Massey products \(\langle y,x,x,\ldots ,x,x,y\rangle\) in Galois cohomology via rational points
A smooth curve \(X\) over a field \(k\) with a choice of basepoint can be embedded into its generalized Jacobian, Jac \(X\). Given a rational point of \(X\) there is a commutative diagram between Spec \(k\), \(X\), and Jac \(X\). Applying the functor \(\pi_1^{et}\) we obtain a diagram where we ask about the lifting of a map. For \(X= P_k^1\setminus\{0,1,\infty\}\) the obstructions for such a lifting are higher order Massey products. The whole study is motivated by Grothendieck's section conjecture, and Belyi's theorem suggests to study such \(X\). The main result is as follows. Theorem. Let \(x\) be an element of \(k^*\setminus \{1\}\). The \(n\)-th order Massey product \[ \langle (1-x)^{-1},x^{-1},\dots,x^{-1},(1-x)^{-1}\rangle \] contains \(0\) and \(-[f]\cup (1-x)^{-1}\) (where \(f\) is a certain cocycle). Let \(x\) be an element of \(k^*\). The \(n\)-th order Massey product \[ \langle -x^{-1},x^{-1},\dots,x^{-1},-x^{-1}\rangle \] contains \(0\) and \(-([f]\cup (-x)^{-1})\).
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