Rice University Department of Mathematics Colloquium

Integral solutions of polynomial equations and reciprocity laws

4:00 pm Thursday, September 1st, 2016
Jennifer Berg (Rice University)

Abstract: One problem of great interest in number theory is determining whether a polynomial equation has a rational or integral solution. A necessary first step is to determine whether the equation is "locally soluble", that is, to find solutions with coordinates in the real numbers and modulo each positive integer. However, local solubility is generally not sufficient to guarantee a "global" (integer or rational) solution. In order for local solutions to come from a common global solution, it turns out that they must satisfy certain compatibility conditions that can arise from quadratic reciprocity and higher reciprocity laws. These conditions are known as the Brauer-Manin obstruction. In this talk, I will provide many examples of equations that fail to have global solutions despite the existence of local solutions, and explain how they fit into this framework. Then I will describe recent work on computing this obstruction for a particular family of surfaces defined over the integers.

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