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.

transfer principle? – Chandan Singh Dalawat Jan 14 '12 at 10:46