On subgroups of semi-abelian varieties defined by difference equations

Chatzidakis, Zoé; Hrushovski, Ehud

doi:10.1090/tran/6924

On subgroups of semi-abelian varieties defined by difference equations
HTML articles powered by AMS MathViewer

by Zoé Chatzidakis and Ehud Hrushovski PDF

Trans. Amer. Math. Soc. 369 (2017), 3673-3705 Request permission

Abstract:

We study the induced structure on definable groups in existentially closed difference fields. If $G$ is a definable subgroup of a semi-abelian variety, orthogonal to every definable field, we show that $G$ is stable and stably embedded; every definable subset of $G^n$ is a Boolean combination of cosets of definable subgroups of $G^n$, and $G^n$ has at most countably many definable subgroups. This generalises to positive characteristic earlier results of the authors.

References

Shreeram Abhyankar, On the ramification of algebraic functions, Amer. J. Math. 77 (1955), 575–592. MR 71851, DOI 10.2307/2372643
Zoé Chatzidakis and Ehud Hrushovski, Model theory of difference fields, Trans. Amer. Math. Soc. 351 (1999), no. 8, 2997–3071. MR 1652269, DOI 10.1090/S0002-9947-99-02498-8
Zoé Chatzidakis, Ehud Hrushovski, and Ya’acov Peterzil, Model theory of difference fields. II. Periodic ideals and the trichotomy in all characteristics, Proc. London Math. Soc. (3) 85 (2002), no. 2, 257–311. MR 1912052, DOI 10.1112/S0024611502013576
Richard M. Cohn, Difference algebra, Interscience Publishers John Wiley & Sons, New York-London-Sydeny, 1965. MR 0205987
Françoise Delon, Hensel fields in equal characteristic $p>0$, Model theory of algebra and arithmetic (Proc. Conf., Karpacz, 1979), Lecture Notes in Math., vol. 834, Springer, Berlin, 1980, pp. 108–116. MR 606782
Otto Endler, Valuation theory, Universitext, Springer-Verlag, New York-Heidelberg, 1972. To the memory of Wolfgang Krull (26 August 1899–12 April 1971). MR 0357379
Antonio J. Engler and Alexander Prestel, Valued fields, Springer Monographs in Mathematics, Springer-Verlag, Berlin, 2005. MR 2183496
Robin Hartshorne, Algebraic geometry, Graduate Texts in Mathematics, No. 52, Springer-Verlag, New York-Heidelberg, 1977. MR 0463157
Ehud Hrushovski, The Manin-Mumford conjecture and the model theory of difference fields, Ann. Pure Appl. Logic 112 (2001), no. 1, 43–115. MR 1854232, DOI 10.1016/S0168-0072(01)00096-3
E. Hrushovski, The first-order theory of the Frobenius, preprint (1996).
Ehud Hrushovski and James Loveys, Strongly and co-strongly minimal abelian structures, J. Symbolic Logic 75 (2010), no. 2, 442–458. MR 2648150, DOI 10.2178/jsl/1268917489
U. Hrushovski and A. Pillay, Weakly normal groups, Logic colloquium ’85 (Orsay, 1985) Stud. Logic Found. Math., vol. 122, North-Holland, Amsterdam, 1987, pp. 233–244. MR 895647, DOI 10.1016/S0049-237X(09)70556-7
Ehud Hrushovski and Anand Pillay, Groups definable in local fields and pseudo-finite fields, Israel J. Math. 85 (1994), no. 1-3, 203–262. MR 1264346, DOI 10.1007/BF02758643
Byunghan Kim and Anand Pillay, Simple theories, Ann. Pure Appl. Logic 88 (1997), no. 2-3, 149–164. Joint AILA-KGS Model Theory Meeting (Florence, 1995). MR 1600895, DOI 10.1016/S0168-0072(97)00019-5
Piotr Kowalski and A. Pillay, A note on groups definable in difference fields, Proc. Amer. Math. Soc. 130 (2002), no. 1, 205–212. MR 1855638, DOI 10.1090/S0002-9939-01-06004-X
Franz-Viktor Kuhlmann, Elimination of ramification I: the generalized stability theorem, Trans. Amer. Math. Soc. 362 (2010), no. 11, 5697–5727. MR 2661493, DOI 10.1090/S0002-9947-2010-04973-6
Serge Lang, Unramified class field theory over function fields in several variables, Ann. of Math. (2) 64 (1956), 285–325. MR 83174, DOI 10.2307/1969975
Serge Lang, Introduction to algebraic geometry, Interscience Publishers, Inc., New York-London, 1958. MR 0100591
Serge Lang and Jean-Pierre Serre, Sur les revêtements non ramifiés des variétés algébriques, Amer. J. Math. 79 (1957), 319–330 (French). MR 87207, DOI 10.2307/2372685
James S. Milne, Étale cohomology, Princeton Mathematical Series, No. 33, Princeton University Press, Princeton, N.J., 1980. MR 559531
David Mumford, The red book of varieties and schemes, Second, expanded edition, Lecture Notes in Mathematics, vol. 1358, Springer-Verlag, Berlin, 1999. Includes the Michigan lectures (1974) on curves and their Jacobians; With contributions by Enrico Arbarello. MR 1748380, DOI 10.1007/b62130
Abraham Robinson, Complete theories, North-Holland Publishing Co., Amsterdam, 1956. MR 0075897
O. F. G. Schilling, The Theory of Valuations, Mathematical Surveys, No. 4, American Mathematical Society, New York, N. Y., 1950. MR 0043776
Igor R. Shafarevich, Basic algebraic geometry. 1, 2nd ed., Springer-Verlag, Berlin, 1994. Varieties in projective space; Translated from the 1988 Russian edition and with notes by Miles Reid. MR 1328833
Volker Weispfenning, Quantifier elimination and decision procedures for valued fields, Models and sets (Aachen, 1983) Lecture Notes in Math., vol. 1103, Springer, Berlin, 1984, pp. 419–472. MR 775704, DOI 10.1007/BFb0099397