Double coset operators and eta-quotients
Fuente:
arXiv
Saved in:
| Main Authors: | , |
|---|---|
| Format: | Preprint |
| Published: |
2021
|
| Subjects: | |
| Online Access: | |
| Tags: |
Add Tag
No Tags, Be the first to tag this record!
|
| _version_ | 1866918043219656704 |
|---|---|
| author | Zhou, Hai-Gang Zhu, Xiao-Jie |
| author_facet | Zhou, Hai-Gang Zhu, Xiao-Jie |
| contents | 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. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2110_06768 |
| institution | arXiv |
| publishDate | 2021 |
| record_format | arxiv |
| spellingShingle | Double coset operators and eta-quotients Zhou, Hai-Gang Zhu, Xiao-Jie Number Theory Primary: 11F25, Secondary: 11F20, 11F11, 11F37, 11F03, 11F30 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. |
| title | Double coset operators and eta-quotients |
| topic | Number Theory Primary: 11F25, Secondary: 11F20, 11F11, 11F37, 11F03, 11F30 |
| url | https://arxiv.org/abs/2110.06768 |