Homotopical Observables and the Langlands Program via $\infty$-Topoi
Fuente:
arXiv
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| Natura: | Preprint |
| Pubblicazione: |
2025
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| _version_ | 1866915349896626176 |
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| author | Galikhanov, Anatoly |
| author_facet | Galikhanov, Anatoly |
| contents | We introduce a pro-étale geometric object $D_\infty$ arising naturally from the tower of Artin-Schreier extensions in characteristic 2, equipped with a canonical endofunctor $O$ whose fixed points correspond to automorphic representations of $\mathrm{GL}_2(\mathbb{A}_{\mathbb{F}_2})$. The main theorem establishes that invariant predicates on $D_\infty$ parametrize cuspidal automorphic representations, preserving $L$-functions. We provide complete proofs using $\infty$-categorical techniques, explicit computations for small cases, and establish connections to discrete conformal field theory. As applications, we resolve the Carlitz-Drinfeld uniformization conjecture for function fields and compute previously unknown motivic cohomology groups. Our approach differs fundamentally from coalgebraic models by working internally in topoi and connecting to arithmetic geometry. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2505_22558 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Homotopical Observables and the Langlands Program via $\infty$-Topoi Galikhanov, Anatoly General Mathematics 11F70, 14G32, 18N50, 55U40 F.4.1; G.0; F.1.1 We introduce a pro-étale geometric object $D_\infty$ arising naturally from the tower of Artin-Schreier extensions in characteristic 2, equipped with a canonical endofunctor $O$ whose fixed points correspond to automorphic representations of $\mathrm{GL}_2(\mathbb{A}_{\mathbb{F}_2})$. The main theorem establishes that invariant predicates on $D_\infty$ parametrize cuspidal automorphic representations, preserving $L$-functions. We provide complete proofs using $\infty$-categorical techniques, explicit computations for small cases, and establish connections to discrete conformal field theory. As applications, we resolve the Carlitz-Drinfeld uniformization conjecture for function fields and compute previously unknown motivic cohomology groups. Our approach differs fundamentally from coalgebraic models by working internally in topoi and connecting to arithmetic geometry. |
| title | Homotopical Observables and the Langlands Program via $\infty$-Topoi |
| topic | General Mathematics 11F70, 14G32, 18N50, 55U40 F.4.1; G.0; F.1.1 |
| url | https://arxiv.org/abs/2505.22558 |