1 Richard M., Cohn, Difference algebra, Interscience Publishers John Wiley & Sons, New York-London-Sydney, 1965.

2 Serge, Lang, Introduction to algebraic geometry, Addison-Wesley Publishing Co., Inc., Reading, Mass., 1973. Fourth printing.Model theoretic results on difference fields:

3 Angus, Macintyre, Generic automorphisms of fields, Annals of Pure and Applied Logic, vol. 88 (1997), no. 2–3, pp. 165–180, Joint AILA-KGSModel TheoryMeeting (Florence, 1995.Google Scholar

4 Zoé, Chatzidakis and Ehud, Hrushovski, Model theory of difference fields, Transactionsof the American Mathematical Society, vol. 351 (1999), no. 8, pp. 2997–3071.

5 Z., Chatzidakis, E., Hrushovski, and Y., Peterzil, Model theory of difference fields, II:Periodic ideals and the trichotomy in all characteristics, preprint 1999, Proceedings of TheLondon Mathematical Society, vol. 85 (2002), pp. 257–311.

6 Ehud, Hrushovski, The Manin-Mumford conjecture and the model theory of differencefields, Annals of Pure and Applied Logic, vol. 112 (2001), no. 1, pp. 43–115.

7 Ehud, Hrushovski, The first-order theory of the Frobenius, preprint, 1996.

8 A., Macintyre, Nonstandard Frobenius, in preparation. The papers [5] to [7] definitely fall into the category of “further advanced reading”. Some of the easy parts of [6] appear in survey papers, see below. Related model-theoretic results on finite fields:

9 James, Ax, The elementary theory of finite fields, Annals of Mathematics (2), vol. 88 (1968), pp. 239–271.Google Scholar

10 E., Hrushovski, Pseudo-finite fields and related structures, manuscript, 1991.

11 Ehud, Hrushovski and Anand, Pillay, Groups definable in local fields and pseudo-finitefields, Israel Journal of Mathematics, vol. 85 (1994), no. 1–3, pp. 203–262.Google Scholar

12 E., Hrushovski and A., Pillay, Definable subgroups of algebraic groups over finite fields, Journal für die Reine und Angewandte Mathematik, vol. 462 (1995), pp. 69–91. The proofs of many of the results of the first fewsections of these notes are essentially translations of proofs appearing in [10]. See also [11]. In [12], one uses results of [11] to obtain very pretty results on groups definable in finite fields.Other model-theoretic results:Google Scholar

13 L., van den Dries and K., Schmidt, Bounds in the theory of polynomial rings over fields. Anapproach, Inventiones Mathematicae, vol. 76 (1984), no. 1, pp. 77–91.Google Scholar

14 E., Hrushovski and A., Pillay, Weakly normal groups, Logic colloquium ‘85 (Orsay, 1985., North-Holland, Amsterdam, 1987. pp. 233–244.

15 Byunghan, Kim and Anand, Pillay, Simple theories, Annals of Pure and Applied Logic, vol. 88 (1997), no. 2–3, pp. 149–164, Joint AILA-KGSModel TheoryMeeting (Florence, 1995.Google Scholar

16 Anand, Pillay, Definability and definable groups in simple theories, The Journal of SymbolicLogic, vol. 63 (1998), no. 3, pp. 788–796. Stability is a vast subject with an abundant literature. Here are some books for the interested reader, but there are many others.Google Scholar

17 Steven, Buechler, Essential stability theory, Perspectives inMathematical Logic, Springer- Verlag, Berlin, 1996.

18 Anand, Pillay, An introduction to stability theory, The Clarendon Press Oxford University Press, New York, 1983.

19 Frank O., Wagner, Simple theories, Kluwer Academic Publishers, Dordrecht, 2000.

20 E., Bouscaren, Théorie des modèles et conjecture de Manin-Mumford, [d'après Ehud Hrushovski], Séminaire Bourbaki vol. 1999/2000, Astérisque, no. 276 (2002), pp. 137–159.Google Scholar

21 Zoé, Chatzidakis, Groups definable in ACFA, Algebraic model theory (Toronto, ON, 1996. (B., Hart et al., editors), Kluwer Academic Publishers, Dordrecht, 1997. pp. 25–52.

22 Zoé, Chatzidakis, A survey on the model theory of difference fields, Model theory, algebra,and geometry (Steinhorn, Haskell, Pillay, editors),MSRI Publications, vol. 39, Cambridge University Press, 2000. pp. 65–96.

23 Anand, Pillay,ACFAand theManin-Mumford conjecture, Algebraic model theory (Toronto,ON, 1996., (B. Hart et al., editors) Kluwer Academic Publishers, Dordrecht, 1997. pp. 195–205. After these notes were written, progress was made by several people. Some results were simplified or generalised (Proposition6.17 in [24], the dichotomy Theorem 8.2 in characteristic 0 in [25]). Other results of interest appear in [26], [27] and [28].

24 Frank O., Wagner, Some remarks on one-basedness, The Journal of Symbolic Logic, vol. 69 (2004), no. 1, pp. 34–38.Google Scholar

25 Anand, Pillay and Martin, Ziegler, Jet spaces of varieties over differential and differencefields, Selecta Mathematica. New Series, vol. 9 (2003), no. 4, pp. 579–599.Google Scholar

26 Thomas, Scanlon, Diophantine geometry of the torsion of a Drinfeld module, Journal ofNumber Theory, vol. 97 (2002), no. 1, pp. 10–25.Google Scholar

27 Richard, Pink and Damian, Roessler, On Hrushovski's proof of the Manin-Mumfordconjecture, Proceedings of the International Congress of Mathematicians, (Beijing, 2002.), vol. I, pp. 539–546, Higher Education Press, Beijing, 2002.

28 Anand, Pillay, Mordell-Lang conjectures for function fields in characteristic zero, revisited, Compositio Mathematica, vol. 140 (2004), no. 1, pp. 64–68.Google Scholar