The geometric fundamental group of the affine line over a finite field
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arXiv
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| Format: | Preprint |
| Publié: |
2024
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| _version_ | 1866909288051507200 |
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| author | Russell, Henrik |
| author_facet | Russell, Henrik |
| contents | The affine line and the punctured affine line over a finite field F are taken as benchmarks for the problem of describing geometric étale fundamental groups. To this end, using a reformulation of Tannaka duality we construct for a projective variety X a (non-commutative) universal affine pro-algebraic group Lu(X), such that for any given affine subvariety U of X any finite and étale Galois covering of U over F is a pull-back of a Galois covering of a quotient Lu(X,U) of Lu(X). Then the geometric fundamental group of U is a completion of the k-points of Lu(X,U), where k is an algebraic closure of F. We obtain explicit descriptions of the universal affine groups Lu(X,U) for U the affine line and the punctured affine line over F. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2408_08175 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | The geometric fundamental group of the affine line over a finite field Russell, Henrik Algebraic Geometry Number Theory 14H30, 14L17, 14G15 The affine line and the punctured affine line over a finite field F are taken as benchmarks for the problem of describing geometric étale fundamental groups. To this end, using a reformulation of Tannaka duality we construct for a projective variety X a (non-commutative) universal affine pro-algebraic group Lu(X), such that for any given affine subvariety U of X any finite and étale Galois covering of U over F is a pull-back of a Galois covering of a quotient Lu(X,U) of Lu(X). Then the geometric fundamental group of U is a completion of the k-points of Lu(X,U), where k is an algebraic closure of F. We obtain explicit descriptions of the universal affine groups Lu(X,U) for U the affine line and the punctured affine line over F. |
| title | The geometric fundamental group of the affine line over a finite field |
| topic | Algebraic Geometry Number Theory 14H30, 14L17, 14G15 |
| url | https://arxiv.org/abs/2408.08175 |