5iv.org/abs/2204.08262

AMS MR: https://mathscinet.ams.org/mathscinet/article?mr=4684413

zbMath: https://zbmath.org/1532.11001

Rong Chen, Xiao-Jie Zhu, Correspondence among congruence families for generalized Frobenius partitions via modular permutations, Jun 2025.

Abstract:

In 2024, Garvan, Sellers and Smoot discovered a remarkable symmetry in the families of congruences for generalized Frobenius partitions \(c\psi_{2,0}\) and \(c\psi_{2,1}\). They also emphasized that the considerations for the general case of \(c\psi_{k,\beta}\) are important for future work. In this paper, for each \(k\) we construct a vector-valued modular form for the generating functions of \(c\psi_{k,\beta}\), and determine an equivalence relation among all \(\beta\). Within each equivalence class, we can identify modular transformations relating the congruences of one \(c\psi_{k,\beta}\) to that of another \(c\psi_{k,\beta'}\). 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 \(c\phi_{3}\), the Andrews' \(3\)-colored Frobenius partition.

Keywords:

Partition congruences, Frobenius partitions, Modular functions, Weil representations, Jacobi forms

MSC:

Primary 11P83, Secondary 11F27; 11F33; 11F37; 20C15

arXiv: https://arxiv.org/abs/2506.16823

ar5iv: https://ar5iv.org/abs/2506.16823

Xiao-Jie Zhu, A simple closed formula for Fourier coefficients of certain eta-quotients, Aug 2024.

Abstract:

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.

Keywords:

Dedekind eta function, eta-quotient, Fourier coefficient, Hecke operator, dimension formula

MSC:

Primary 11F20, 11F30, Secondary 11F25, 11F11

arXiv: https://arxiv.org/abs/2408.09480

ar5iv: https://ar5iv.org/abs/2408.09480

Xiao-Jie Zhu, Dimension formulas for modular form spaces of rational weights, the classification of eta-quotient characters and an extension of Martin's theorem, Aug 2024.

Abstract:

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 \(\Gamma_0(4)\) for any fixed weight. We also provide SageMath programs on checking the theorems and generating the tables.

Keywords:

modular form, dimension formula, rational weight, Dedekind eta function, Hecke operator, multiplicative eta-quotient

MSC:

Primary 11F12; Secondary 11F20, 11F25, 11F30, 11L05, 30F10

arXiv: https://arxiv.org/abs/2408.00246

ar5iv: https://ar5iv.org/abs/2408.00246

Rong Chen, Xiao-Jie Zhu, Transformation properties of Andrews-Beck \(NT\) functions and generalized Appell-Lerch series, Jul 2024.

Abstract:

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 \(M_\omega(r,m,n)\). As a related result, some new properties of generalized Appell-Lerch series are given.

Keywords:

Andrews-Beck \(NT\) function, Dyson's rank function, partitions, non-holomorphic modular forms, Appell-Lerch series

MSC:

Primary 11P84; Secondary 05A17, 11F11, 11F30, 11F37, 11P83

arXiv: https://arxiv.org/abs/2407.20790

ar5iv: https://ar5iv.org/abs/2407.20790