Newspaces with Nebentypus: An Explicit Dimension Formula and Classification of Trivial Newspaces
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arXiv
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| Format: | Preprint |
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2024
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| _version_ | 1866916768090423296 |
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| author | Ross, Erick |
| author_facet | Ross, Erick |
| contents | Consider $N \geq 1$, $k \geq 2$, and $χ$ a Dirichlet character modulo $N$ such that $χ(-1) = (-1)^k$. For any bound $B$, one can show that $\dim S_k(Γ_0(N),χ) \le B$ for only finitely many triples $(N,k,χ)$. It turns out that this property does not extend to the newspace; there exists an infinite family of triples $(N,k,χ)$ for which $\dim S_k^{\text{new}}(Γ_0(N),χ) = 0$. However, we classify this case entirely. We also show that excluding the infinite family for which $\dim S_k^{\text{new}}(Γ_0(N),χ) = 0$, $\dim S_k^{\text{new}}(Γ_0(N),χ) \leq B$ for only finitely many triples $(N,k,χ)$. In order to show these results, we derive an explicit dimension formula for the newspace $S_k^{\text{new}}(Γ_0(N),χ)$. We also use this explicit dimension formula to prove a character equidistribution property and disprove a conjecture from Greg Martin that $\dim S_2^{\text{new}}(Γ_0(N))$ takes on all possible non-negative integers. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2407_08881 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Newspaces with Nebentypus: An Explicit Dimension Formula and Classification of Trivial Newspaces Ross, Erick Number Theory 11F11 (Primary), 11F06 (Secondary) Consider $N \geq 1$, $k \geq 2$, and $χ$ a Dirichlet character modulo $N$ such that $χ(-1) = (-1)^k$. For any bound $B$, one can show that $\dim S_k(Γ_0(N),χ) \le B$ for only finitely many triples $(N,k,χ)$. It turns out that this property does not extend to the newspace; there exists an infinite family of triples $(N,k,χ)$ for which $\dim S_k^{\text{new}}(Γ_0(N),χ) = 0$. However, we classify this case entirely. We also show that excluding the infinite family for which $\dim S_k^{\text{new}}(Γ_0(N),χ) = 0$, $\dim S_k^{\text{new}}(Γ_0(N),χ) \leq B$ for only finitely many triples $(N,k,χ)$. In order to show these results, we derive an explicit dimension formula for the newspace $S_k^{\text{new}}(Γ_0(N),χ)$. We also use this explicit dimension formula to prove a character equidistribution property and disprove a conjecture from Greg Martin that $\dim S_2^{\text{new}}(Γ_0(N))$ takes on all possible non-negative integers. |
| title | Newspaces with Nebentypus: An Explicit Dimension Formula and Classification of Trivial Newspaces |
| topic | Number Theory 11F11 (Primary), 11F06 (Secondary) |
| url | https://arxiv.org/abs/2407.08881 |