The image of the generalized Dedekind sum
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arXiv
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| Main Authors: | , , , |
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| Format: | Preprint |
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2025
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| _version_ | 1866913733560762368 |
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| author | Knight, Evelyne S. Matos, Carlos Alexov Sefidi, Amira Young, Matthew P. |
| author_facet | Knight, Evelyne S. Matos, Carlos Alexov Sefidi, Amira Young, Matthew P. |
| contents | The newform Dedekind sum $S_{χ_1, χ_2}$ associated to a pair of primitive Dirichlet characters $χ_1$, $χ_2$ of respective conductors $q_1$, $q_2$, is a group homomorphism from $Γ_1(q_1 q_2)$ into the number field $F_{χ_1, χ_2}$ generated by the values of the characters. It is a basic question to identify the image of this map, which is known to be a lattice $L_{χ_1, χ_2}$ in $F_{χ_1, χ_2}$. It has recently been conjectured that when $χ_1$ and $χ_2$ are quadratic, then $ L_{χ_1, χ_2} = 2 \mathbb{Z}$.
In this paper, we make some progress towards this conjecture by exhibiting an explicit lattice in which $L_{χ_1, χ_2}$ is contained; in particular, when the characters are quadratic, the $q_i$ are coprime, odd, and sufficiently large, then $L_{χ_1, χ_2} \subseteq \mathbb{Z}$. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2503_09741 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | The image of the generalized Dedekind sum Knight, Evelyne S. Matos, Carlos Alexov Sefidi, Amira Young, Matthew P. Number Theory The newform Dedekind sum $S_{χ_1, χ_2}$ associated to a pair of primitive Dirichlet characters $χ_1$, $χ_2$ of respective conductors $q_1$, $q_2$, is a group homomorphism from $Γ_1(q_1 q_2)$ into the number field $F_{χ_1, χ_2}$ generated by the values of the characters. It is a basic question to identify the image of this map, which is known to be a lattice $L_{χ_1, χ_2}$ in $F_{χ_1, χ_2}$. It has recently been conjectured that when $χ_1$ and $χ_2$ are quadratic, then $ L_{χ_1, χ_2} = 2 \mathbb{Z}$. In this paper, we make some progress towards this conjecture by exhibiting an explicit lattice in which $L_{χ_1, χ_2}$ is contained; in particular, when the characters are quadratic, the $q_i$ are coprime, odd, and sufficiently large, then $L_{χ_1, χ_2} \subseteq \mathbb{Z}$. |
| title | The image of the generalized Dedekind sum |
| topic | Number Theory |
| url | https://arxiv.org/abs/2503.09741 |