{"paper":{"title":"An explicit theory of $\\pi_{1}^{\\un,\\crys}(\\mathbb{P}^{1} - \\{0,\\mu_{N},\\infty\\})$ - V-1 : The Frobenius extended to $\\pi_{1}^{\\un,\\DR}(\\mathbb{P}^{1} - \\{0,\\mu_{p^{\\alpha}N},\\infty\\})$","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.NT","authors_text":"David Jarossay","submitted_at":"2017-08-26T18:39:37Z","abstract_excerpt":"Let $p$ a prime number. For all $N \\in \\mathbb{N}^{\\ast}$ prime to $p$, let $k_{N}$ be a finite field of characteristic $p$ containing a primitive $N$-th root of unity. Let $X_{k_{N},N}=\\text{ }\\mathbb{P}^{1} - (\\{0,\\infty\\} \\cup \\mu_{N})\\text{ }/\\text{ }k_{N}$. This work is an explicit theory of the crystalline pro-unipotent fundamental groupoid $(\\pi_{1}^{\\un,\\crys})$ of $X_{k_{N},N}$. In the parts I to IV, we have considered each possible value of $N$ separately. The purpose of part V is to study the role of the morphisms relating $\\pi_{1}^{\\un}(\\mathbb{P}^{1} - \\{0,\\mu_{N_{1}},\\infty\\})$ a"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1708.08009","kind":"arxiv","version":1},"verdict":{"id":null,"model_set":{},"created_at":null,"strongest_claim":"","one_line_summary":"","pipeline_version":null,"weakest_assumption":"","pith_extraction_headline":""},"references":{"count":0,"sample":[],"resolved_work":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57","internal_anchors":0},"formal_canon":{"evidence_count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"author_claims":{"count":0,"strong_count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"builder_version":"pith-number-builder-2026-05-17-v1"}