Ido Efrat’s research while affiliated with Ben-Gurion University of the Negev and other places

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Publications (60)


Linking invariants for valuations and orderings on fields
  • Article
  • Full-text available

October 2024

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8 Reads

Research in the Mathematical Sciences

Ido Efrat

The mod-2 arithmetic Milnor invariants, introduced by Morishita, provide a decomposition law for primes in canonical Galois extensions of Q\mathbb {Q} with unitriangular Galois groups and contain the Legendre and Rédei symbols as special cases. Morishita further proposed a notion of mod-q arithmetic Milnor invariants, where q is a prime power, for number fields containing the qth roots of unity and satisfying certain class field theory assumptions. We extend this theory from the number field context to general fields, by introducing a notion of a linking invariant for discrete valuations and orderings. We further express it as a Magnus homomorphism coefficient and relate it to Massey product elements in Galois cohomology.

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The kernel generating condition and absolute Galois groups

November 2023

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6 Reads

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1 Citation

Israel Journal of Mathematics

For a list L{\cal L} of finite groups and for a profinite group G, we consider the intersection T(G) of all open normal subgroups N of G with G/N in L{\cal L}. We give a cohomological characterization of the epimorphisms π:S → G of profinite groups (satisfying some additional requirements) such that π[T(S)] = T(G). For p prime, this is used to describe cohomologically the profinite groups G whose nth term G(n,p) (resp., G(n,p)) in the p-Zassenhaus filtration (resp., lower p-central filtration) is an intersection of this form. When G = GF is the absolute Galois group of a field F containing a root of unity of order p, we recover as special cases results by Mináč, Spira and the author, describing G(3,p) and G(3,p) as T(G) for appropriate lists L{\cal L}.


The symbol length for elementary type pro-p groups and Massey products

December 2022

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17 Reads

For a prime number p and an integer m2m\geq2, we prove that the symbol length of all elements of m-fold Massey products in H2(G,Fp)H^2(G,\mathbb{F}_p), for pro-p groups G of elementary type, is bounded by (m2/4)+m(m^2/4)+m. Assuming the Elementary Type Conjecture, this applies to all finitely generated maximal pro-p Galois groups G=GF(p)G=G_F(p) of fields F which contain a root of unity of order p. More generally, we provide such a uniform bound for the symbol length of all pullbacks ρ(ωˉ)\rho^*(\bar\omega) of a given cohomology element ωˉHn(Gˉ,Fp)\bar\omega\in H^n(\bar G,\mathbb{F}_p), where Gˉ\bar G is a finite p-group, n2n\geq2, and ρ ⁣:GGˉ\rho\colon G\to \bar G is a pro-p group homomorphism.


Generalized Steinberg relations

October 2022

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14 Reads

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2 Citations

Research in Number Theory

We consider a field F and positive integers n, m, such that m is not divisible by Char(F)\mathrm {Char}(F) and is prime to n!. The absolute Galois group GFG_F acts on the group Un(Z/m)\mathbb {U}_n(\mathbb {Z}/m) of all (n+1)×(n+1)(n+1)\times (n+1) unipotent upper-triangular matrices over Z/m\mathbb {Z}/m cyclotomically. Given 0,1zF0,1\ne z\in F and an arbitrary list w of n Kummer elements (z)F(z)_F, (1z)F(1-z)_F in H1(GF,μm)H^1(G_F,\mu _m), we construct in a canonical way a quotient Uw\mathbb {U}_w of Un(Z/m)\mathbb {U}_n(\mathbb {Z}/m) and a cohomology element ρz\rho ^z in H1(GF,Uw)H^1(G_F,\mathbb {U}_w) whose projection to the superdiagonal is the prescribed list. This extends results by Wickelgren, and in the case n=2 recovers the Steinberg relation in Galois cohomology, proved by Tate.


Cohomology and the Combinatorics of Words for Magnus Formations

May 2022

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33 Reads

For a prime number p and a free pro-p group G on a totally ordered basis X, we consider closed normal subgroups GΦG^\Phi of G which are generated by p-powers of iterated commutators associated with Lyndon words in the alphabet X. We express the profinite cohomology group H2(G/GΦ)H^2(G/G^\Phi) combinatorically, in terms of the shuffle algebra on X. This partly extends existing results for the lower p-central and p-Zassenhaus filtrations of G.


THE p -ZASSENHAUS FILTRATION OF A FREE PROFINITE GROUP AND SHUFFLE RELATIONS

September 2021

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30 Reads

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4 Citations

Journal of the Institute of Mathematics of Jussieu

For a prime number p and a free profinite group S on the basis X , let S(n,p)S_{\left (n,p\right )} , n=1,2,,n=1,2,\dotsc , be the p -Zassenhaus filtration of S . For p>np>n , we give a word-combinatorial description of the cohomology group H2(S/S(n,p),Z/p)H^2\left (S/S_{\left (n,p\right )},\mathbb {Z}/p\right ) in terms of the shuffle algebra on X . We give a natural linear basis for this cohomology group, which is constructed by means of unitriangular representations arising from Lyndon words.


Generalized Steinberg Relations

September 2021

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26 Reads

We consider a field F and positive integers n, m, such that m is not divisible by Char(F)\mathrm{Char}(F) and is prime to n!. The absolute Galois group GFG_F acts on the group Un(Z/m)\mathbb{U}_n(\mathbb{Z}/m) of all (n+1)×(n+1)(n+1)\times(n+1) unipotent upper-triangular matrices over Z/m\mathbb{Z}/m cyclotomically. Given 0,1zF0,1\neq z\in F and an arbitrary list w of n Kummer elements (z)F(z)_F, (1z)F(1-z)_F in H1(GF,μm)H^1(G_F,\mu_m), we construct in a canonical way a quotient Uw\mathbb{U}_w of Un(Z/m)\mathbb{U}_n(\mathbb{Z}/m) and a cohomology element ρz\rho^z in H1(GF,Uw)H^1(G_F,\mathbb{U}_w) whose projection to the superdiagonal is the prescribed list. This extends results by Wickelgren, and in the case n=2 recovers the Steinberg relation in Galois cohomology, proved by Tate.


The kernel generating condition and absolute Galois groups

June 2021

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23 Reads

For a list L\cal{L} of finite groups and for a profinite group G, we consider the intersection T(G) of all open normal subgroups N of G with G/N in L\cal{L}. We give a cohomological characterization of the epimorphisms π ⁣:SG\pi\colon S\to G of profinite groups (satisfying some additional requirements) such that π[T(S)]=T(G)\pi[T(S)]=T(G). For p prime, this is used to describe cohomologically the profinite groups G whose nth term G(n,p)G_{(n,p)} (resp., G(n,p)G^{(n,p)}) in the p-Zassenhaus filtration (resp., lower p-central filtration) is an intersection of this form. When G=GFG=G_F is the absolute Galois group of a field F containing a root of unity of order p, we recover as special cases results by Minac, Spira and the author, describing G(3,p)G_{(3,p)} and G(3,p)G^{(3,p)} as T(G) for appropriate lists L\cal{L}.


3-fold Massey products in Galois cohomology -- The non-prime case

October 2020

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12 Reads

For m2m\geq2, let F be a field of characteristic prime to m and containing the roots of unity of order m, and let GFG_F be its absolute Galois group. We show that the 3-fold Massey products χ1,χ2,χ3\langle\chi_1,\chi_2,\chi_3\rangle, with χ1,χ2,χ3H1(GF,Z/m)\chi_1,\chi_2,\chi_3\in H^1(G_F,\mathbb{Z}/m) and χ1,χ3\chi_1,\chi_3 Z/m\mathbb{Z}/m-linearly independent, are non-essential. This was earlier proved for m prime. Our proof is based on the study of unitriangular representations of GFG_F.


Citations (35)


... The pullbackρ * (ω m ) corresponds to the upper central extension [12, Remark 6.1]. One hasπ | N ∈ H 1 (N ) G and trg(π | N ) = ρ * (ω m ), by [11,Prop. 4.1 and Remark 4.2(4)] (which is based on results of Hoechsmann [16]). The assertion now follows from the definition of the pairing (·, ·). ...

Reference:

Linking invariants for valuations and orderings on fields
The kernel generating condition and absolute Galois groups
  • Citing Article
  • November 2023

Israel Journal of Mathematics

... In the case n = 3, it refines results by Vogel [37, §2]. The paper builds upon our earlier work [8], supplemented by [9], where we proved analogous results for the lower p-central filtration, defined inductively by G (1,p) = G and G (n,p) = G (n−1,p) p G,G (n−1,p) for n ≥ 2. In many respects, this filtration and the p-Zassenhaus filtration are the opposite extremes among the filtrations related to mod-p cohomology. ...

The lower p-central series of a free profinite group and the Shuffle algebra
  • Citing Article
  • October 2019

Journal of Pure and Applied Algebra

... Presentations of (prop) groups via generators and relations have played an important role in the development of group theory (see [21,Chapter 2] and [22]), and more generally in the current theory of profinite groups and especially prop groups. These methods, combined with cohomological results, are also used to detect Galois groups of p-extensions, see for instance Koch [15], Mináč-Rogelstad-Tân [27] and [26], and Efrat-Quadrelli [7]. ...

The Kummerian Property and Maximal Pro-p Galois Groups

Journal of Algebra

... Then, for a more general profinite group G (such as G F ), one takes a profinite presentation, i.e., a continuous epimorphism π : S → G, where S is a free profinite group, and transfers the equality (1.1) from S to G. The first part is purely group-theoretic, and is usually proved using Magnus theory, i.e., by viewing the elements of G = S as formal power series. A general machinery to obtain such results in the free profinite case is given in [Efr14b] (see also [CE16]). ...

Filtrations of free groups arising from the lower central series
  • Citing Article
  • January 2016

Journal of Group Theory

... The most classical result in this direction is due to Artin and Schreier [Art24,AS27], who proved that every non-trivial finite subgroup of an absolute Galois group is cyclic of order 2. A deeper necessary condition is the Bloch-Kato conjecture, now a theorem due to Voevodsky and Rost [HW19], which in particular implies that the mod p cohomology ring of an absolute Galois group of a field containing a primitive p-th root of unity is generated in degree 1 with relations in degree 2. As a more recent example, the Massey vanishing conjecture of Mináč and Tân [MT17] predicts that all non-empty Massey products of n ≥ 3 elements in the mod p Galois cohomology of fields contain the zero element. This conjecture has been proved in several cases [HW15,MT15,MT16,EM17,HW23,MS22,MS23], and each of these theorems provides new restrictions on the cohomology of absolute Galois groups of fields. ...

Triple Massey products and absolute Galois groups

Journal of the European Mathematical Society

... Mináč and Tân extended this to the case where m = 2 and F is an arbitrary field [MT17], as well as to the case where m = p is an arbitrary prime and F is global or local [MT15b]. Alternative proofs of the latter two results were given in [EfMa15]. In fact, it was noted in [EfMa15] that the result for m = 2 is a cohomological reformulation of a classical characterization of bi-quaternionic algebras, due to Albert [Alb39]. ...

Vanishing of Massey Products and Brauer Groups

Canadian mathematical bulletin = Bulletin canadien de mathématiques