Fundamental Fourier coefficients of Siegel modular forms of higher degrees and levels
Fuente:
arXiv
Saved in:
| Main Authors: | , |
|---|---|
| Format: | Preprint |
| Published: |
2025
|
| Subjects: | |
| Online Access: | |
| Tags: |
Add Tag
No Tags, Be the first to tag this record!
|
| _version_ | 1866914315193286656 |
|---|---|
| author | Anamby, Pramath Das, Soumya |
| author_facet | Anamby, Pramath Das, Soumya |
| contents | We prove the following statement about any Siegel modular form $F$ of degree $n$ and arbitrary odd level $N$ on the group $Γ_{0}^{(n)}(N)$. Let $A(F,T)$ denote the Fourier coefficients of $F$ and write $T=(T(i,j))$.
Suppose that $F$ has a non-zero Fourier coefficient $A(F,T_0)$ such that $(T_0(n,n),N)=1$. Then there exist infinitely many odd and square-free (and thus fundamental) integers $m$ such that $m=\mathrm{discriminant}(T)$ and $A(F,T)\neq 0$. In the case of odd degrees, we prove a stronger result by replacing odd and square-free with odd and prime. We also prove quantitative results in this direction.
As a consequence, we can show in particular that the statement of the main result in arXiv:2408.03442 about the algebraicity of certain critical values (and the expected functional equation) of the spinor $L$-functions of holomorphic newforms (in the ambit of Deligne's conjectures) on congruence subgroups of $\mathrm{GSp}(3)$ is unconditional. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2509_12148 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Fundamental Fourier coefficients of Siegel modular forms of higher degrees and levels Anamby, Pramath Das, Soumya Number Theory Primary 11F30, 11F46, Secondary 11F50, 11F37 We prove the following statement about any Siegel modular form $F$ of degree $n$ and arbitrary odd level $N$ on the group $Γ_{0}^{(n)}(N)$. Let $A(F,T)$ denote the Fourier coefficients of $F$ and write $T=(T(i,j))$. Suppose that $F$ has a non-zero Fourier coefficient $A(F,T_0)$ such that $(T_0(n,n),N)=1$. Then there exist infinitely many odd and square-free (and thus fundamental) integers $m$ such that $m=\mathrm{discriminant}(T)$ and $A(F,T)\neq 0$. In the case of odd degrees, we prove a stronger result by replacing odd and square-free with odd and prime. We also prove quantitative results in this direction. As a consequence, we can show in particular that the statement of the main result in arXiv:2408.03442 about the algebraicity of certain critical values (and the expected functional equation) of the spinor $L$-functions of holomorphic newforms (in the ambit of Deligne's conjectures) on congruence subgroups of $\mathrm{GSp}(3)$ is unconditional. |
| title | Fundamental Fourier coefficients of Siegel modular forms of higher degrees and levels |
| topic | Number Theory Primary 11F30, 11F46, Secondary 11F50, 11F37 |
| url | https://arxiv.org/abs/2509.12148 |