Imaginaries, invariant types and pseudo 𝑝-adically closed fields

Montenegro, Samaria; Rideau-Kikuchi, Silvain

doi:10.1090/tran/8052

Imaginaries, invariant types and pseudo $p$-adically closed fields

Authors: Samaria Montenegro and Silvain Rideau-Kikuchi
Journal: Trans. Amer. Math. Soc. 374 (2021), 803-828
MSC (2020): Primary 03C60; Secondary 03C45, 03C98, 12J12
DOI: https://doi.org/10.1090/tran/8052
Published electronically: November 18, 2020
MathSciNet review: 4196378
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Abstract: In this paper, we give a general criterion for elimination of imaginaries using an abstract independence relation. We also study germs of definable functions at certain well-behaved invariant types. Finally we apply these results to prove the elimination of imaginaries in bounded pseudo $p$-adically closed fields.

References

Hans Adler, Enrique Casanovas, and Anand Pillay, Generic stability and stability, J. Symb. Log. 79 (2014), no. 1, 179–185. MR 3226018, DOI https://doi.org/10.1017/jsl.2013.29
James Ax, The elementary theory of finite fields, Ann. of Math. (2) 88 (1968), 239–271. MR 229613, DOI https://doi.org/10.2307/1970573
Şerban A. Basarab, Definite functions on algebraic varieties over ordered fields, Rev. Roumaine Math. Pures Appl. 29 (1984), no. 7, 527–535. MR 759511
Zoé Chatzidakis, Properties of forking in $\omega $-free pseudo-algebraically closed fields, J. Symbolic Logic 67 (2002), no. 3, 957–996. MR 1925952, DOI https://doi.org/10.2178/jsl/1190150143
Zoé Chatzidakis and Ehud Hrushovski, Model theory of difference fields, Trans. Amer. Math. Soc. 351 (1999), no. 8, 2997–3071. MR 1652269, DOI https://doi.org/10.1090/S0002-9947-99-02498-8
Ido Efrat and Moshe Jarden, Free pseudo $p$-adically closed fields of finite corank, J. Algebra 133 (1990), no. 1, 132–150. MR 1063387, DOI https://doi.org/10.1016/0021-8693%2890%2990074-X
Yu. L. Ershov, Relatively regularly closed fields, Dokl. Akad. Nauk 332 (1993), no. 3, 286–288 (Russian); English transl., Russian Acad. Sci. Dokl. Math. 48 (1994), no. 2, 300–303. MR 1257019
Michael D. Fried and Moshe Jarden, Field arithmetic, 3rd ed., Ergebnisse der Mathematik und ihrer Grenzgebiete. 3. Folge. A Series of Modern Surveys in Mathematics [Results in Mathematics and Related Areas. 3rd Series. A Series of Modern Surveys in Mathematics], vol. 11, Springer-Verlag, Berlin, 2008. Revised by Jarden. MR 2445111

Camilla Grob, Die entscheidbarkeit der theorie der maximalen pseudo p-adisch abgeschlossenen körper, Konstanzer Dissertationen, vol. 202, Hartung-Gorre, 1987.

Dan Haran and Moshe Jarden, The absolute Galois group of a pseudo $p$-adically closed field, J. Reine Angew. Math. 383 (1988), 147–206. MR 921990, DOI https://doi.org/10.1515/crll.1988.383.147

Deirdre Haskell, Ehud Hrushovski, and Dugald Macpherson, Definable sets in algebraically closed valued fields: elimination of imaginaries, J. Reine Angew. Math. 597 (2006), 175–236. MR 2264318, DOI https://doi.org/10.1515/CRELLE.2006.066

Ehud Hrushovski, Pseudo-finite fields and related structures, Model theory and applications, Quad. Mat., vol. 11, Aracne, Rome, 2002, pp. 151–212. MR 2159717

Ehud Hrushovski, Imaginaries and definable types in algebraically closed valued fields, Valuation theory in interaction, EMS Ser. Congr. Rep., Eur. Math. Soc., Zürich, 2014, pp. 297–319. MR 3329038

Ehud Hrushovski, Ben Martin, and Silvain Rideau, Definable equivalence relations and zeta functions of groups, J. Eur. Math. Soc. (JEMS) 20 (2018), no. 10, 2467–2537. With an appendix by Raf Cluckers. MR 3852185, DOI https://doi.org/10.4171/JEMS/817

Ehud Hrushovski and Anand Pillay, On NIP and invariant measures, J. Eur. Math. Soc. (JEMS) 13 (2011), no. 4, 1005–1061. MR 2800483, DOI https://doi.org/10.4171/JEMS/274

Ehud Hrushovski and Silvain Rideau-Kikuchi, Valued fields, metastable groups, Selecta Math. (N.S.) 25 (2019), no. 3, Paper No. 47, 58. MR 3984105, DOI https://doi.org/10.1007/s00029-019-0491-x

Moshe Jarden, Algebraic realization of $p$-adically projective groups, Compositio Math. 79 (1991), no. 1, 21–62. MR 1112279

Will Johnson, On the proof of elimination of imaginaries in algebraically closed valued fields, arXiv: 1406.3654.

Angus Macintyre, On definable subsets of $p$-adic fields, J. Symbolic Logic 41 (1976), no. 3, 605–610. MR 485335, DOI https://doi.org/10.2307/2272038

Samaria Montenegro, Imaginaries in bounded pseudo real closed fields, Ann. Pure Appl. Logic 168 (2017), no. 10, 1866–1877. MR 3670075, DOI https://doi.org/10.1016/j.apal.2017.04.003

Samaria Montenegro, Pseudo real closed fields, pseudo $p$-adically closed fields and ${\rm NTP}_2$, Ann. Pure Appl. Logic 168 (2017), no. 1, 191–232. MR 3564381, DOI https://doi.org/10.1016/j.apal.2016.09.004

Bruno Poizat, Une théorie de Galois imaginaire, J. Symbolic Logic 48 (1983), no. 4, 1151–1170 (1984) (French). MR 727805, DOI https://doi.org/10.2307/2273680

Florian Pop, Embedding problems over large fields, Ann. of Math. (2) 144 (1996), no. 1, 1–34. MR 1405941, DOI https://doi.org/10.2307/2118581

Alexander Prestel, Pseudo real closed fields, Set theory and model theory (Bonn, 1979) Lecture Notes in Math., vol. 872, Springer, Berlin-New York, 1981, pp. 127–156. MR 645909

Alexander Prestel and Martin Ziegler, Model-theoretic methods in the theory of topological fields, J. Reine Angew. Math. 299(300) (1978), 318–341. MR 491852

Silvain Rideau, Imaginaries and invariant types in existentially closed valued differential fields, J. Reine Angew. Math. 750 (2019), 157–196. MR 3943320, DOI https://doi.org/10.1515/crelle-2016-0036

Pierre Simon and Sergei Starchenko, On forking and definability of types in some DP-minimal theories, J. Symb. Log. 79 (2014), no. 4, 1020–1024. MR 3343527, DOI https://doi.org/10.1017/jsl.2014.45

Katrin Tent and Martin Ziegler, A course in model theory, Lecture Notes in Logic, vol. 40, Association for Symbolic Logic, La Jolla, CA; Cambridge University Press, Cambridge, 2012. MR 2908005

Lou van den Dries, Model theory of fields: decidability, and bounds for polynomial ideals, Ph.D. thesis, Rijksuniversiteit Utrecht, 1978.