Homomorphisms of Multiplicative Groups of Fields
Homomorphisms of Multiplicative Groups of Fields
Motivation
Let X be an algebraic variety over a field L. When is X(L) 6= ∅?
Motivation
Let X be an algebraic variety over a field L. When is X(L) 6= ∅? Frequently, L is a number field, or a function of a curve. We have local obstructions: X(Lν ) global obstructions: Brauer group,... When is X (stably) rational over L?
Geometric framework
We work over a ground field k. Let π : X→Y be a fibration, with connected generic fiber, dim(Y ) ≥ 2, K := k(X),
When does π admit a section?
L := k(Y ).
Geometric framework
We work over a ground field k. Let π : X→Y be a fibration, with connected generic fiber, dim(Y ) ≥ 2, K := k(X),
L := k(Y ).
When does π admit a section? For example, if π is a quadric bundle over a surface.
Section conjecture
Let GK , GL be the Galois groups of K := k(X) and L := k(Y ). We have π ∗ : k(Y ) ,→ k(X),
π∗ : GK →GL .
Section conjecture
Let GK , GL be the Galois groups of K := k(X) and L := k(Y ). We have π ∗ : k(Y ) ,→ k(X),
π∗ : GK →GL .
Conjecture If π∗ admits a section then π : X→Y admits a section.
Background The Section conjecture is part of the Anabelian Geometry Program of Grothendieck: a variety (or its function field) should be encoded in its algebraic fundamental group (Galois group), functorially, e.g., Homk (L, K) = HomGk (GK , GL ). There are many results in this direction, e.g., Nakamura, Tamagawa, Mochizuki, Pop: if k is a finite extension of Qp or a finitely generated extension over Q, Qp K¨onigsman, Ellenberg, Stix, Esnault–Hai, Esnault-Wittenberg, Wickelgren, Saidi-Tamagawa, ... — mostly curves
Almost abelian anabelian geometry Consider k containing all `n -roots of 1, char(k) 6= `. GK
GK ,
Galois group of the maximal pro-`-extension. / GK /[GK , GK ]
GK /[GK , [GK , GK ]]
c GK
$
/ Ga K
Almost abelian anabelian geometry Consider k containing all `n -roots of 1, char(k) 6= `. GK
GK ,
Galois group of the maximal pro-`-extension. / GK /[GK , GK ]
GK /[GK , [GK , GK ]]
c GK
$
/ Ga K
a is a torsion-free Z -module of infinite rank. The abelianization GK `
Almost abelian anabelian geometry Consider k containing all `n -roots of 1, char(k) 6= `. GK
GK ,
Galois group of the maximal pro-`-extension. / GK /[GK , GK ]
GK /[GK , [GK , GK ]]
c GK
$
/ Ga K
a is a torsion-free Z -module of infinite rank. The abelianization GK ` a such that $ −1 (σ) is abelian. Let ΣK := {σ}, noncyclic σ ⊂ GK
Bogomolov’s program
a , Σ ) determines K. The pair (GK K
Almost abelian anabelian geometry Bogomolov-T. 2009, Pop 2011 ¯ p (X) and L = F ¯ p (Y ), trdegk (K) ≥ 2 and ` 6= p. If there Let K = F exists an isomorphism a ψ : GK ' GLa , inducing a bijection of sets ΣK = ΣL , then, for some c ∈ Z∗` , cψ is induced by an isomorphism of purely inseparable closures of K and L.
Almost abelian anabelian geometry Bogomolov-T. 2009, Pop 2011 ¯ p (X) and L = F ¯ p (Y ), trdegk (K) ≥ 2 and ` 6= p. If there Let K = F exists an isomorphism a ψ : GK ' GLa , inducing a bijection of sets ΣK = ΣL , then, for some c ∈ Z∗` , cψ is induced by an isomorphism of purely inseparable closures of K and L. Further developments: Pop (2014): extension to arbitrary k = k¯ of characteristic 6= `
Almost abelian anabelian geometry Bogomolov-T. 2009, Pop 2011 ¯ p (X) and L = F ¯ p (Y ), trdegk (K) ≥ 2 and ` 6= p. If there Let K = F exists an isomorphism a ψ : GK ' GLa , inducing a bijection of sets ΣK = ΣL , then, for some c ∈ Z∗` , cψ is induced by an isomorphism of purely inseparable closures of K and L. Further developments: Pop (2014): extension to arbitrary k = k¯ of characteristic 6= ` Topaz (2014, 2015): mod-` versions
Almost abelian Anabelian Section conjecture Assume that a πa : GK → GLa
admits a section ξa such that ξa (ΣL ) ⊂ ΣK . Then, modulo purely inseparable extensions, there is a rational map ξ : Y → X, such that ξ ∗ ◦ π ∗ (L) = L, i.e., ξ(Y ) is a section over Y .
Goal
Explain the main ideas of Bogomolov’s program.
Goal
Explain the main ideas of Bogomolov’s program. Explain a version of the Section conjecture in this context.
Projective geometry
Projective structure: (S, L) = points and lines, such that P1 ∃ s ∈ S and l ∈ L such that s ∈ / l; P2 ∀ l ∈ L ∃ distinct s, s0 , s00 ∈ l; P3 ∀ s, s0 ∈ S ∃! l = l(s, s0 ) ∈ L, such that s, s0 ∈ l; P4 ∀ s, s0 , t, t0 ∈ S l(s, s0 ) ∩ l(t, t0 ) 6= ∅ ⇒ l(s, t) ∩ l(s0 , t0 ) 6= ∅. P5 Pappus axiom
Projective geometry
Projective geometry
Projective structure: (S, L) = points and lines, such that P1 ∃ s ∈ S and l ∈ L such that s ∈ / l; P2 ∀ l ∈ L ∃ distinct s, s0 , s00 ∈ l; P3 ∀ s, s0 ∈ S ∃! l = l(s, s0 ) ∈ L, such that s, s0 ∈ l; P4 ∀ s, s0 , t, t0 ∈ S l(s, s0 ) ∩ l(t, t0 ) 6= ∅ ⇒ l(s, t) ∩ l(s0 , t0 ) 6= ∅. P5 Pappus axiom A morphism ρ : (S, L)→(S 0 , L0 ) is a map ρ : S → S 0 preserving lines.
Projective geometry
Projective geometry
Reconstruction Let (S, L) be a projective structure of dimension n ≥ 2. Then there exist a vector space V over a field k and an isomorphism ∼
σ : Pk (V ) −→ S.
Projective geometry
Projective geometry
Reconstruction Let (S, L) be a projective structure of dimension n ≥ 2. Then there exist a vector space V over a field k and an isomorphism ∼
σ : Pk (V ) −→ S.
Example Pn (k), n ≥ 2, has an abstract projective structure.
Projective geometry
Projective geometry
Let K/k be an extension of fields. Then S := Pk (K) = (K \ 0)/k × has a projective structure, preserved by multiplication in K × /k × .
Projective geometry
Main theorem
Reconstructing fields Let K/k and K 0 /k 0 be field extensions of degree ≥ 3 and ψ¯ : S = Pk (K)→Pk0 (K 0 ) = S 0 an injective homomorphism of abelian groups compatible with projective structures. Then k ' k 0 and K is isomorphic to a subfield of K 0 .
Projective geometry
Starting point: Reconstruction of fields
Recall that × KM 1 (K) = K
Projective geometry
Starting point: Reconstruction of fields
Recall that × KM 1 (K) = K
and that we have a surjection M M σK : K M 1 (K) ⊗ K1 (K) → K2 (K),
with Ker(σK ) generated by symbols (x, 1 − x), for x ∈ K × \ 1.
Projective geometry
Starting point: Reconstruction of fields
Recall that × KM 1 (K) = K
and that we have a surjection M M σK : K M 1 (K) ⊗ K1 (K) → K2 (K),
with Ker(σK ) generated by symbols (x, 1 − x), for x ∈ K × \ 1.
Bogomolov-T. 2008 KM 1 (K) and Ker(σK ) determine K, up to purely-inseparable extensions.
Projective geometry
Sketch of proof
The ground field = divisible elements An element f ∈ K × = KM 1 (K) is (infinitely) divisible if and only if × M f ∈ k . In particular, K1 (K) determines K × /k × = Pk (K), as an abelian group.
Projective geometry
Sketch of proof
1-dimensional subfields Given f, g ∈ K × /k × , we have trdegk k(f, g) = 1 iff (f, g) divisible ∈ KM 2 (K).
Projective geometry
Sketch of proof
1-dimensional subfields Given f, g ∈ K × /k × , we have trdegk k(f, g) = 1 iff (f, g) divisible ∈ KM 2 (K). Thus, KM 2 (K) determines {Pk (E) ⊂ Pk (K)}tr degk (E)=1 .
Projective geometry
Sketch of proof Reconstructing lines: Functional equations Assume that trdegk k(x, y) = 2 and chose ×
p ∈ k(x) ,
×
q ∈ k(y)
so that x, y, p, q are multiplicatively independent in K × /k × .
Projective geometry
Sketch of proof Reconstructing lines: Functional equations Assume that trdegk k(x, y) = 2 and chose ×
p ∈ k(x) ,
×
q ∈ k(y)
so that x, y, p, q are multiplicatively independent in K × /k × . Every nonconstant × × Π ∈ k(x/y) · y ∩ k(p/q) · q, arising from infinitely many p, q as above, is, modulo k × , Π = Πκ,δ (x, y) := (xδ − κy δ )δ , with κ ∈ k × and δ = ±1.
Projective geometry
(1)
Sketch of proof Reconstructing lines: Functional equations Assume that trdegk k(x, y) = 2 and chose ×
p ∈ k(x) ,
×
q ∈ k(y)
so that x, y, p, q are multiplicatively independent in K × /k × . Every nonconstant × × Π ∈ k(x/y) · y ∩ k(p/q) · q, arising from infinitely many p, q as above, is, modulo k × , Π = Πκ,δ (x, y) := (xδ − κy δ )δ , with κ ∈ k × and δ = ±1. These are projective lines, or their “inverses”. Projective geometry
(1)
Torelli version (Topaz 2017)
Consider X/C, put K := C(X), and Hi (K, Z(j)) := lim Hi (U (C), Z(j)), −→
U ⊂ X,
(infinite-rank) mixed Hodge structure. We have the cup-product σK : H1 (K, Z(1)) ⊗ H1 (K, Z(1))→H2 (K, Z(2)).
Projective geometry
Torelli version (Topaz 2017)
Consider X/C, put K := C(X), and Hi (K, Z(j)) := lim Hi (U (C), Z(j)), −→
U ⊂ X,
(infinite-rank) mixed Hodge structure. We have the cup-product σK : H1 (K, Z(1)) ⊗ H1 (K, Z(1))→H2 (K, Z(2)). H1 (K) and Ker(σK ) determine the field, up to isomorphism.
Projective geometry
Sketch of proof
Basics on MHS K-theory reconstruction Projective geometry
Projective geometry
Group cohomology
Let Hi (G, M ) be the i-cohomology group of a finite or profinite group G, with coefficients in a G-module M . Recall: H0 (G, M ) = M G , the submodule of G-invariants;
Galois cohomology
Group cohomology
Let Hi (G, M ) be the i-cohomology group of a finite or profinite group G, with coefficients in a G-module M . Recall: H0 (G, M ) = M G , the submodule of G-invariants; H1 (G, M ) = Hom(G, M ), provided M has trivial G-action;
Galois cohomology
Group cohomology
Let Hi (G, M ) be the i-cohomology group of a finite or profinite group G, with coefficients in a G-module M . Recall: H0 (G, M ) = M G , the submodule of G-invariants; H1 (G, M ) = Hom(G, M ), provided M has trivial G-action; H2 (G, M ) classifies central extensions ˜ 1→M →G→G→1,
Galois cohomology
Galois cohomology
¯ with char(k) 6= `. Let K = k(X) be a function field over k = k, n Kummer theory: H1 (GK ) = Hom(GK , Z/`n ) = KM 1 (K)/`
Galois cohomology
Galois cohomology
¯ with char(k) 6= `. Let K = k(X) be a function field over k = k, n Kummer theory: H1 (GK ) = Hom(GK , Z/`n ) = KM 1 (K)/` n Merkuriev-Suslin: H2 (GK ) = KM 2 (K)/`
Galois cohomology
Galois cohomology
¯ with char(k) 6= `. Let K = k(X) be a function field over k = k, n Kummer theory: H1 (GK ) = Hom(GK , Z/`n ) = KM 1 (K)/` n Merkuriev-Suslin: H2 (GK ) = KM 2 (K)/` n Voevodsky, Rost, Weibel: Hn (GK ) = KM n (K)/`
Galois cohomology
Idea of proof: Isomorphism version
a , Σ ) determines K: (GK K
Proof: main steps
Idea of proof: Isomorphism version
a , Σ ) determines K: (GK K a is dual to K ˆ × , thus GK a GK ' GLa
Proof: main steps
⇒
ˆ× ' L ˆ ×. K
Idea of proof: Isomorphism version
a , Σ ) determines K: (GK K a is dual to K ˆ × , thus GK a GK ' GLa
⇒
ˆ× ' L ˆ ×. K
c is ‘dual’ (?) to completed KM (K), and encodes algebraic GK 2 ˆ ×. dependence of formal functions fˆ, gˆ ∈ K
Proof: main steps
Idea of proof From c a 1→ZK →GK →GK →1.
we have a ∧2 (GK )→ZK ,
Put
(γ, γ 0 ) 7→ [˜ γ , γ˜ 0 ].
c ) := Ker(∧2 (G a )→Z ) R(GK K K
and let
c ) ⊆ R(G c ), R∧ (GK K
generated by hγ, γ 0 i, where γ, γ 0 is a ‘commuting pair’.
Bogomolov ¯ p (X) then If K = F c c ))∨ . Brnr (K) = (R(GK )/R∧ (GK
Proof: main steps
Idea of proof From c a 1→ZK →GK →GK →1.
we have a ∧2 (GK )→ZK ,
Put
(γ, γ 0 ) 7→ [˜ γ , γ˜ 0 ].
c ) := Ker(∧2 (G a )→Z ) R(GK K K
and let
c ) ⊆ R(G c ), R∧ (GK K
generated by hγ, γ 0 i, where γ, γ 0 is a ‘commuting pair’.
Bogomolov ¯ p (X) then If K = F c c ))∨ . Brnr (K) = (R(GK )/R∧ (GK
Thus ΣK carries information about the Brauer group and KM 2 (K). Proof: main steps
Valuations How to see a (sub) variety in a Galois group? They are centers of valuations on varying projective models of the function field.
Valuations
Valuations How to see a (sub) variety in a Galois group? They are centers of valuations on varying projective models of the function field. A (nonarchimedean) valuation on a field K is a pair ν = (ν, Γν ) consisting of a totally ordered (torsion-free) abelian group Γν and a map ν : K→Γν,∞ = Γν ∪ ∞ such that ν : K ∗ →Γν is a surjective homomorphism; ν(κ + κ0 ) ≥ min(ν(κ), ν(κ0 )) for all κ, κ0 ∈ K; ν(0) = ∞. Let o = oν be the valuation ring, kν the residue field, and VK the set of valuations of K. Valuations
Valuations and projective geometry
A 3-coloring of P2 (k) is a surjective map c : P2 (k)→{•, ◦, ?} such that every l ⊂ P2 (k) is colored in exactly two colors. It is called trivial of type I: if there exists an l ⊂ P2 such that c is constant on P2 (k) \ l(k) II: if there exists a point q ∈ P2 (k) such that for all l ⊂ P2 containing q, c is constant on l \ q.
Valuations
Valuations and projective geometry
Hales–Strauss 1982 Assume that P2 (k) carries a 3-coloring c. Then there exists a ν ∈ Vk such that c is induced from a trivial covering cν : P2 (kν )→{•, ◦, ?}, for some ρ : P2 (k)→P2 (kν ).
Valuations
Example ¯ We decompose Let K = k(X), with k = k. (K × /k × \ 1) = tj∈J Tj , into algebraic dependency classes: f, f 0 ∈ Tj are algebraically dependent, and fj ∈ Tj , fj 0 ∈ Tj 0 are not, for j 6= j 0 . Now split J = J2 t J3 and consider P(K) = S1 t S2 t S3 with S1 = 1, S2 = tj∈J2 Tj , S3 = tj∈J3 Tj .
Valuations
Example ¯ We decompose Let K = k(X), with k = k. (K × /k × \ 1) = tj∈J Tj , into algebraic dependency classes: f, f 0 ∈ Tj are algebraically dependent, and fj ∈ Tj , fj 0 ∈ Tj 0 are not, for j 6= j 0 . Now split J = J2 t J3 and consider P(K) = S1 t S2 t S3 with S1 = 1, S2 = tj∈J2 Tj , S3 = tj∈J3 Tj . Then c : P2 (k) → {1, 2, 3} is a 3-coloring, on every P2 (k) ⊂ P(K), containing 1 and at least two nonconstant algebraically independent elements. Valuations
Valuations and projective geometry We study (not necessarily injective) homomorphisms ψ : K × /k × →L× /k × , preserving algebraic dependence. Let u := ∪l(1, x) ⊂ Pk (K), over lines on which ψ is injective. Assume that u contains x1 , x2 such that trdegk (k(ψ(x1 ), ψ(x2 )) = 2.
Valuations
Valuations and projective geometry We study (not necessarily injective) homomorphisms ψ : K × /k × →L× /k × , preserving algebraic dependence. Let u := ∪l(1, x) ⊂ Pk (K), over lines on which ψ is injective. Assume that u contains x1 , x2 such that trdegk (k(ψ(x1 ), ψ(x2 )) = 2. There exists a ν ∈ VK such that o× ν = u · u ⊂ Pk (K). Valuations
Theorem (Bogomolov–Rovinsky–T.)
¯ char(k) > 0. Assume that there exists Let K, L be fields over k = k, a (not necessarily injective) homomorphism ψ : K × /k × →L× /k × , such that ψ preserves algebraic dependence, there exist y1 , y2 ∈ ψ(K × /k × ),
Valuations
such that
trdegk (k(y1 , y2 )) ≥ 2.
Theorem (Bogomolov–Rovinsky–T.) Then one of the following holds: (P) ∃ k ( F ⊂ K such that ψ factors through K × /k × −→ → K × /F × , (V) ∃ ν ∈ VK such that the restriction of ψ to × × × o× K,ν /ok,ν ⊆ K /k
is trivial on (1 + mν )× /o× k,ν and factors through the reduction map × × × × o× → K× ν /kν →L /k , K,ν /ok,ν −→
(VP) ∃ ν ∈ VK and F ν ⊂ K ν such that the restriction of ψ to × o× K,ν /ok,ν factors through × × × × o× → K× ν /F ν →L /k . K,ν /ok,ν −→ Valuations
Theorem (Bogomolov–Rovinsky–T.)
¯ p (X), the center of ν in case (V) is, birationally, the When K = F image of the section. The theorem is a rational version of the Section conjecture.
Valuations
Valuations A key step is the reconstruction of valuations. K-theoretic versions go back to Aranson–Elman–Jacob (1987), Efrat (1999), and others.
Valuations
Valuations A key step is the reconstruction of valuations. K-theoretic versions go back to Aranson–Elman–Jacob (1987), Efrat (1999), and others. We focus on Galois-type versions.
Valuations
Valuations A key step is the reconstruction of valuations. K-theoretic versions go back to Aranson–Elman–Jacob (1987), Efrat (1999), and others. We focus on Galois-type versions. We have
1
Valuations
1
/ o×
/ K×
/ Γν
/ (1 + mν )×
/ o×
/ K× ν
/1
ν
ν
/1
Weil groups ¯ p (X), R a (topological) ring with torsion coprime to p. K=F Abelian Weil group: a (R) := Hom(K × /k × , R). WK
Valuations
Weil groups ¯ p (X), R a (topological) ring with torsion coprime to p. K=F Abelian Weil group: a (R) := Hom(K × /k × , R). WK
By Kummer theory, a a WK (Z` ) = GK .
Valuations
Weil groups ¯ p (X), R a (topological) ring with torsion coprime to p. K=F Abelian Weil group: a (R) := Hom(K × /k × , R). WK
By Kummer theory, a a WK (Z` ) = GK .
Other interesting cases: R = Z or R = Z/`.
Valuations
Weil groups ¯ p (X), R a (topological) ring with torsion coprime to p. K=F Abelian Weil group: a (R) := Hom(K × /k × , R). WK
By Kummer theory, a a WK (Z` ) = GK .
Other interesting cases: R = Z or R = Z/`. Define a (R) | µ trivial on (1 + m )× }, Daν (R) = {µ ∈ WK ν a (R) | ι trivial on o× }. Iaν (R) = {ι ∈ WK ν
Valuations
Weil groups ¯ p (X), R a (topological) ring with torsion coprime to p. K=F Abelian Weil group: a (R) := Hom(K × /k × , R). WK
By Kummer theory, a a WK (Z` ) = GK .
Other interesting cases: R = Z or R = Z/`. Define a (R) | µ trivial on (1 + m )× }, Daν (R) = {µ ∈ WK ν a (R) | ι trivial on o× }. Iaν (R) = {ι ∈ WK ν
For R = Z` these are the usual decomposition and inertia a corresponding to ν. subgroups in GK Valuations
Commuting pairs
For E ⊂ K we have the restriction homomorphism a a ρE : WK (R)→WK (R)
Definition a (R) is a c-subgroup if σ ⊂ WK
ρE (σ)
is cyclic ∀E ⊂ K, trdegk (E) = 1.
a (R). Let ΣK (R) be the set of c-subgroups of WK
Valuations
Commuting pairs
Theorem ˆ Z, Z/`n , or Z` . Then Let R = Z, ∀ c-subgroup σ has R-rank ≤ trdeg(K); ∀ c-subgroup σ ∃ ν ∈ VK such that σ is trivial on (1 + mν )× ⊂ K × , ∃ a maximal σ 0 ⊆ σ of R-corank ≤ 1 such that a (R). σ 0 ⊆ Hom(Γν , R) ⊂ Hom(K × , R) = WK
Valuations
Commuting pairs
σ 0 = Iaν (R),
Valuations
for some
ν ∈ VK
Commuting pairs
σ 0 = Iaν (R),
for some
ν ∈ VK
∪Iaν (R)⊆σ σ = Daν (R).
Valuations
Projective geometry of the Weil group Let γ, γ 0 ∈ σ ∈ ΣK (R) be be nonproportional. Then, for every nonconstant f ∈ K × , the restrictions of γ, γ 0 to l(1, f ) are proportional (modulo addition of constants).
Valuations
Projective geometry of the Weil group Let γ, γ 0 ∈ σ ∈ ΣK (R) be be nonproportional. Then, for every nonconstant f ∈ K × , the restrictions of γ, γ 0 to l(1, f ) are proportional (modulo addition of constants). Consider the map K × /k × = Pk (K) → f
Valuations
A2 (R)
7→ (γ(f ), γ 0 (f ))
Projective geometry of the Weil group Let γ, γ 0 ∈ σ ∈ ΣK (R) be be nonproportional. Then, for every nonconstant f ∈ K × , the restrictions of γ, γ 0 to l(1, f ) are proportional (modulo addition of constants). Consider the map K × /k × = Pk (K) → f
A2 (R)
7→ (γ(f ), γ 0 (f ))
This maps every projective line into an affine line, a collineation.
Valuations
Projective geometry of the Weil group Let γ, γ 0 ∈ σ ∈ ΣK (R) be be nonproportional. Then, for every nonconstant f ∈ K × , the restrictions of γ, γ 0 to l(1, f ) are proportional (modulo addition of constants). Consider the map K × /k × = Pk (K) → f
A2 (R)
7→ (γ(f ), γ 0 (f ))
This maps every projective line into an affine line, a collineation.
Images of planes The image of every P2 (k) in A2 (R) is contained in a union of an affine line and a point.
Valuations
Geometry of collineations
Given a map φγ,γ 0 : P2 (k)→{•, ◦, ?} ⊂ A2 (F2 ) such that every l ⊂ P2 (k) is contained in an “affine line” (any subset of two points) one of the following γ, γ 0 , γ + γ 0 is a flag map.
Valuations
Projective geometry of the Weil group
Proposition ∀ σ ∈ ΣK (R) ∃ an inertia element ι = ιν ∈ σ, for some ν ∈ VK .
Valuations
Projective geometry of the Weil group
Proposition ∀ σ ∈ ΣK (R) ∃ an inertia element ι = ιν ∈ σ, for some ν ∈ VK . The elements “commuting” with ι form Daν (R).
Valuations
Projective geometry of the Weil group
In the reconstruction of function fields from their `-Galois groups, i.e., when R = Z` , an isomorphism pairs a (GK , ΣK ) ' (GLa , ΣL )
allows to identify relations between valuations, and the projective structures of Pk (K) and Pl (L); thus a field isomorphism.
Valuations
Application: universal spaces for unramified cohomology ¯ p , ` 6= p, K = k(X), dim(X) ≥ 2. Let k = F
Bogomolov–T. 2014 For α ∈ Hinr (GK , Z/`n ), i ≥ 2, there exist a surjective GK →Ga onto a finite abelian `-group, projective Ga -representations P(Vj ) over k, an explicit open Ga -stable subset Y P◦ ⊂ P := P(Vj ), j
and a rational map % : X→P◦ /Ga such that α is induced from P◦ /Ga . Valuations
Next steps
Arbitrary ground fields, e.g, k = C?
Valuations
Next steps
Arbitrary ground fields, e.g, k = C? c ? Section conjecture for GK
Valuations
Next steps
Arbitrary ground fields, e.g, k = C? c ? Section conjecture for GK
Are all birational types captured by V /Gc , for Gc a central extension of an abelian group?
Valuations
Next steps
Arbitrary ground fields, e.g, k = C? c ? Section conjecture for GK
Are all birational types captured by V /Gc , for Gc a central extension of an abelian group? a , Σ ) encode, e.g., ampleness of the How does the pair (GK K canonical class?
Valuations
Next steps
Arbitrary ground fields, e.g, k = C? c ? Section conjecture for GK
Are all birational types captured by V /Gc , for Gc a central extension of an abelian group? a , Σ ) encode, e.g., ampleness of the How does the pair (GK K canonical class?
Valuations
Motivation
Let X be an algebraic variety over a field L. When is X(L) 6= ∅?
Motivation
Let X be an algebraic variety over a field L. When is X(L) 6= ∅? Frequently, L is a number field, or a function of a curve. We have local obstructions: X(Lν ) global obstructions: Brauer group,... When is X (stably) rational over L?
Geometric framework
We work over a ground field k. Let π : X→Y be a fibration, with connected generic fiber, dim(Y ) ≥ 2, K := k(X),
When does π admit a section?
L := k(Y ).
Geometric framework
We work over a ground field k. Let π : X→Y be a fibration, with connected generic fiber, dim(Y ) ≥ 2, K := k(X),
L := k(Y ).
When does π admit a section? For example, if π is a quadric bundle over a surface.
Section conjecture
Let GK , GL be the Galois groups of K := k(X) and L := k(Y ). We have π ∗ : k(Y ) ,→ k(X),
π∗ : GK →GL .
Section conjecture
Let GK , GL be the Galois groups of K := k(X) and L := k(Y ). We have π ∗ : k(Y ) ,→ k(X),
π∗ : GK →GL .
Conjecture If π∗ admits a section then π : X→Y admits a section.
Background The Section conjecture is part of the Anabelian Geometry Program of Grothendieck: a variety (or its function field) should be encoded in its algebraic fundamental group (Galois group), functorially, e.g., Homk (L, K) = HomGk (GK , GL ). There are many results in this direction, e.g., Nakamura, Tamagawa, Mochizuki, Pop: if k is a finite extension of Qp or a finitely generated extension over Q, Qp K¨onigsman, Ellenberg, Stix, Esnault–Hai, Esnault-Wittenberg, Wickelgren, Saidi-Tamagawa, ... — mostly curves
Almost abelian anabelian geometry Consider k containing all `n -roots of 1, char(k) 6= `. GK
GK ,
Galois group of the maximal pro-`-extension. / GK /[GK , GK ]
GK /[GK , [GK , GK ]]
c GK
$
/ Ga K
Almost abelian anabelian geometry Consider k containing all `n -roots of 1, char(k) 6= `. GK
GK ,
Galois group of the maximal pro-`-extension. / GK /[GK , GK ]
GK /[GK , [GK , GK ]]
c GK
$
/ Ga K
a is a torsion-free Z -module of infinite rank. The abelianization GK `
Almost abelian anabelian geometry Consider k containing all `n -roots of 1, char(k) 6= `. GK
GK ,
Galois group of the maximal pro-`-extension. / GK /[GK , GK ]
GK /[GK , [GK , GK ]]
c GK
$
/ Ga K
a is a torsion-free Z -module of infinite rank. The abelianization GK ` a such that $ −1 (σ) is abelian. Let ΣK := {σ}, noncyclic σ ⊂ GK
Bogomolov’s program
a , Σ ) determines K. The pair (GK K
Almost abelian anabelian geometry Bogomolov-T. 2009, Pop 2011 ¯ p (X) and L = F ¯ p (Y ), trdegk (K) ≥ 2 and ` 6= p. If there Let K = F exists an isomorphism a ψ : GK ' GLa , inducing a bijection of sets ΣK = ΣL , then, for some c ∈ Z∗` , cψ is induced by an isomorphism of purely inseparable closures of K and L.
Almost abelian anabelian geometry Bogomolov-T. 2009, Pop 2011 ¯ p (X) and L = F ¯ p (Y ), trdegk (K) ≥ 2 and ` 6= p. If there Let K = F exists an isomorphism a ψ : GK ' GLa , inducing a bijection of sets ΣK = ΣL , then, for some c ∈ Z∗` , cψ is induced by an isomorphism of purely inseparable closures of K and L. Further developments: Pop (2014): extension to arbitrary k = k¯ of characteristic 6= `
Almost abelian anabelian geometry Bogomolov-T. 2009, Pop 2011 ¯ p (X) and L = F ¯ p (Y ), trdegk (K) ≥ 2 and ` 6= p. If there Let K = F exists an isomorphism a ψ : GK ' GLa , inducing a bijection of sets ΣK = ΣL , then, for some c ∈ Z∗` , cψ is induced by an isomorphism of purely inseparable closures of K and L. Further developments: Pop (2014): extension to arbitrary k = k¯ of characteristic 6= ` Topaz (2014, 2015): mod-` versions
Almost abelian Anabelian Section conjecture Assume that a πa : GK → GLa
admits a section ξa such that ξa (ΣL ) ⊂ ΣK . Then, modulo purely inseparable extensions, there is a rational map ξ : Y → X, such that ξ ∗ ◦ π ∗ (L) = L, i.e., ξ(Y ) is a section over Y .
Goal
Explain the main ideas of Bogomolov’s program.
Goal
Explain the main ideas of Bogomolov’s program. Explain a version of the Section conjecture in this context.
Projective geometry
Projective structure: (S, L) = points and lines, such that P1 ∃ s ∈ S and l ∈ L such that s ∈ / l; P2 ∀ l ∈ L ∃ distinct s, s0 , s00 ∈ l; P3 ∀ s, s0 ∈ S ∃! l = l(s, s0 ) ∈ L, such that s, s0 ∈ l; P4 ∀ s, s0 , t, t0 ∈ S l(s, s0 ) ∩ l(t, t0 ) 6= ∅ ⇒ l(s, t) ∩ l(s0 , t0 ) 6= ∅. P5 Pappus axiom
Projective geometry
Projective geometry
Projective structure: (S, L) = points and lines, such that P1 ∃ s ∈ S and l ∈ L such that s ∈ / l; P2 ∀ l ∈ L ∃ distinct s, s0 , s00 ∈ l; P3 ∀ s, s0 ∈ S ∃! l = l(s, s0 ) ∈ L, such that s, s0 ∈ l; P4 ∀ s, s0 , t, t0 ∈ S l(s, s0 ) ∩ l(t, t0 ) 6= ∅ ⇒ l(s, t) ∩ l(s0 , t0 ) 6= ∅. P5 Pappus axiom A morphism ρ : (S, L)→(S 0 , L0 ) is a map ρ : S → S 0 preserving lines.
Projective geometry
Projective geometry
Reconstruction Let (S, L) be a projective structure of dimension n ≥ 2. Then there exist a vector space V over a field k and an isomorphism ∼
σ : Pk (V ) −→ S.
Projective geometry
Projective geometry
Reconstruction Let (S, L) be a projective structure of dimension n ≥ 2. Then there exist a vector space V over a field k and an isomorphism ∼
σ : Pk (V ) −→ S.
Example Pn (k), n ≥ 2, has an abstract projective structure.
Projective geometry
Projective geometry
Let K/k be an extension of fields. Then S := Pk (K) = (K \ 0)/k × has a projective structure, preserved by multiplication in K × /k × .
Projective geometry
Main theorem
Reconstructing fields Let K/k and K 0 /k 0 be field extensions of degree ≥ 3 and ψ¯ : S = Pk (K)→Pk0 (K 0 ) = S 0 an injective homomorphism of abelian groups compatible with projective structures. Then k ' k 0 and K is isomorphic to a subfield of K 0 .
Projective geometry
Starting point: Reconstruction of fields
Recall that × KM 1 (K) = K
Projective geometry
Starting point: Reconstruction of fields
Recall that × KM 1 (K) = K
and that we have a surjection M M σK : K M 1 (K) ⊗ K1 (K) → K2 (K),
with Ker(σK ) generated by symbols (x, 1 − x), for x ∈ K × \ 1.
Projective geometry
Starting point: Reconstruction of fields
Recall that × KM 1 (K) = K
and that we have a surjection M M σK : K M 1 (K) ⊗ K1 (K) → K2 (K),
with Ker(σK ) generated by symbols (x, 1 − x), for x ∈ K × \ 1.
Bogomolov-T. 2008 KM 1 (K) and Ker(σK ) determine K, up to purely-inseparable extensions.
Projective geometry
Sketch of proof
The ground field = divisible elements An element f ∈ K × = KM 1 (K) is (infinitely) divisible if and only if × M f ∈ k . In particular, K1 (K) determines K × /k × = Pk (K), as an abelian group.
Projective geometry
Sketch of proof
1-dimensional subfields Given f, g ∈ K × /k × , we have trdegk k(f, g) = 1 iff (f, g) divisible ∈ KM 2 (K).
Projective geometry
Sketch of proof
1-dimensional subfields Given f, g ∈ K × /k × , we have trdegk k(f, g) = 1 iff (f, g) divisible ∈ KM 2 (K). Thus, KM 2 (K) determines {Pk (E) ⊂ Pk (K)}tr degk (E)=1 .
Projective geometry
Sketch of proof Reconstructing lines: Functional equations Assume that trdegk k(x, y) = 2 and chose ×
p ∈ k(x) ,
×
q ∈ k(y)
so that x, y, p, q are multiplicatively independent in K × /k × .
Projective geometry
Sketch of proof Reconstructing lines: Functional equations Assume that trdegk k(x, y) = 2 and chose ×
p ∈ k(x) ,
×
q ∈ k(y)
so that x, y, p, q are multiplicatively independent in K × /k × . Every nonconstant × × Π ∈ k(x/y) · y ∩ k(p/q) · q, arising from infinitely many p, q as above, is, modulo k × , Π = Πκ,δ (x, y) := (xδ − κy δ )δ , with κ ∈ k × and δ = ±1.
Projective geometry
(1)
Sketch of proof Reconstructing lines: Functional equations Assume that trdegk k(x, y) = 2 and chose ×
p ∈ k(x) ,
×
q ∈ k(y)
so that x, y, p, q are multiplicatively independent in K × /k × . Every nonconstant × × Π ∈ k(x/y) · y ∩ k(p/q) · q, arising from infinitely many p, q as above, is, modulo k × , Π = Πκ,δ (x, y) := (xδ − κy δ )δ , with κ ∈ k × and δ = ±1. These are projective lines, or their “inverses”. Projective geometry
(1)
Torelli version (Topaz 2017)
Consider X/C, put K := C(X), and Hi (K, Z(j)) := lim Hi (U (C), Z(j)), −→
U ⊂ X,
(infinite-rank) mixed Hodge structure. We have the cup-product σK : H1 (K, Z(1)) ⊗ H1 (K, Z(1))→H2 (K, Z(2)).
Projective geometry
Torelli version (Topaz 2017)
Consider X/C, put K := C(X), and Hi (K, Z(j)) := lim Hi (U (C), Z(j)), −→
U ⊂ X,
(infinite-rank) mixed Hodge structure. We have the cup-product σK : H1 (K, Z(1)) ⊗ H1 (K, Z(1))→H2 (K, Z(2)). H1 (K) and Ker(σK ) determine the field, up to isomorphism.
Projective geometry
Sketch of proof
Basics on MHS K-theory reconstruction Projective geometry
Projective geometry
Group cohomology
Let Hi (G, M ) be the i-cohomology group of a finite or profinite group G, with coefficients in a G-module M . Recall: H0 (G, M ) = M G , the submodule of G-invariants;
Galois cohomology
Group cohomology
Let Hi (G, M ) be the i-cohomology group of a finite or profinite group G, with coefficients in a G-module M . Recall: H0 (G, M ) = M G , the submodule of G-invariants; H1 (G, M ) = Hom(G, M ), provided M has trivial G-action;
Galois cohomology
Group cohomology
Let Hi (G, M ) be the i-cohomology group of a finite or profinite group G, with coefficients in a G-module M . Recall: H0 (G, M ) = M G , the submodule of G-invariants; H1 (G, M ) = Hom(G, M ), provided M has trivial G-action; H2 (G, M ) classifies central extensions ˜ 1→M →G→G→1,
Galois cohomology
Galois cohomology
¯ with char(k) 6= `. Let K = k(X) be a function field over k = k, n Kummer theory: H1 (GK ) = Hom(GK , Z/`n ) = KM 1 (K)/`
Galois cohomology
Galois cohomology
¯ with char(k) 6= `. Let K = k(X) be a function field over k = k, n Kummer theory: H1 (GK ) = Hom(GK , Z/`n ) = KM 1 (K)/` n Merkuriev-Suslin: H2 (GK ) = KM 2 (K)/`
Galois cohomology
Galois cohomology
¯ with char(k) 6= `. Let K = k(X) be a function field over k = k, n Kummer theory: H1 (GK ) = Hom(GK , Z/`n ) = KM 1 (K)/` n Merkuriev-Suslin: H2 (GK ) = KM 2 (K)/` n Voevodsky, Rost, Weibel: Hn (GK ) = KM n (K)/`
Galois cohomology
Idea of proof: Isomorphism version
a , Σ ) determines K: (GK K
Proof: main steps
Idea of proof: Isomorphism version
a , Σ ) determines K: (GK K a is dual to K ˆ × , thus GK a GK ' GLa
Proof: main steps
⇒
ˆ× ' L ˆ ×. K
Idea of proof: Isomorphism version
a , Σ ) determines K: (GK K a is dual to K ˆ × , thus GK a GK ' GLa
⇒
ˆ× ' L ˆ ×. K
c is ‘dual’ (?) to completed KM (K), and encodes algebraic GK 2 ˆ ×. dependence of formal functions fˆ, gˆ ∈ K
Proof: main steps
Idea of proof From c a 1→ZK →GK →GK →1.
we have a ∧2 (GK )→ZK ,
Put
(γ, γ 0 ) 7→ [˜ γ , γ˜ 0 ].
c ) := Ker(∧2 (G a )→Z ) R(GK K K
and let
c ) ⊆ R(G c ), R∧ (GK K
generated by hγ, γ 0 i, where γ, γ 0 is a ‘commuting pair’.
Bogomolov ¯ p (X) then If K = F c c ))∨ . Brnr (K) = (R(GK )/R∧ (GK
Proof: main steps
Idea of proof From c a 1→ZK →GK →GK →1.
we have a ∧2 (GK )→ZK ,
Put
(γ, γ 0 ) 7→ [˜ γ , γ˜ 0 ].
c ) := Ker(∧2 (G a )→Z ) R(GK K K
and let
c ) ⊆ R(G c ), R∧ (GK K
generated by hγ, γ 0 i, where γ, γ 0 is a ‘commuting pair’.
Bogomolov ¯ p (X) then If K = F c c ))∨ . Brnr (K) = (R(GK )/R∧ (GK
Thus ΣK carries information about the Brauer group and KM 2 (K). Proof: main steps
Valuations How to see a (sub) variety in a Galois group? They are centers of valuations on varying projective models of the function field.
Valuations
Valuations How to see a (sub) variety in a Galois group? They are centers of valuations on varying projective models of the function field. A (nonarchimedean) valuation on a field K is a pair ν = (ν, Γν ) consisting of a totally ordered (torsion-free) abelian group Γν and a map ν : K→Γν,∞ = Γν ∪ ∞ such that ν : K ∗ →Γν is a surjective homomorphism; ν(κ + κ0 ) ≥ min(ν(κ), ν(κ0 )) for all κ, κ0 ∈ K; ν(0) = ∞. Let o = oν be the valuation ring, kν the residue field, and VK the set of valuations of K. Valuations
Valuations and projective geometry
A 3-coloring of P2 (k) is a surjective map c : P2 (k)→{•, ◦, ?} such that every l ⊂ P2 (k) is colored in exactly two colors. It is called trivial of type I: if there exists an l ⊂ P2 such that c is constant on P2 (k) \ l(k) II: if there exists a point q ∈ P2 (k) such that for all l ⊂ P2 containing q, c is constant on l \ q.
Valuations
Valuations and projective geometry
Hales–Strauss 1982 Assume that P2 (k) carries a 3-coloring c. Then there exists a ν ∈ Vk such that c is induced from a trivial covering cν : P2 (kν )→{•, ◦, ?}, for some ρ : P2 (k)→P2 (kν ).
Valuations
Example ¯ We decompose Let K = k(X), with k = k. (K × /k × \ 1) = tj∈J Tj , into algebraic dependency classes: f, f 0 ∈ Tj are algebraically dependent, and fj ∈ Tj , fj 0 ∈ Tj 0 are not, for j 6= j 0 . Now split J = J2 t J3 and consider P(K) = S1 t S2 t S3 with S1 = 1, S2 = tj∈J2 Tj , S3 = tj∈J3 Tj .
Valuations
Example ¯ We decompose Let K = k(X), with k = k. (K × /k × \ 1) = tj∈J Tj , into algebraic dependency classes: f, f 0 ∈ Tj are algebraically dependent, and fj ∈ Tj , fj 0 ∈ Tj 0 are not, for j 6= j 0 . Now split J = J2 t J3 and consider P(K) = S1 t S2 t S3 with S1 = 1, S2 = tj∈J2 Tj , S3 = tj∈J3 Tj . Then c : P2 (k) → {1, 2, 3} is a 3-coloring, on every P2 (k) ⊂ P(K), containing 1 and at least two nonconstant algebraically independent elements. Valuations
Valuations and projective geometry We study (not necessarily injective) homomorphisms ψ : K × /k × →L× /k × , preserving algebraic dependence. Let u := ∪l(1, x) ⊂ Pk (K), over lines on which ψ is injective. Assume that u contains x1 , x2 such that trdegk (k(ψ(x1 ), ψ(x2 )) = 2.
Valuations
Valuations and projective geometry We study (not necessarily injective) homomorphisms ψ : K × /k × →L× /k × , preserving algebraic dependence. Let u := ∪l(1, x) ⊂ Pk (K), over lines on which ψ is injective. Assume that u contains x1 , x2 such that trdegk (k(ψ(x1 ), ψ(x2 )) = 2. There exists a ν ∈ VK such that o× ν = u · u ⊂ Pk (K). Valuations
Theorem (Bogomolov–Rovinsky–T.)
¯ char(k) > 0. Assume that there exists Let K, L be fields over k = k, a (not necessarily injective) homomorphism ψ : K × /k × →L× /k × , such that ψ preserves algebraic dependence, there exist y1 , y2 ∈ ψ(K × /k × ),
Valuations
such that
trdegk (k(y1 , y2 )) ≥ 2.
Theorem (Bogomolov–Rovinsky–T.) Then one of the following holds: (P) ∃ k ( F ⊂ K such that ψ factors through K × /k × −→ → K × /F × , (V) ∃ ν ∈ VK such that the restriction of ψ to × × × o× K,ν /ok,ν ⊆ K /k
is trivial on (1 + mν )× /o× k,ν and factors through the reduction map × × × × o× → K× ν /kν →L /k , K,ν /ok,ν −→
(VP) ∃ ν ∈ VK and F ν ⊂ K ν such that the restriction of ψ to × o× K,ν /ok,ν factors through × × × × o× → K× ν /F ν →L /k . K,ν /ok,ν −→ Valuations
Theorem (Bogomolov–Rovinsky–T.)
¯ p (X), the center of ν in case (V) is, birationally, the When K = F image of the section. The theorem is a rational version of the Section conjecture.
Valuations
Valuations A key step is the reconstruction of valuations. K-theoretic versions go back to Aranson–Elman–Jacob (1987), Efrat (1999), and others.
Valuations
Valuations A key step is the reconstruction of valuations. K-theoretic versions go back to Aranson–Elman–Jacob (1987), Efrat (1999), and others. We focus on Galois-type versions.
Valuations
Valuations A key step is the reconstruction of valuations. K-theoretic versions go back to Aranson–Elman–Jacob (1987), Efrat (1999), and others. We focus on Galois-type versions. We have
1
Valuations
1
/ o×
/ K×
/ Γν
/ (1 + mν )×
/ o×
/ K× ν
/1
ν
ν
/1
Weil groups ¯ p (X), R a (topological) ring with torsion coprime to p. K=F Abelian Weil group: a (R) := Hom(K × /k × , R). WK
Valuations
Weil groups ¯ p (X), R a (topological) ring with torsion coprime to p. K=F Abelian Weil group: a (R) := Hom(K × /k × , R). WK
By Kummer theory, a a WK (Z` ) = GK .
Valuations
Weil groups ¯ p (X), R a (topological) ring with torsion coprime to p. K=F Abelian Weil group: a (R) := Hom(K × /k × , R). WK
By Kummer theory, a a WK (Z` ) = GK .
Other interesting cases: R = Z or R = Z/`.
Valuations
Weil groups ¯ p (X), R a (topological) ring with torsion coprime to p. K=F Abelian Weil group: a (R) := Hom(K × /k × , R). WK
By Kummer theory, a a WK (Z` ) = GK .
Other interesting cases: R = Z or R = Z/`. Define a (R) | µ trivial on (1 + m )× }, Daν (R) = {µ ∈ WK ν a (R) | ι trivial on o× }. Iaν (R) = {ι ∈ WK ν
Valuations
Weil groups ¯ p (X), R a (topological) ring with torsion coprime to p. K=F Abelian Weil group: a (R) := Hom(K × /k × , R). WK
By Kummer theory, a a WK (Z` ) = GK .
Other interesting cases: R = Z or R = Z/`. Define a (R) | µ trivial on (1 + m )× }, Daν (R) = {µ ∈ WK ν a (R) | ι trivial on o× }. Iaν (R) = {ι ∈ WK ν
For R = Z` these are the usual decomposition and inertia a corresponding to ν. subgroups in GK Valuations
Commuting pairs
For E ⊂ K we have the restriction homomorphism a a ρE : WK (R)→WK (R)
Definition a (R) is a c-subgroup if σ ⊂ WK
ρE (σ)
is cyclic ∀E ⊂ K, trdegk (E) = 1.
a (R). Let ΣK (R) be the set of c-subgroups of WK
Valuations
Commuting pairs
Theorem ˆ Z, Z/`n , or Z` . Then Let R = Z, ∀ c-subgroup σ has R-rank ≤ trdeg(K); ∀ c-subgroup σ ∃ ν ∈ VK such that σ is trivial on (1 + mν )× ⊂ K × , ∃ a maximal σ 0 ⊆ σ of R-corank ≤ 1 such that a (R). σ 0 ⊆ Hom(Γν , R) ⊂ Hom(K × , R) = WK
Valuations
Commuting pairs
σ 0 = Iaν (R),
Valuations
for some
ν ∈ VK
Commuting pairs
σ 0 = Iaν (R),
for some
ν ∈ VK
∪Iaν (R)⊆σ σ = Daν (R).
Valuations
Projective geometry of the Weil group Let γ, γ 0 ∈ σ ∈ ΣK (R) be be nonproportional. Then, for every nonconstant f ∈ K × , the restrictions of γ, γ 0 to l(1, f ) are proportional (modulo addition of constants).
Valuations
Projective geometry of the Weil group Let γ, γ 0 ∈ σ ∈ ΣK (R) be be nonproportional. Then, for every nonconstant f ∈ K × , the restrictions of γ, γ 0 to l(1, f ) are proportional (modulo addition of constants). Consider the map K × /k × = Pk (K) → f
Valuations
A2 (R)
7→ (γ(f ), γ 0 (f ))
Projective geometry of the Weil group Let γ, γ 0 ∈ σ ∈ ΣK (R) be be nonproportional. Then, for every nonconstant f ∈ K × , the restrictions of γ, γ 0 to l(1, f ) are proportional (modulo addition of constants). Consider the map K × /k × = Pk (K) → f
A2 (R)
7→ (γ(f ), γ 0 (f ))
This maps every projective line into an affine line, a collineation.
Valuations
Projective geometry of the Weil group Let γ, γ 0 ∈ σ ∈ ΣK (R) be be nonproportional. Then, for every nonconstant f ∈ K × , the restrictions of γ, γ 0 to l(1, f ) are proportional (modulo addition of constants). Consider the map K × /k × = Pk (K) → f
A2 (R)
7→ (γ(f ), γ 0 (f ))
This maps every projective line into an affine line, a collineation.
Images of planes The image of every P2 (k) in A2 (R) is contained in a union of an affine line and a point.
Valuations
Geometry of collineations
Given a map φγ,γ 0 : P2 (k)→{•, ◦, ?} ⊂ A2 (F2 ) such that every l ⊂ P2 (k) is contained in an “affine line” (any subset of two points) one of the following γ, γ 0 , γ + γ 0 is a flag map.
Valuations
Projective geometry of the Weil group
Proposition ∀ σ ∈ ΣK (R) ∃ an inertia element ι = ιν ∈ σ, for some ν ∈ VK .
Valuations
Projective geometry of the Weil group
Proposition ∀ σ ∈ ΣK (R) ∃ an inertia element ι = ιν ∈ σ, for some ν ∈ VK . The elements “commuting” with ι form Daν (R).
Valuations
Projective geometry of the Weil group
In the reconstruction of function fields from their `-Galois groups, i.e., when R = Z` , an isomorphism pairs a (GK , ΣK ) ' (GLa , ΣL )
allows to identify relations between valuations, and the projective structures of Pk (K) and Pl (L); thus a field isomorphism.
Valuations
Application: universal spaces for unramified cohomology ¯ p , ` 6= p, K = k(X), dim(X) ≥ 2. Let k = F
Bogomolov–T. 2014 For α ∈ Hinr (GK , Z/`n ), i ≥ 2, there exist a surjective GK →Ga onto a finite abelian `-group, projective Ga -representations P(Vj ) over k, an explicit open Ga -stable subset Y P◦ ⊂ P := P(Vj ), j
and a rational map % : X→P◦ /Ga such that α is induced from P◦ /Ga . Valuations
Next steps
Arbitrary ground fields, e.g, k = C?
Valuations
Next steps
Arbitrary ground fields, e.g, k = C? c ? Section conjecture for GK
Valuations
Next steps
Arbitrary ground fields, e.g, k = C? c ? Section conjecture for GK
Are all birational types captured by V /Gc , for Gc a central extension of an abelian group?
Valuations
Next steps
Arbitrary ground fields, e.g, k = C? c ? Section conjecture for GK
Are all birational types captured by V /Gc , for Gc a central extension of an abelian group? a , Σ ) encode, e.g., ampleness of the How does the pair (GK K canonical class?
Valuations
Next steps
Arbitrary ground fields, e.g, k = C? c ? Section conjecture for GK
Are all birational types captured by V /Gc , for Gc a central extension of an abelian group? a , Σ ) encode, e.g., ampleness of the How does the pair (GK K canonical class?
Valuations
