Geometric realizations of representations for $\text{PSL}(2, \mathbb{F}_p)$ and Galois representations arising from defining ideals of modular curves
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| Formato: | Preprint |
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2024
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| _version_ | 1866918017888157696 |
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| author | Yang, Lei |
| author_facet | Yang, Lei |
| contents | We construct a geometric realization of representations for $\text{PSL}(2, \mathbb{F}_p)$ by the defining ideals of rational models $\mathcal{L}(X(p))$ of modular curves $X(p)$ over $\mathbb{Q}$, which gives rise to a Rosetta stone for geometric representations of $\text{PSL}(2, \mathbb{F}_p)$. The defining ideal of a modular curve, i.e., an anabelian counterpart of the Eisenstein ideal, is the anabelianization of the Jacobian of this modular curve and is a reification of the fundamental group $π_1$. We show that there exists a correspondence among the defining ideals of modular curves over $\mathbb{Q}$, reducible $\mathbb{Q}(ζ_p)$-rational representations $π_p: \text{PSL}(2, \mathbb{F}_p) \rightarrow \text{Aut}(\mathcal{L}(X(p)))$ of $\text{PSL}(2, \mathbb{F}_p)$, and $\mathbb{Q}(ζ_p)$-rational Galois representations $ρ_p: \text{Gal}(\overline{\mathbb{Q}}/\mathbb{Q}) \rightarrow \text{Aut}(\mathcal{L}(X(p)))$ as well as their modular and surjective realization. It is an anabelian counterpart of the global Langlands correspondence for $\text{GL}(2, \mathbb{Q})$ by the étale cohomology of modular curves as well as an anabelian counterpart of Artin's conjecture, Serre's modularity conjecture and the Fontaine-Mazur conjecture. It is an ideal theoretic (i.e. nonlinear) counterpart of Grothendieck's section conjecture and an ideal theoretic (i.e. nonlinear) reification of ``arithmetic theory of $π_1$'' expected by Weil for modular curves. It is also an anabelian counterpart of the theory of Kubert-Lang and Mazur-Wiles on the cuspidal divisor class groups and the Eisenstein ideals of modular curves. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2409_02589 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Geometric realizations of representations for $\text{PSL}(2, \mathbb{F}_p)$ and Galois representations arising from defining ideals of modular curves Yang, Lei Number Theory Algebraic Geometry Representation Theory 11R39, 11F80, 20C33, 14C25, 14H42, 11G18, 14L30, 13A50, 14H37, 11G32 We construct a geometric realization of representations for $\text{PSL}(2, \mathbb{F}_p)$ by the defining ideals of rational models $\mathcal{L}(X(p))$ of modular curves $X(p)$ over $\mathbb{Q}$, which gives rise to a Rosetta stone for geometric representations of $\text{PSL}(2, \mathbb{F}_p)$. The defining ideal of a modular curve, i.e., an anabelian counterpart of the Eisenstein ideal, is the anabelianization of the Jacobian of this modular curve and is a reification of the fundamental group $π_1$. We show that there exists a correspondence among the defining ideals of modular curves over $\mathbb{Q}$, reducible $\mathbb{Q}(ζ_p)$-rational representations $π_p: \text{PSL}(2, \mathbb{F}_p) \rightarrow \text{Aut}(\mathcal{L}(X(p)))$ of $\text{PSL}(2, \mathbb{F}_p)$, and $\mathbb{Q}(ζ_p)$-rational Galois representations $ρ_p: \text{Gal}(\overline{\mathbb{Q}}/\mathbb{Q}) \rightarrow \text{Aut}(\mathcal{L}(X(p)))$ as well as their modular and surjective realization. It is an anabelian counterpart of the global Langlands correspondence for $\text{GL}(2, \mathbb{Q})$ by the étale cohomology of modular curves as well as an anabelian counterpart of Artin's conjecture, Serre's modularity conjecture and the Fontaine-Mazur conjecture. It is an ideal theoretic (i.e. nonlinear) counterpart of Grothendieck's section conjecture and an ideal theoretic (i.e. nonlinear) reification of ``arithmetic theory of $π_1$'' expected by Weil for modular curves. It is also an anabelian counterpart of the theory of Kubert-Lang and Mazur-Wiles on the cuspidal divisor class groups and the Eisenstein ideals of modular curves. |
| title | Geometric realizations of representations for $\text{PSL}(2, \mathbb{F}_p)$ and Galois representations arising from defining ideals of modular curves |
| topic | Number Theory Algebraic Geometry Representation Theory 11R39, 11F80, 20C33, 14C25, 14H42, 11G18, 14L30, 13A50, 14H37, 11G32 |
| url | https://arxiv.org/abs/2409.02589 |