Equidistribution of integers represented by standard quadratic form under arithmetic constraints
Fuente:
arXiv
Saved in:
| Main Author: | |
|---|---|
| Format: | Preprint |
| Published: |
2025
|
| Subjects: | |
| Online Access: | |
| Tags: |
Add Tag
No Tags, Be the first to tag this record!
|
| _version_ | 1866909525694480384 |
|---|---|
| author | Ma, Yefei |
| author_facet | Ma, Yefei |
| contents | We study the equidistribution of integers of the form $n= x_1^2 + \cdots + x_d^2$ under the arithmetic constraints given by $(\mathbb{Z}/p\mathbb{Z})^d$. The first step in addressing this problem is to construct modular forms whose Fourier expansion coefficients correspond to the counting problem over the quadric in $\mathbb{Z}^d$ induced by the standard quadratic form, subject to the aforementioned arithmetic constraints. The weak modular property of these modular forms allows us to use representation theory to identify the congruence subgroup to which our modular forms correspond. We then establish a necessary and sufficient condition for functions on $(\mathbb{Z}/p\mathbb{Z})^d$ that defines a cusp form. Finally, we conclude that the equidistribution phenomenon occurs locally on $p+1$ orbits for $d \geq 4$. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2503_03873 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Equidistribution of integers represented by standard quadratic form under arithmetic constraints Ma, Yefei Number Theory Representation Theory We study the equidistribution of integers of the form $n= x_1^2 + \cdots + x_d^2$ under the arithmetic constraints given by $(\mathbb{Z}/p\mathbb{Z})^d$. The first step in addressing this problem is to construct modular forms whose Fourier expansion coefficients correspond to the counting problem over the quadric in $\mathbb{Z}^d$ induced by the standard quadratic form, subject to the aforementioned arithmetic constraints. The weak modular property of these modular forms allows us to use representation theory to identify the congruence subgroup to which our modular forms correspond. We then establish a necessary and sufficient condition for functions on $(\mathbb{Z}/p\mathbb{Z})^d$ that defines a cusp form. Finally, we conclude that the equidistribution phenomenon occurs locally on $p+1$ orbits for $d \geq 4$. |
| title | Equidistribution of integers represented by standard quadratic form under arithmetic constraints |
| topic | Number Theory Representation Theory |
| url | https://arxiv.org/abs/2503.03873 |