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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:

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

  2. 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.)

  3. Is there any relationship between the transfer principle and Hasse principle?

Please give me any advice.

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Maybe this gives some hints wwwmath.uni-muenster.de/u/serpe/arbeiten-en.html. –  user5831 Jan 14 '12 at 10:39
    
What is the transfer principle ? –  Chandan Singh Dalawat Jan 14 '12 at 10:46
    
>Bora Yalkinoglu Thanks! These papers are interesting, but it seems to me that one has to be familiar with nonstandard methods in algebra (or algebraic geometry) in order to read them. Isn't there any more introductory papers? –  Hiro Jan 14 '12 at 11:54
    
>Chandan Singh Dalawat I meant by "transfer principle(s)" statements of the following form: all statements of some language that are true for some structure are true for another structure. I felt that there are some relations between this and Hasse principle, which connects global statements and local statements. –  Hiro Jan 14 '12 at 12:01
    
You might be interested by my answer to this question: mathoverflow.net/questions/6628 –  Denis-Charles Cisinski Apr 16 '12 at 10:38
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4 Answers

up vote 4 down vote accepted

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.

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I did not know the last PDF file ("Complex Spaces and Nonstandard schemes"). It seems introductory to me, so I will try this one first of all. Thank you so much! –  Hiro Jan 14 '12 at 11:45
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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.

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I could not find the article online, so I will find it in the library later. Thank you for recommendation! –  Hiro Jan 14 '12 at 13:34
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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.

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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)

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