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LDM70

TOPICS ON GROUPS AND THEIR REPRESENTATIONS

The Kummerian Property for pro-p-groups

CLAUDIO QUADRELLI with Ido Efrat Ben-Gurion Univ. University of MILANO BICOCCA

October 10th 2017 Garda Lake - Palazzo Feltrinelli

Profinite groups Representations of dim 1 The Kummerian property Cohomology References

Vladimir Voevodsky

4 Jun. 1966 – 30 Sep. 2017 Fields medalist in 2002

Claudio Quadrelli Universit` a Milano-Bicocca

Profinite groups Representations of dim 1 The Kummerian property Cohomology References

Profinite groups

A profinite group is a topological group which (1) is compact; (2) is totally disconnected; (3) has a basis {Ni G} of open neighb.hds of 1 Equivalently: G = lim ← −i Gi, with |Gi| < ∞; or G ≃ Gal(E | F) for some Galois extension E | F

Holy Grail of Galois Theory

Which profinite G is Gal(¯ Fs | F)? We have some restrictions: if such G is finite, then |G| = 2 (Artin-Schreier Theorem)

Claudio Quadrelli Universit` a Milano-Bicocca

Profinite groups Representations of dim 1 The Kummerian property Cohomology References

Pro-p-groups

G profinite is pro-p if [G : U] = pn for every open U ⊆ G; equivalently G = lim ← −i Pi with |Pi| = pni They behave very nicely — almost like finite p-groups

Examples

finite p-groups Zp = {a0 + a1p + a2p2 + . . .} = 1 is the completion of Z w.r.t. the topology induced by {pnZp, n ≥ 1} a free pro-p group F is the completion of Fabs w.r.t. the topology induced by {U ⊆ Fabs | [Fabs : U] = pn} the Nottingham group GF(p), the maximal pro-p quotient of Gal(¯ Fs | F)

Claudio Quadrelli Universit` a Milano-Bicocca

Profinite groups Representations of dim 1 The Kummerian property Cohomology References

p-adic representations of dimension 1

For G pro-p and θ a p-adic (continuous) representation θ: G − → GL1(Zp) ≃ Z×

p

let Vp denote Zp edowed with the G-action induced by θ, and set N = ker(θ). Since either im(θ) ≃ Zp or is trivial, the extension 1

N G Γ

• 1

splits, and G ≃ N ⋊ Γ.

Claudio Quadrelli Universit` a Milano-Bicocca

Profinite groups Representations of dim 1 The Kummerian property Cohomology References

The cyclotomic character

For a field F, set µp∞ =

• α ∈ ¯

Fs | αpn = 1 for some n ≥ 0

• Then Aut(µp∞) ≃ GL1(Zp), and the action GF(p) µp∞ induces

naturally a representation θF : GF(p) − → GL1(Zp) ≃ Aut(µp∞) called the cyclotomic character

Idea

Study the couples (G, θ) where θ “behaves” like the cyclotomic character, to understand better GF(p)

Claudio Quadrelli Universit` a Milano-Bicocca

Profinite groups Representations of dim 1 The Kummerian property Cohomology References

The Kummer subgroup

For G endowed with θ: G → GL1(Zp), the Kummer subgroup is Kθ =

• h−θ(g) · ghg−1 | g ∈ G, h ∈ N
• G

Kθ ⊆ N and Kθ ⊆ G p[G, G] Kθ ⊇ [N, N], so N/Kθ is abelian G/Kθ ≃ (N/Kθ) ⋊ Γ

Definition

(G, θ) is Kummerian if N/Kθ is torsion-free — i.e., N/Kθ ≃ Vθ ⊕ . . . ⊕ Vθ E.E. Kummer

Claudio Quadrelli Universit` a Milano-Bicocca

Profinite groups Representations of dim 1 The Kummerian property Cohomology References

Examples

free pro-p-groups are Kummerian for every representation θ If θ ≡ 1, then (G, θ) is Kummerian ⇔ G/[G, G] is torsion-free Z/pn is NOT Kummerian for pn = 2... ... whereas (Z/2, θ) IS Kummerian for Im(θ){±1} (Artin-Schreier Theorem)

Proposition

For F containing a root of 1 of order p, GF(p) with the cyclotomic character IS Kummerian, since by Kummer theory GF(p)/KθF = Gal

• p∞

√ F

• F
• Claudio Quadrelli

Universit` a Milano-Bicocca

Profinite groups Representations of dim 1 The Kummerian property Cohomology References

The relations of Kummerian groups

Given a minimal presentation 1

R F

π

G 1

(i.e. F is free s.t. F/F p[F, F] ≃ G/G p[G, G]), set ˆ θ = θ ◦ π, ˆ θ: F → GL1(Zp)

Theorem

For im(θ) torsion-free the following are equivalent (G, θ) is Kummer G/Kθ is torsion-free for any minimal presentation G ≃ F/R one has F/Kˆ

θ ≃ G/Kθ

Consequence: (G, θ) is Kummer ⇔ R ⊆ Kˆ

θ

Claudio Quadrelli Universit` a Milano-Bicocca

Profinite groups Representations of dim 1 The Kummerian property Cohomology References

The group H1(G, V )

Let V be a (continuous) G-module. A 1-cocycle is a (continuous) map c : G → V such that c(g1g2) = c(g1) + g1.c(g2) ∀ g1, g2 ∈ G

Definition

H1(G, V ) := Z 1(G, V )/B1(G, V ), where Z 1(G, V ) = {c : G → V | c is a 1-cocycle} B1(G, V ) = {g → (g − 1).v, v ∈ V } A morphism V → W induces H1(G, V ) → H1(G, W ) If G V is trivial then H1(G, V ) = HomG(G, V ) If G is pro-p then H1(G, Z/p) = (G/G p[G, G])∗

Claudio Quadrelli Universit` a Milano-Bicocca

Profinite groups Representations of dim 1 The Kummerian property Cohomology References

The Kummerian property and H1(G, Vθ)

Theorem

For im(θ) torsion-free the following are equivalent (1) (G, θ) is Kummerian (2) Kθ =

c : G→Vθ ker(c) (ker(c) = c−1(0))

(3) The projection Vθ → Z/p induces an epimorphism H1(G, Vθ)

H1(G, Z/p)

(4) For {xi, i ∈ I} minimally generating G and vi ∈ Zp, there is a unique 1-cocycle c : G → Vθ s.t. c(xi) = vi for all i ∈ I The equivalence (3)⇔(4) is due to J. Labute (1967)1.

1N.D. Tˆ

an deserves my gratitude for putting this result under my attention and therefore inspiring some of these results

Claudio Quadrelli Universit` a Milano-Bicocca

Profinite groups Representations of dim 1 The Kummerian property Cohomology References

Hunting “bad” relations 1

Given a pro-p group G with minimal presentation 1

R F G 1

and a representation θ, (G, θ) is Kummerian if and only if R ⊆

• c : F→Vˆ

θ

ker(c) Recipe: take {xi, i ∈ I} minimally generating F and G. (1) Let ˆ θ: F → GL1(Zp) be arbitrary s.t. R ⊆ ker(ˆ θ) (2) Assign “smart” values vi ∈ Zp to each xi, and compute the 1-cocycle c : F → Vˆ

θ induced by such values

(3) If there are some r ∈ R s.t. c(r) = 0, then (G, θ) is NOT Kummerian for any θ: G → GL1(Zp)

Claudio Quadrelli Universit` a Milano-Bicocca

Profinite groups Representations of dim 1 The Kummerian property Cohomology References

Hunting “bad” relations 2

Examples

Set G = x1, x2, x3 | R and let θ: G → GL1(Zp) be arbitrary. Assume r = xq

1 [x2, x3] ∈ R, whith q = pn

(1) θ(r) = 1 ⇒ θ(xq

1 [x2, x3]) = θ(xq 1 ) = 1 ⇒ θ(x1) = 1

(2) Set v1 = c(x1) = 1, v2 = c(x2) = v3 = c(x3) = 0 (3) c(r) = c(xq

1 ) + θ(xq 1 ) · c([x2, x3]) = q · c(x1) + 0 = q = 0

Assume r1 = xq1

1 [x1, x3] ∈ R and r2 = xq2 2 [x2, x3] ∈ R, q1 = q2

(1) θ(ri) = 1 ⇒ θ(x1) = θ(x2) = 1 (2) Set v1 = c(x1) = v2 = c(x2) = 1 and v3 = c(x3) = 0 (3) c(ri) = c(xqi

i ) + θ(xq 1 ) · c([x2, x3]) = qi + (θ(x3)−1 − 1), and if

c(r1) = c(r2) = 0 then q1 = q2, a contradiction In both cases (G, θ) is NOT Kummer

Claudio Quadrelli Universit` a Milano-Bicocca

Profinite groups Representations of dim 1 The Kummerian property Cohomology References

More “bad” relations

Consider a pro-p group G with minimal presentation 1

R F G 1 ,

with {x1, x2, . . .} minimally generating F, and assume that r ∈ R with... r = xq

1 s, 0 = q ∈ pZp and s ∈ x2, x3, . . . =: F1, or

r = xq

1 st, q ∈ pZk pZp, k ≥ 2, s ∈ F1 ∩ [F, F], t ∈ γk+1(F)

Then2 (G, θ) is NOT Kummerian for any θ: G → GL1(Zp).

2Similar results have been obtained recently and independently by J. Min´

aˇ c,

• M. Rogelstad and N.D. Tˆ

an too, employing the equivalence (3)⇔(4) as well

Claudio Quadrelli Universit` a Milano-Bicocca

Profinite groups Representations of dim 1 The Kummerian property Cohomology References

Further research

Theorem (Weigel-Q., 2017)

Given F containing a root of 1 of order p, for every i ≥ 1, V ⊗i

θF → Z/p induces an epimorphism

Hi(GF(p), V ⊗i

θF )

Hi(GF(p), Z/p)

This is a consequence of Rost-Voevodsky theorem. What can we say on the structure of G? (Group-theoretic translation in case i = 2?) Also, the Bogomolov Conjecture (1995) can be translated as follows: Given F containing a root of 1 of order p, is KθF free?

Claudio Quadrelli Universit` a Milano-Bicocca

Profinite groups Representations of dim 1 The Kummerian property Cohomology References

References

F.A. Bogomolov, On the structure of Galois groups of the fields of rational functions, Proc. Sympos. Pure Math., vol. 58, part 2, American Mathematical Society, Rhode Island, 1995,

• pp. 83–88.
• I. Efrat, C. Quadrelli, The Kummerian property and maximal

pro-p-Galois groups, preprint (2017), arXiv:1707.07018.

• J. Labute, Classification of Demuˇ

skin groups, Canad. J. Math 19 (1967), 106–132.

• J. Min´

aˇ c, M. Rogelstad, N.D. Tˆ an, Relations in the maximal pro-p-quotient of an absolute Galois group, in preparation.

• C. Quadrelli, Th. Weigel, Profinite groups with a cyclotomic

p-orientation, preprint (2017).

Claudio Quadrelli Universit` a Milano-Bicocca

Profinite groups Representations of dim 1 The Kummerian property Cohomology References

“Future is not SECS” 3 (L. Di Martino) THANK YOU!

3SECS = Statistical Economical Computational Sciences Claudio Quadrelli Universit` a Milano-Bicocca