Modular Symbols over Function Fields of Elliptic Curves
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arXiv
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| Format: | Preprint |
| Publié: |
2024
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| _version_ | 1866910006851403776 |
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| author | Jung, Seong Eun |
| author_facet | Jung, Seong Eun |
| contents | Let $k =\mathbb{F}_q$ be the finite field of $q$ elements and $E$ an elliptic curve over $k$. Let $F = k(E)$ be the function field over $E$ and let $\mathcal{O} = k[E]$ be the ring of integers. We fix the place at $\infty$ of $F$ and let $F_{\infty}$ be the completion. The group $\overlineΓ= {\rm{GL}}_2(\mathcal{O})$ acts on $\mathcal{T}$, the Bruhat-Tits building of ${\rm{PGL}}_2(F_{\infty})$. In this article, we construct the group of modular symbols over $Γ$, a congruence subgroup of $\overlineΓ$. We prove that this space is given by an explicit set of generators and relations among them. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2408_04330 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Modular Symbols over Function Fields of Elliptic Curves Jung, Seong Eun Number Theory Let $k =\mathbb{F}_q$ be the finite field of $q$ elements and $E$ an elliptic curve over $k$. Let $F = k(E)$ be the function field over $E$ and let $\mathcal{O} = k[E]$ be the ring of integers. We fix the place at $\infty$ of $F$ and let $F_{\infty}$ be the completion. The group $\overlineΓ= {\rm{GL}}_2(\mathcal{O})$ acts on $\mathcal{T}$, the Bruhat-Tits building of ${\rm{PGL}}_2(F_{\infty})$. In this article, we construct the group of modular symbols over $Γ$, a congruence subgroup of $\overlineΓ$. We prove that this space is given by an explicit set of generators and relations among them. |
| title | Modular Symbols over Function Fields of Elliptic Curves |
| topic | Number Theory |
| url | https://arxiv.org/abs/2408.04330 |