The Elementary Theory of the Frobenius Automorphisms
A Frobenius difference field is an algebraically closed field of characteristic $p>0$, enriched with a symbol for $x \mapsto x^{p^m}$. We study a sentence or formula in the language of fields with a distinguished automorphism, interpreted in Frobenius difference fields with $p$ or $m$ tending...
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| Format: | Journal article |
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2004
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| _version_ | 1826305755144257536 |
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| author | Hrushovski, E |
| author_facet | Hrushovski, E |
| author_sort | Hrushovski, E |
| collection | OXFORD |
| description | A Frobenius difference field is an algebraically closed field of characteristic $p>0$, enriched with a symbol for $x \mapsto x^{p^m}$. We study a sentence or formula in the language of fields with a distinguished automorphism, interpreted in Frobenius difference fields with $p$ or $m$ tending to infinity. In particular, a decision procedure is found to determine when a sentence is true in almost every Frobenius difference field. This generalizes Cebotarev's density theorem and Weil's Riemann hypothesis for curves (both in qualitative versions), but hinges on a result going slightly beyond the latter. The setting for the proof is the geometry of difference varieties of transformal dimension zero; these generalize algebraic varieties, and are shown to have a rich structure, only partly explicated here. Some applications are given, in particular to finite simple groups, and to the Jacobi bound for difference equations. |
| first_indexed | 2024-03-07T06:37:39Z |
| format | Journal article |
| id | oxford-uuid:f8341253-3d79-4b1b-92e9-7294026decff |
| institution | University of Oxford |
| last_indexed | 2024-03-07T06:37:39Z |
| publishDate | 2004 |
| record_format | dspace |
| spelling | oxford-uuid:f8341253-3d79-4b1b-92e9-7294026decff2022-03-27T12:48:31ZThe Elementary Theory of the Frobenius AutomorphismsJournal articlehttp://purl.org/coar/resource_type/c_dcae04bcuuid:f8341253-3d79-4b1b-92e9-7294026decffSymplectic Elements at Oxford2004Hrushovski, EA Frobenius difference field is an algebraically closed field of characteristic $p>0$, enriched with a symbol for $x \mapsto x^{p^m}$. We study a sentence or formula in the language of fields with a distinguished automorphism, interpreted in Frobenius difference fields with $p$ or $m$ tending to infinity. In particular, a decision procedure is found to determine when a sentence is true in almost every Frobenius difference field. This generalizes Cebotarev's density theorem and Weil's Riemann hypothesis for curves (both in qualitative versions), but hinges on a result going slightly beyond the latter. The setting for the proof is the geometry of difference varieties of transformal dimension zero; these generalize algebraic varieties, and are shown to have a rich structure, only partly explicated here. Some applications are given, in particular to finite simple groups, and to the Jacobi bound for difference equations. |
| spellingShingle | Hrushovski, E The Elementary Theory of the Frobenius Automorphisms |
| title | The Elementary Theory of the Frobenius Automorphisms |
| title_full | The Elementary Theory of the Frobenius Automorphisms |
| title_fullStr | The Elementary Theory of the Frobenius Automorphisms |
| title_full_unstemmed | The Elementary Theory of the Frobenius Automorphisms |
| title_short | The Elementary Theory of the Frobenius Automorphisms |
| title_sort | elementary theory of the frobenius automorphisms |
| work_keys_str_mv | AT hrushovskie theelementarytheoryofthefrobeniusautomorphisms AT hrushovskie elementarytheoryofthefrobeniusautomorphisms |