Massey products 〈y,x,x,…,x,x,y〉 in Galois cohomology via rational points

Abstract

Abstract. For x an element of a field other than 0 or 1, we compute the order n Massey productsof n − 2 factors of x −1 and two factors of (1 − x) −1 by embedding P 1 − {0, 1, ∞} into its Picard variety and constructing Gal(k s /k) equivariant maps from π et 1 applied to this embedding to unipotent matrix groups. This method produces obstructions to π 1 -sections of P 1 − {0, 1, ∞}, partial computations of obstructions of Jordan Ellenberg, and also computes the Massey products

Generalized Steinberg Relations

“…), and we recover Wickelgren's result in [Wic12b]. Finally, when w = (yxx • • • xxy) we have U w = U n (Z/m), recovering the above-mentioned result from [Wic17].…”

“…As in [Wic12b] and [Wic17], we also obtain an analogous result, where the Kummer elements (z) F , (1−z) F are replaced by the elements (z) F , (−z) F -See Remark 7.3. When n = 2 this recovers the identity (−z) F ∪ (z) F = 0, which is a formal consequence of Tate's relation [Efr06,Prop.…”

“…The next discussion follows closely [Wic12a, §12.2.1], [Wic12b], and [Wic17], where one can find more details. In particular, we refer to these papers, as well as to [Del89, §15], [Nak99], for the notion of tangential base points.…”

“…Remark 5.1. [Wic12b] and [Wic17] work with α x = 1 for every x ∈ X, which requires the assumption that n! is invertible in A.…”

“…and is (z) F for all other values of 1 ≤ i ≤ n. Here it is assumed that Char(F ) = 0 and that m is prime to n!. In a subsequent work [Wic17], she also constructs (in an arbitrary characteristic not dividing m) a 1-cocycle ρ z such that (pr i,i+1 ) * [ρ z ] is (1 − z) F for i = 1 and i = n and is (z) F for 2 ≤ i ≤ n − 1. However this time the canonical continuous 1-cocycle ρ z is into a quotient group U n (Z/m) of U n (Z/m), which consists of all unipotent upper-triangular (n+1)×(n+1)-matrices over Z/m but with the (1, n+1)entry omitted.…”