200701Xu Fei
 
Title: Brauer-Manin obstruction for integral points.

Time&Place: Jan. 24 2:00am-4:00am (Beijing Normal University, 北师大教八楼(数学楼)214室)

Speaker: Xu Fei (Academic China)

Abstract: Most classical methods, for example circle method, are purely local. Namely the local-global principle holds. However one can not expect this principle is true in general. In this talk, I'll explain how to refine the local-principle by using the reciprocity law. This is simply analogue to study the rational points case developed by Manin, Swinnerton-Dyer, Sansuc, Colliot-Thelene, Borovoi, Skorobogatov, Harari ... and so on. Our main result is that the integral Brauer-Manin obstruction is the only obstruction from local to global for the integral points of schemes of finite type over ring of integers of number fields, whose generic fiber are the homogenous spaces of semi-simple, simply connected algebraic groups of non-compact type. Some interesting examples will be provided. As application, the sum of three integral squares over imaginary quadratic fields and cyclotomic fields are determined. This is a joint work with Colliot-Thelene and Dasheng Wei.

Lecture Notes

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发件人 Xu Fei 2007-1