Etale descent obstruction and anabelian geometry of curves over finite fields
Fuente:
arXiv
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| Autori principali: | , |
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| Natura: | Preprint |
| Pubblicazione: |
2023
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| _version_ | 1866910258829459456 |
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| author | Creutz, Brendan Voloch, Jose Felipe |
| author_facet | Creutz, Brendan Voloch, Jose Felipe |
| contents | Let $C$ and $D$ be smooth, proper and geometrically integral curves over a finite field $F$. Any morphism from $D$ to $C$ induces a morphism of their étale fundamental groups. The anabelian philosophy proposed by Grothendieck suggests that, when $C$ has genus at least $2$, all open homomorphisms between the étale fundamental groups should arise in this way from a nonconstant morphism of curves. We relate this expectation to the arithmetic of the curve $C_K$ over the global function field $K = F(D)$. Specifically, we show that there is a bijection between the set of conjugacy classes of well-behaved morphism of fundamental groups and locally constant adelic points of $C_K$ that survive étale descent. We use this to provide further evidence for the anabelian conjecture by relating it to another recent conjecture by Sutherland and the second author. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2306_04844 |
| institution | arXiv |
| publishDate | 2023 |
| record_format | arxiv |
| spellingShingle | Etale descent obstruction and anabelian geometry of curves over finite fields Creutz, Brendan Voloch, Jose Felipe Number Theory Algebraic Geometry 11G20, 11G30, 14G05, 14G15 Let $C$ and $D$ be smooth, proper and geometrically integral curves over a finite field $F$. Any morphism from $D$ to $C$ induces a morphism of their étale fundamental groups. The anabelian philosophy proposed by Grothendieck suggests that, when $C$ has genus at least $2$, all open homomorphisms between the étale fundamental groups should arise in this way from a nonconstant morphism of curves. We relate this expectation to the arithmetic of the curve $C_K$ over the global function field $K = F(D)$. Specifically, we show that there is a bijection between the set of conjugacy classes of well-behaved morphism of fundamental groups and locally constant adelic points of $C_K$ that survive étale descent. We use this to provide further evidence for the anabelian conjecture by relating it to another recent conjecture by Sutherland and the second author. |
| title | Etale descent obstruction and anabelian geometry of curves over finite fields |
| topic | Number Theory Algebraic Geometry 11G20, 11G30, 14G05, 14G15 |
| url | https://arxiv.org/abs/2306.04844 |