Nonstandard Methods ( or Model Theory ) and Arithmetic Geometry I hear that the nonstandard methods are applied to many problems in various fields of mathematics such as functional analysis, topology, probability theory and so on.
So, I have become interested in using nonstandard methods to my research areas, which are in and around arithmetic geometry.
Questions:


*

*What kind of useful applications of nonstandard methods to arithmetic geometry exist?

*Is there any recommendation of introductory textbook or PDF file to study nonstandard methods in arithmetic geometry? (I heve studied the "nonstandard analysis" to a certain extent: construction of ultraproducts, the transfer principle etc. But I have few knowledge of nonstandard methods for algebra or algebraic geometry.)

*Is there any relationship between the transfer principle and Hasse principle?
Please give me any advice.
 A: One "classical" application of nonstandard methods to number theory is the approach of Robinson and Roquette to the Siegel-Mahler-Thue Theorem.
Robinson, A.; Roquette, P.
On the finiteness theorem of Siegel and Mahler concerning Diophantine equations. 
J. Number Theory 7 (1975), 121–176. 
A: Try this: www.dpmms.cam.ac.uk/~cb496/nsag1.pdf
 and also this http://wwwmath.uni-muenster.de/u/serpe/documents/ultramath2008serpe-nonstandard-handout.pdf, logicandanalysis.org/index.php/jla/article/view/77/29
 and references therein.
A: Since you also mention algebraic geometry, you may want to take a look at "Model Theory and Algebraic Goemetry" edited by Elizabeth Bouscaren (Springer LNM 1696). Its primary purpose is to introduce Hrushovki's proof of Mordell-Lang for function fields. Except for some elementary model theory, it is self-contained, and shows that deep results in model theory can be used to prove nontrivial statements. Although this might be more geometric than arithmetic, perhaps some of the techniques could still be useful to you.
A: An ongoing conference on that: http://www.mpim-bonn.mpg.de/node/3515
(It would be great if slides or texts from it were put online)
