Model theory of difference fields with an additive character on the fixed field
Fuente:
arXiv
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| Natura: | Preprint |
| Pubblicazione: |
2025
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| _version_ | 1866915669256175616 |
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| author | Ludwig, Stefan Marian |
| author_facet | Ludwig, Stefan Marian |
| contents | Following a research line proposed by Hrushovski in his work on pseudofinite fields with an additive character, we investigate the theory $\mathrm{ACFA}^{+}$ which is the model companion of the theory of difference fields with an additive character on the fixed field added as a continuous logic predicate. $\mathrm{ACFA}^{+}$ is the common theory (in characteristic $0$) of the algebraic closure of finite fields with the Frobenius automorphism and the standard character on the fixed field and turns out to be a simple theory. We fully characterise 3-amalgamation and deduce that the connected component of the Kim-Pillay group (for any completion of $\mathrm{ACFA}^{+}$) is abelian as conjectured by Hrushovski. Finally, we describe a natural expansion of $\mathrm{ACFA}^{+}$ in which geometric elimination of continuous logic imaginaries holds. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2512_10583 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Model theory of difference fields with an additive character on the fixed field Ludwig, Stefan Marian Logic Primary 03C66, 03C60, Secondary 12L10 Following a research line proposed by Hrushovski in his work on pseudofinite fields with an additive character, we investigate the theory $\mathrm{ACFA}^{+}$ which is the model companion of the theory of difference fields with an additive character on the fixed field added as a continuous logic predicate. $\mathrm{ACFA}^{+}$ is the common theory (in characteristic $0$) of the algebraic closure of finite fields with the Frobenius automorphism and the standard character on the fixed field and turns out to be a simple theory. We fully characterise 3-amalgamation and deduce that the connected component of the Kim-Pillay group (for any completion of $\mathrm{ACFA}^{+}$) is abelian as conjectured by Hrushovski. Finally, we describe a natural expansion of $\mathrm{ACFA}^{+}$ in which geometric elimination of continuous logic imaginaries holds. |
| title | Model theory of difference fields with an additive character on the fixed field |
| topic | Logic Primary 03C66, 03C60, Secondary 12L10 |
| url | https://arxiv.org/abs/2512.10583 |