Toric ranks and component groups of modular curves
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arXiv
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| Format: | Preprint |
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2024
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| _version_ | 1866916313707839488 |
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| author | Helminck, Paul Alexander |
| author_facet | Helminck, Paul Alexander |
| contents | Let $p\neq{2,3}$ be a prime number and let $Γ\subset \mathrm{SL}_{2}(\mathbb{Z})$ be a congruence subgroup with modular curve $X_Γ/K$ and Jacobian $J(X_Γ)$. In this paper we give an explicit group-theoretic description of the semistable toric rank and component group of $J(X_Γ)$ at the finite places of $K$ lying over $p$. We first produce a suitable deformation retract of the minimal Berkovich skeleton of $X_Γ$ in terms of Hecke-Iwahori double coset spaces. We call this deformation retract the pruned skeleton of the curve. Our description of this skeleton includes a group-theoretic formula for the edge lengths, allowing us to give the component group of the modular curve as the quotient of a lattice using the monodromy pairing. For $X_{0}(N)$, $X_{1}(N)$, $X_{sp}(N)$ and $X_{sp}^{+}(N)$, we explicitly determine the pruned skeleta using a set of coset schemes over $\mathbb{Z}$. This in particular recovers results by Deligne-Rapoport, Edixhoven, Coleman-McMurdy and Tsushima on the semistable reduction type of $X_{0}(p^{n})$ for $n\leq{4}$. Finally, we determine the geometric Tamagawa number and the prime-to-$2$ structure of the component group of $X_{0}(N)$ over the extension given by Krir's theorem. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2403_09995 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Toric ranks and component groups of modular curves Helminck, Paul Alexander Number Theory Algebraic Geometry Let $p\neq{2,3}$ be a prime number and let $Γ\subset \mathrm{SL}_{2}(\mathbb{Z})$ be a congruence subgroup with modular curve $X_Γ/K$ and Jacobian $J(X_Γ)$. In this paper we give an explicit group-theoretic description of the semistable toric rank and component group of $J(X_Γ)$ at the finite places of $K$ lying over $p$. We first produce a suitable deformation retract of the minimal Berkovich skeleton of $X_Γ$ in terms of Hecke-Iwahori double coset spaces. We call this deformation retract the pruned skeleton of the curve. Our description of this skeleton includes a group-theoretic formula for the edge lengths, allowing us to give the component group of the modular curve as the quotient of a lattice using the monodromy pairing. For $X_{0}(N)$, $X_{1}(N)$, $X_{sp}(N)$ and $X_{sp}^{+}(N)$, we explicitly determine the pruned skeleta using a set of coset schemes over $\mathbb{Z}$. This in particular recovers results by Deligne-Rapoport, Edixhoven, Coleman-McMurdy and Tsushima on the semistable reduction type of $X_{0}(p^{n})$ for $n\leq{4}$. Finally, we determine the geometric Tamagawa number and the prime-to-$2$ structure of the component group of $X_{0}(N)$ over the extension given by Krir's theorem. |
| title | Toric ranks and component groups of modular curves |
| topic | Number Theory Algebraic Geometry |
| url | https://arxiv.org/abs/2403.09995 |