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Archive for November, 2014

The problem of finding solutions to Diophantine equations can be recast in the following abstract form. Let R be a commutative ring, which in the most classical case might be a number field like \mathbb{Q} or the ring of integers in a number field like \mathbb{Z}. Suppose we want to find solutions, over R, of a system of polynomial equations

\displaystyle f_1 = \dots = f_m = 0, f_i \in R[x_1, \dots x_n].

Then it’s not hard to see that this problem is equivalent to the problem of finding R-algebra homomorphisms from S = R[x_1, \dots x_n]/(f_1, \dots f_m) to R. This is equivalent to the problem of finding left inverses to the morphism

\displaystyle R \to S

of commutative rings making S an R-algebra, or more geometrically equivalent to the problem of finding right inverses, or sections, of the corresponding map

\displaystyle \text{Spec } S \to \text{Spec } R

of affine schemes. Allowing \text{Spec } S to be a more general scheme over \text{Spec } R can also capture more general Diophantine problems.

The problem of finding sections of a morphism – call it the section problem – is a problem that can be stated in any category, and the goal of this post is to say some things about the corresponding problem for spaces. That is, rather than try to find sections of a map between affine schemes, we’ll try to find sections of a map f : E \to B between spaces; this amounts, very roughly speaking, to solving a “topological Diophantine equation.” The notation here is meant to evoke a particularly interesting special case, namely that of fiber bundles.

We’ll try to justify the section problem for spaces both as an interesting problem in and of itself, capable of encoding many other nontrivial problems in topology, and as a possible source of intuition about Diophantine equations. In particular we’ll discuss what might qualify as topological analogues of the Hasse principle and the Brauer-Manin obstruction.

(more…)

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