Applications of projective geometry January 2009 Applications of projective geometry
Applications of projective geometry
January 2009
Applications of projective geometry
Euclid (Elements, Book I)
Applications of projective geometry
An old result
Dans certains pays d’Europe, dont la France, le theoreme deThales designe un theoreme de geometrie qui affirme que, dans unplan, une droite parallele a l’un des cotes d’un triangle sectionne cedernier en un triangle semblable. Dans d’autres langues,notamment en anglais, ce resultat est connu sous le nom detheoreme d’intersection.
En anglais et allemand, le theoreme de Thales designe un autretheoreme de geometrie qui affirme qu’un triangle inscrit dans uncercle et dont un cote est un diametre est un triangle rectangle.
Le “theoreme de Thales suisse” exprime par contre le carre de lahauteur dans un triangle rectangle.
Applications of projective geometry
An old result
Dans certains pays d’Europe, dont la France, le theoreme deThales designe un theoreme de geometrie qui affirme que, dans unplan, une droite parallele a l’un des cotes d’un triangle sectionne cedernier en un triangle semblable.
Dans d’autres langues,notamment en anglais, ce resultat est connu sous le nom detheoreme d’intersection.
En anglais et allemand, le theoreme de Thales designe un autretheoreme de geometrie qui affirme qu’un triangle inscrit dans uncercle et dont un cote est un diametre est un triangle rectangle.
Le “theoreme de Thales suisse” exprime par contre le carre de lahauteur dans un triangle rectangle.
Applications of projective geometry
An old result
Dans certains pays d’Europe, dont la France, le theoreme deThales designe un theoreme de geometrie qui affirme que, dans unplan, une droite parallele a l’un des cotes d’un triangle sectionne cedernier en un triangle semblable. Dans d’autres langues,notamment en anglais, ce resultat est connu sous le nom detheoreme d’intersection.
En anglais et allemand, le theoreme de Thales designe un autretheoreme de geometrie qui affirme qu’un triangle inscrit dans uncercle et dont un cote est un diametre est un triangle rectangle.
Le “theoreme de Thales suisse” exprime par contre le carre de lahauteur dans un triangle rectangle.
Applications of projective geometry
An old result
Dans certains pays d’Europe, dont la France, le theoreme deThales designe un theoreme de geometrie qui affirme que, dans unplan, une droite parallele a l’un des cotes d’un triangle sectionne cedernier en un triangle semblable. Dans d’autres langues,notamment en anglais, ce resultat est connu sous le nom detheoreme d’intersection.
En anglais et allemand, le theoreme de Thales designe un autretheoreme de geometrie qui affirme qu’un triangle inscrit dans uncercle et dont un cote est un diametre est un triangle rectangle.
Le “theoreme de Thales suisse” exprime par contre le carre de lahauteur dans un triangle rectangle.
Applications of projective geometry
An old result
Dans certains pays d’Europe, dont la France, le theoreme deThales designe un theoreme de geometrie qui affirme que, dans unplan, une droite parallele a l’un des cotes d’un triangle sectionne cedernier en un triangle semblable. Dans d’autres langues,notamment en anglais, ce resultat est connu sous le nom detheoreme d’intersection.
En anglais et allemand, le theoreme de Thales designe un autretheoreme de geometrie qui affirme qu’un triangle inscrit dans uncercle et dont un cote est un diametre est un triangle rectangle.
Le “theoreme de Thales suisse” exprime par contre le carre de lahauteur dans un triangle rectangle.
Applications of projective geometry
Multiplication
Applications of projective geometry
More on projective geometry: axiomatization
Steiner 1832: Durch gehorige Aneignung der wenigen
Grundbeziehungen macht man sich zum Herrn des ganzen
Gegenstandes; es tritt Ordnung in das Chaos ein, und man sieht,
wie alle Theile naturgemass ineinander greifen, in schonster
Ordnung sich in Reihen stellen ...
Staudt 1847: Ich habe in dieser Schrift versucht, die Geometrie der
Lage zu einer selbststandigen Wissenschaft zu machen, welche des
Messens nicht bedarf.
Klein, Pasch, Pieri, ...
Schur: proof of the fundamental theorem of projectivegeometry from incidence axioms, Desargues and Pappus axiom
Hessenberg: Desargues follows from Pappus
Veblen 1910: What we call general projective geometry is,
analytically, the geometry associated with a general number field.
Hilbert ... Klein: When people run out of ideas they start
axiomatizing.
Applications of projective geometry
More on projective geometry: axiomatization
Steiner 1832: Durch gehorige Aneignung der wenigen
Grundbeziehungen macht man sich zum Herrn des ganzen
Gegenstandes; es tritt Ordnung in das Chaos ein, und man sieht,
wie alle Theile naturgemass ineinander greifen, in schonster
Ordnung sich in Reihen stellen ...
Staudt 1847: Ich habe in dieser Schrift versucht, die Geometrie der
Lage zu einer selbststandigen Wissenschaft zu machen, welche des
Messens nicht bedarf.
Klein, Pasch, Pieri, ...
Schur: proof of the fundamental theorem of projectivegeometry from incidence axioms, Desargues and Pappus axiom
Hessenberg: Desargues follows from Pappus
Veblen 1910: What we call general projective geometry is,
analytically, the geometry associated with a general number field.
Hilbert ... Klein: When people run out of ideas they start
axiomatizing.
Applications of projective geometry
More on projective geometry: axiomatization
Steiner 1832: Durch gehorige Aneignung der wenigen
Grundbeziehungen macht man sich zum Herrn des ganzen
Gegenstandes; es tritt Ordnung in das Chaos ein, und man sieht,
wie alle Theile naturgemass ineinander greifen, in schonster
Ordnung sich in Reihen stellen ...
Staudt 1847: Ich habe in dieser Schrift versucht, die Geometrie der
Lage zu einer selbststandigen Wissenschaft zu machen, welche des
Messens nicht bedarf.
Klein, Pasch, Pieri, ...
Schur: proof of the fundamental theorem of projectivegeometry from incidence axioms, Desargues and Pappus axiom
Hessenberg: Desargues follows from Pappus
Veblen 1910: What we call general projective geometry is,
analytically, the geometry associated with a general number field.
Hilbert ... Klein: When people run out of ideas they start
axiomatizing.
Applications of projective geometry
More on projective geometry: axiomatization
Steiner 1832: Durch gehorige Aneignung der wenigen
Grundbeziehungen macht man sich zum Herrn des ganzen
Gegenstandes; es tritt Ordnung in das Chaos ein, und man sieht,
wie alle Theile naturgemass ineinander greifen, in schonster
Ordnung sich in Reihen stellen ...
Staudt 1847: Ich habe in dieser Schrift versucht, die Geometrie der
Lage zu einer selbststandigen Wissenschaft zu machen, welche des
Messens nicht bedarf.
Klein, Pasch, Pieri, ...
Schur: proof of the fundamental theorem of projectivegeometry from incidence axioms, Desargues and Pappus axiom
Hessenberg: Desargues follows from Pappus
Veblen 1910: What we call general projective geometry is,
analytically, the geometry associated with a general number field.
Hilbert ... Klein: When people run out of ideas they start
axiomatizing.
Applications of projective geometry
More on projective geometry: axiomatization
Steiner 1832: Durch gehorige Aneignung der wenigen
Grundbeziehungen macht man sich zum Herrn des ganzen
Gegenstandes; es tritt Ordnung in das Chaos ein, und man sieht,
wie alle Theile naturgemass ineinander greifen, in schonster
Ordnung sich in Reihen stellen ...
Staudt 1847: Ich habe in dieser Schrift versucht, die Geometrie der
Lage zu einer selbststandigen Wissenschaft zu machen, welche des
Messens nicht bedarf.
Klein, Pasch, Pieri, ...
Schur: proof of the fundamental theorem of projectivegeometry from incidence axioms, Desargues and Pappus axiom
Hessenberg: Desargues follows from Pappus
Veblen 1910: What we call general projective geometry is,
analytically, the geometry associated with a general number field.
Hilbert ... Klein: When people run out of ideas they start
axiomatizing.
Applications of projective geometry
More on projective geometry: axiomatization
Steiner 1832: Durch gehorige Aneignung der wenigen
Grundbeziehungen macht man sich zum Herrn des ganzen
Gegenstandes; es tritt Ordnung in das Chaos ein, und man sieht,
wie alle Theile naturgemass ineinander greifen, in schonster
Ordnung sich in Reihen stellen ...
Staudt 1847: Ich habe in dieser Schrift versucht, die Geometrie der
Lage zu einer selbststandigen Wissenschaft zu machen, welche des
Messens nicht bedarf.
Klein, Pasch, Pieri, ...
Schur: proof of the fundamental theorem of projectivegeometry from incidence axioms, Desargues and Pappus axiom
Hessenberg: Desargues follows from Pappus
Veblen 1910: What we call general projective geometry is,
analytically, the geometry associated with a general number field.
Hilbert ... Klein: When people run out of ideas they start
axiomatizing.
Applications of projective geometry
More on projective geometry: axiomatization
Steiner 1832: Durch gehorige Aneignung der wenigen
Grundbeziehungen macht man sich zum Herrn des ganzen
Gegenstandes; es tritt Ordnung in das Chaos ein, und man sieht,
wie alle Theile naturgemass ineinander greifen, in schonster
Ordnung sich in Reihen stellen ...
Staudt 1847: Ich habe in dieser Schrift versucht, die Geometrie der
Lage zu einer selbststandigen Wissenschaft zu machen, welche des
Messens nicht bedarf.
Klein, Pasch, Pieri, ...
Schur: proof of the fundamental theorem of projectivegeometry from incidence axioms, Desargues and Pappus axiom
Hessenberg: Desargues follows from Pappus
Veblen 1910: What we call general projective geometry is,
analytically, the geometry associated with a general number field.
Hilbert ... Klein: When people run out of ideas they start
axiomatizing.
Applications of projective geometry
Fano plane
Applications of projective geometry
Universality theorems
Configuration spaces: moduli of finitely many points with specifiedalignments.
Mnev 1988
Any scheme over Z arises as a configuration space of points in P2.
Lafforgue 2002: singularities of certain strata in some modulispaces arising in the Geometric Langlands Program, e.g.,compactifications of PGLn+1
r /PGLr .
Vakil: Murphy’s law - badly behaved moduli spaces, e.g.,Hilbert schemes of smooth curves in projective space, surfacesin P4, etc.
Applications of projective geometry
Universality theorems
Configuration spaces: moduli of finitely many points with specifiedalignments.
Mnev 1988
Any scheme over Z arises as a configuration space of points in P2.
Lafforgue 2002: singularities of certain strata in some modulispaces arising in the Geometric Langlands Program, e.g.,compactifications of PGLn+1
r /PGLr .
Vakil: Murphy’s law - badly behaved moduli spaces, e.g.,Hilbert schemes of smooth curves in projective space, surfacesin P4, etc.
Applications of projective geometry
Universality theorems
Configuration spaces: moduli of finitely many points with specifiedalignments.
Mnev 1988
Any scheme over Z arises as a configuration space of points in P2.
Lafforgue 2002: singularities of certain strata in some modulispaces arising in the Geometric Langlands Program, e.g.,compactifications of PGLn+1
r /PGLr .
Vakil: Murphy’s law - badly behaved moduli spaces, e.g.,Hilbert schemes of smooth curves in projective space, surfacesin P4, etc.
Applications of projective geometry
Universality theorems
Configuration spaces: moduli of finitely many points with specifiedalignments.
Mnev 1988
Any scheme over Z arises as a configuration space of points in P2.
Lafforgue 2002: singularities of certain strata in some modulispaces arising in the Geometric Langlands Program, e.g.,compactifications of PGLn+1
r /PGLr .
Vakil: Murphy’s law - badly behaved moduli spaces, e.g.,Hilbert schemes of smooth curves in projective space, surfacesin P4, etc.
Applications of projective geometry
Universality theorems
Configuration spaces: moduli of finitely many points with specifiedalignments.
Mnev 1988
Any scheme over Z arises as a configuration space of points in P2.
Lafforgue 2002: singularities of certain strata in some modulispaces arising in the Geometric Langlands Program, e.g.,compactifications of PGLn+1
r /PGLr .
Vakil: Murphy’s law - badly behaved moduli spaces, e.g.,Hilbert schemes of smooth curves in projective space, surfacesin P4, etc.
Applications of projective geometry
Axioms
Definition
A projective structure is a pair (S ,L) where S is a (nonempty) set(of points) and L a collection of subsets l ⊂ S (lines) such that
P1 there exist an s ∈ S and an l ∈ L such that s /∈ l;
P2 for every l ∈ L there exist at least three distinct s, s ′, s ′′ ∈ l;
P3 for every pair of distinct s, s ′ ∈ S there exists exactly one
l = l(s, s ′) ∈ L
such that s, s ′ ∈ l;
P4 for every quadruple of pairwise distinct s, s ′, t, t ′ ∈ S one has
l(s, s ′) ∩ l(t, t ′) 6= ∅ ⇒ l(s, t) ∩ l(s ′, t ′) 6= ∅.
Applications of projective geometry
Axioms
A morphism of projective structures ρ : (S ,L)→(S ′,L′) is a mapof sets ρ : S → S ′ preserving lines, i.e., ρ(l) ∈ L′, for all l ∈ L.
A projective structure (S ,L) satisfies Pappus’ axiom if
PA for all 2-dimensional subspaces and every configuration of sixpoints and lines in these subspaces as below
the intersections are collinear.
Applications of projective geometry
Axioms
A morphism of projective structures ρ : (S ,L)→(S ′,L′) is a mapof sets ρ : S → S ′ preserving lines, i.e., ρ(l) ∈ L′, for all l ∈ L.
A projective structure (S ,L) satisfies Pappus’ axiom if
PA for all 2-dimensional subspaces and every configuration of sixpoints and lines in these subspaces as below
the intersections are collinear.
Applications of projective geometry
Fundamental theorem
Reconstruction
Let (S ,L) be a projective structure of dimension n ≥ 2 whichsatisfies Pappus’ axiom. Then there exists a vector space V over afield k and an isomorphism
σ : Pk(V )∼−→ S .
Moreover, for any two such triples (V , k , σ) and (V ′, k ′, σ′) thereis an isomorphism
V /k∼−→ V ′/k ′
compatible with σ, σ′ and unique up to homothety v 7→ λv ,λ ∈ k∗.
Applications of projective geometry
Main example
Let k be a field and Pn the usual projective space over k ofdimension n ≥ 2. Then Pn(k) carries a projective structure: linesare the usual projective lines P1(k) ⊂ Pn(k).
Let K/k be an extension of fields. Then
S := Pk(K ) = (K \ 0)/k∗
carries a natural (possibly, infinite-dimensional) projectivestructure. Multiplication in K ∗/k∗ preserves this structure.
Applications of projective geometry
Main example
Let k be a field and Pn the usual projective space over k ofdimension n ≥ 2. Then Pn(k) carries a projective structure: linesare the usual projective lines P1(k) ⊂ Pn(k).
Let K/k be an extension of fields. Then
S := Pk(K ) = (K \ 0)/k∗
carries a natural (possibly, infinite-dimensional) projectivestructure.
Multiplication in K ∗/k∗ preserves this structure.
Applications of projective geometry
Main example
Let k be a field and Pn the usual projective space over k ofdimension n ≥ 2. Then Pn(k) carries a projective structure: linesare the usual projective lines P1(k) ⊂ Pn(k).
Let K/k be an extension of fields. Then
S := Pk(K ) = (K \ 0)/k∗
carries a natural (possibly, infinite-dimensional) projectivestructure. Multiplication in K ∗/k∗ preserves this structure.
Applications of projective geometry
Main theorem
Reconstructing fields
Let K/k and K ′/k ′ be field extensions of degree ≥ 3 and
ψ : S = Pk(K )→Pk ′(K ′) = S ′
a bijection of sets which is an isomorphism of abelian groups andof projective structures. Then
k ' k ′ and K ' K ′.
Applications of projective geometry
Main theorem
Reconstructing field homomorphisms
Let K/k and K ′/k ′ be field extensions of degree ≥ 3 and
ψ : S = Pk(K )→Pk ′(K ′) = S ′
an injective homomorphism of abelian groups compatible withprojective structures. Then k ' k ′ and K is isomorphic to asubfield of K ′.
Applications of projective geometry
Pregeometries and geometries
A combinatorial pregeometry (finitary matroid) is a pair (P, cl)where P is a set and
cl : Subsets(P)→ Subsets(P),
such that for all a, b ∈ P and all Y ,Z ⊆ P one has:
Y ⊆ cl(Y ),
if Y ⊆ Z , then cl(Y ) ⊆ cl(Z ),
cl(cl(Y )) = cl(Y ),
if a ∈ cl(Y ), then there is a finite subset Y ′ ⊂ Y such thata ∈ cl(Y ′) (finite character),
(exchange condition) if a ∈ cl(Y ∪ {b}) \ cl(Y ), thenb ∈ cl(Y ∪ {a}).
A geometry is a pregeometry such that cl(a) = a, for all a ∈ P,and cl(∅) = ∅.
Applications of projective geometry
Pregeometries and geometries
A combinatorial pregeometry (finitary matroid) is a pair (P, cl)where P is a set and
cl : Subsets(P)→ Subsets(P),
such that for all a, b ∈ P and all Y ,Z ⊆ P one has:
Y ⊆ cl(Y ),
if Y ⊆ Z , then cl(Y ) ⊆ cl(Z ),
cl(cl(Y )) = cl(Y ),
if a ∈ cl(Y ), then there is a finite subset Y ′ ⊂ Y such thata ∈ cl(Y ′) (finite character),
(exchange condition) if a ∈ cl(Y ∪ {b}) \ cl(Y ), thenb ∈ cl(Y ∪ {a}).
A geometry is a pregeometry such that cl(a) = a, for all a ∈ P,and cl(∅) = ∅.
Applications of projective geometry
Pregeometries and geometries
A combinatorial pregeometry (finitary matroid) is a pair (P, cl)where P is a set and
cl : Subsets(P)→ Subsets(P),
such that for all a, b ∈ P and all Y ,Z ⊆ P one has:
Y ⊆ cl(Y ),
if Y ⊆ Z , then cl(Y ) ⊆ cl(Z ),
cl(cl(Y )) = cl(Y ),
if a ∈ cl(Y ), then there is a finite subset Y ′ ⊂ Y such thata ∈ cl(Y ′) (finite character),
(exchange condition) if a ∈ cl(Y ∪ {b}) \ cl(Y ), thenb ∈ cl(Y ∪ {a}).
A geometry is a pregeometry such that cl(a) = a, for all a ∈ P,and cl(∅) = ∅.
Applications of projective geometry
Examples
1 P = V /k , a vector space over a field k and cl(Y ) the k-spanof Y ⊂ P
2 P = Pk(V ), the usual projective space over a k
3 P = Pk(K ), a field K containing an algebraically closedsubfield k and cl(Y ) - the normal closure of k(Y ) in K ; ageometry is obtained after factoring by x ∼ y iff cl(x) = cl(y).
Applications of projective geometry
Examples
1 P = V /k , a vector space over a field k and cl(Y ) the k-spanof Y ⊂ P
2 P = Pk(V ), the usual projective space over a k
3 P = Pk(K ), a field K containing an algebraically closedsubfield k and cl(Y ) - the normal closure of k(Y ) in K ; ageometry is obtained after factoring by x ∼ y iff cl(x) = cl(y).
Applications of projective geometry
Examples
1 P = V /k , a vector space over a field k and cl(Y ) the k-spanof Y ⊂ P
2 P = Pk(V ), the usual projective space over a k
3 P = Pk(K ), a field K containing an algebraically closedsubfield k and cl(Y ) - the normal closure of k(Y ) in K ;
ageometry is obtained after factoring by x ∼ y iff cl(x) = cl(y).
Applications of projective geometry
Examples
1 P = V /k , a vector space over a field k and cl(Y ) the k-spanof Y ⊂ P
2 P = Pk(V ), the usual projective space over a k
3 P = Pk(K ), a field K containing an algebraically closedsubfield k and cl(Y ) - the normal closure of k(Y ) in K ; ageometry is obtained after factoring by x ∼ y iff cl(x) = cl(y).
Applications of projective geometry
Combinatorial geometries of field extensions
Evans–Hrushovski 1991 / Gismatullin 2008
Let k and k ′ be algebraically closed fields, K/k and K ′/k ′ fieldextensions of transcendence degree ≥ 5 over k , resp. k ′. Then,every isomorphism of combinatorial geometries
Pk(K )→ Pk ′(K ′)
is induced by an isomorphism of purely inseparable closures
K → K ′.
Applications of projective geometry
K-theory
Let KMi (K ) be i-th Milnor K-group of a field K . Recall that
KM1 (K ) = K ∗
and that there is a canonical surjective homomorphism
σK : KM1 (K )⊗KM
1 (K )→KM2 (K )
whose kernel is generated by symbols (x , 1− x), for x ∈ K ∗ \ 1.Let
KMi (K ) := KM
i (K )/infinitely divisible, i = 1, 2,
be the component generated by nondivisible elements.
Applications of projective geometry
K-theory
Let KMi (K ) be i-th Milnor K-group of a field K . Recall that
KM1 (K ) = K ∗
and that there is a canonical surjective homomorphism
σK : KM1 (K )⊗KM
1 (K )→KM2 (K )
whose kernel is generated by symbols (x , 1− x), for x ∈ K ∗ \ 1.
LetKM
i (K ) := KMi (K )/infinitely divisible, i = 1, 2,
be the component generated by nondivisible elements.
Applications of projective geometry
K-theory
Let KMi (K ) be i-th Milnor K-group of a field K . Recall that
KM1 (K ) = K ∗
and that there is a canonical surjective homomorphism
σK : KM1 (K )⊗KM
1 (K )→KM2 (K )
whose kernel is generated by symbols (x , 1− x), for x ∈ K ∗ \ 1.Let
KMi (K ) := KM
i (K )/infinitely divisible, i = 1, 2,
be the component generated by nondivisible elements.
Applications of projective geometry
Reconstructing fields
Let K and L be function fields of algebraic varieties of dimension≥ 2 over algebraically closed fields k and l , respectively. Assumethat there exist isomorphisms
ψi : KMi (K )→ KM
i (L), i = 1, 2,
of abelian groups with a commutative diagram
KM1 (K )⊗KM
1 (K )ψ1⊗ψ1 //
σK
��
KM1 (L)⊗KM
1 (L)
σL
��KM
2 (K )ψ2
// KM2 (L).
Applications of projective geometry
Reconstructing fields
Bogomolov-T. 2008
Then there exists an isomorphism of fields
ψ : K → L,
compatible with ψ1.
Applications of projective geometry
Reconstructing fields
Assume that there exist isomorphisms
ψi : KMi (K )→ KM
i (L), i = 1, 2,
of abelian groups with a commutative diagram
KM1 (K )⊗ KM
1 (K )ψ1⊗ψ1 //
σK
��
KM1 (L)⊗ KM
1 (L)
σL
��KM
2 (K )ψ2
// KM2 (L).
Then there exists a (compatible) isomorphism of fields
ψ : K → L.
Applications of projective geometry
Reconstructing fields
Assume that there exist isomorphisms
ψi : KMi (K )→ KM
i (L), i = 1, 2,
of abelian groups with a commutative diagram
KM1 (K )⊗ KM
1 (K )ψ1⊗ψ1 //
σK
��
KM1 (L)⊗ KM
1 (L)
σL
��KM
2 (K )ψ2
// KM2 (L).
Then there exists a (compatible) isomorphism of fields
ψ : K → L.
Applications of projective geometry
K-groups of function fields
Let K and L be function fields of transcendence degree ≥ 2 overan algebraically closed field k , resp. l . Let
ψ1 : KM1 (K )→KM
1 (L)
be an injective homomorphism.
Assume that there is acommutative diagram
KM1 (K )⊗ KM
1 (K )ψ1⊗ψ1 //
σK
��
KM1 (L)⊗ KM
1 (L)
σL
��KM
2 (K )ψ2
// KM2 (L).
Assume that ψ1(K ∗/k∗) 6⊆ E ∗/l∗, for 1-dimensional E ⊂ L.
Applications of projective geometry
K-groups of function fields
Let K and L be function fields of transcendence degree ≥ 2 overan algebraically closed field k , resp. l . Let
ψ1 : KM1 (K )→KM
1 (L)
be an injective homomorphism. Assume that there is acommutative diagram
KM1 (K )⊗ KM
1 (K )ψ1⊗ψ1 //
σK
��
KM1 (L)⊗ KM
1 (L)
σL
��KM
2 (K )ψ2
// KM2 (L).
Assume that ψ1(K ∗/k∗) 6⊆ E ∗/l∗, for 1-dimensional E ⊂ L.
Applications of projective geometry
K-groups of function fields
Let K and L be function fields of transcendence degree ≥ 2 overan algebraically closed field k , resp. l . Let
ψ1 : KM1 (K )→KM
1 (L)
be an injective homomorphism. Assume that there is acommutative diagram
KM1 (K )⊗ KM
1 (K )ψ1⊗ψ1 //
σK
��
KM1 (L)⊗ KM
1 (L)
σL
��KM
2 (K )ψ2
// KM2 (L).
Assume that ψ1(K ∗/k∗) 6⊆ E ∗/l∗, for 1-dimensional E ⊂ L.
Applications of projective geometry
Reconstructing field homomorphisms
Theorem
Then there exist an r ∈ Q and a homomorphism of fields
ψ : K→L
such that the induced map on K ∗/k∗ coincides with ψr1.
Outlook:
existence of sections of fibrations X → B, e.g., uniqueness ofthe Brauer obstruction to the existence of points.
birational invariants of quotients V /G , where G is a finitegroup and V its representation
Applications of projective geometry
Reconstructing field homomorphisms
Theorem
Then there exist an r ∈ Q and a homomorphism of fields
ψ : K→L
such that the induced map on K ∗/k∗ coincides with ψr1.
Outlook:
existence of sections of fibrations X → B, e.g., uniqueness ofthe Brauer obstruction to the existence of points.
birational invariants of quotients V /G , where G is a finitegroup and V its representation
Applications of projective geometry
Reconstructing field homomorphisms
Theorem
Then there exist an r ∈ Q and a homomorphism of fields
ψ : K→L
such that the induced map on K ∗/k∗ coincides with ψr1.
Outlook:
existence of sections of fibrations X → B, e.g., uniqueness ofthe Brauer obstruction to the existence of points.
birational invariants of quotients V /G , where G is a finitegroup and V its representation
Applications of projective geometry
Sketch of proof
The ground field: Infinitely divisible elements
An element f ∈ K ∗ = KM1 (K ) is infinitely divisible if and only if
f ∈ k∗. In particular,
KM1 (K ) = K ∗/k∗.
Applications of projective geometry
Sketch of proof
1-dimensional subfields
Given a nonconstant f1 ∈ K ∗/k∗, we have
Ker2(f1) = E ∗/k∗,
where E = k(f1)K
is the normal closure in K of the 1-dimensionalfield generated by f1 and
Ker2(f ) := { g ∈ K ∗/k∗ = KM1 (K ) | (f , g) = 0 ∈ KM
2 (K ) }.
Applications of projective geometry
Sketch of proof
Reconstructing lines: Functional equations
Assume that x , y ∈ K ∗ are algebraically independent and that ifboth xb, yb ∈ K ∗ then b ∈ Z.
Let p ∈ k(x)∗, q ∈ k(y)
∗be such
that x , y , p, q are multiplicatively independent in K ∗/k∗. Assumethat there is a nonconstant
Π ∈ k(x/y)∗ · y ∩ k(p/q)
∗ · q.
Assume moreover that this Π arises from infinitely many, moduloscalars, elements p, q as above. Then, modulo k∗,
Π = Πκ,δ(x , y) := (xδ − κy δ)δ, (1)
with κ ∈ k∗ and δ = ±1.
Applications of projective geometry
Sketch of proof
Reconstructing lines: Functional equations
Assume that x , y ∈ K ∗ are algebraically independent and that ifboth xb, yb ∈ K ∗ then b ∈ Z. Let p ∈ k(x)
∗, q ∈ k(y)
∗be such
that x , y , p, q are multiplicatively independent in K ∗/k∗.
Assumethat there is a nonconstant
Π ∈ k(x/y)∗ · y ∩ k(p/q)
∗ · q.
Assume moreover that this Π arises from infinitely many, moduloscalars, elements p, q as above. Then, modulo k∗,
Π = Πκ,δ(x , y) := (xδ − κy δ)δ, (1)
with κ ∈ k∗ and δ = ±1.
Applications of projective geometry
Sketch of proof
Reconstructing lines: Functional equations
Assume that x , y ∈ K ∗ are algebraically independent and that ifboth xb, yb ∈ K ∗ then b ∈ Z. Let p ∈ k(x)
∗, q ∈ k(y)
∗be such
that x , y , p, q are multiplicatively independent in K ∗/k∗. Assumethat there is a nonconstant
Π ∈ k(x/y)∗ · y ∩ k(p/q)
∗ · q.
Assume moreover that this Π arises from infinitely many, moduloscalars, elements p, q as above.
Then, modulo k∗,
Π = Πκ,δ(x , y) := (xδ − κy δ)δ, (1)
with κ ∈ k∗ and δ = ±1.
Applications of projective geometry
Sketch of proof
Reconstructing lines: Functional equations
Assume that x , y ∈ K ∗ are algebraically independent and that ifboth xb, yb ∈ K ∗ then b ∈ Z. Let p ∈ k(x)
∗, q ∈ k(y)
∗be such
that x , y , p, q are multiplicatively independent in K ∗/k∗. Assumethat there is a nonconstant
Π ∈ k(x/y)∗ · y ∩ k(p/q)
∗ · q.
Assume moreover that this Π arises from infinitely many, moduloscalars, elements p, q as above. Then, modulo k∗,
Π = Πκ,δ(x , y) := (xδ − κy δ)δ, (1)
with κ ∈ k∗ and δ = ±1.
Applications of projective geometry
Sketch of proof
Reconstructing lines: Functional equations
The corresponding p and q are given by
pκx ,1(x) = x + κx , qκy ,1(y) = y + κy
pκx ,−1(x) = (x−1 + κx)−1, qκx ,−1(y) = (y−1 + κy )−1
withκxκy = κ.
Applications of projective geometry
Anabelian geometry
Grothendieck’s Anabelian program
The Galois group of a function field determines the field.
Two group operations, + and ·, are encoded in one group.
Let K be a field with absolute Galois group GK := Gal(K/K ).
Let GK be the pro-`-completion of GK , for ` 6= char(K ) a prime.
Uchida, Tamagawa, Mochizuki, Pop, Konigsmann, Zaidi ...:reconstruction of function fields from the full GK or GK .
Applications of projective geometry
Anabelian geometry
Grothendieck’s Anabelian program
The Galois group of a function field determines the field.
Two group operations, + and ·, are encoded in one group.
Let K be a field with absolute Galois group GK := Gal(K/K ).
Let GK be the pro-`-completion of GK , for ` 6= char(K ) a prime.
Uchida, Tamagawa, Mochizuki, Pop, Konigsmann, Zaidi ...:reconstruction of function fields from the full GK or GK .
Applications of projective geometry
Anabelian geometry
Grothendieck’s Anabelian program
The Galois group of a function field determines the field.
Two group operations, + and ·, are encoded in one group.
Let K be a field with absolute Galois group GK := Gal(K/K ).
Let GK be the pro-`-completion of GK , for ` 6= char(K ) a prime.
Uchida, Tamagawa, Mochizuki, Pop, Konigsmann, Zaidi ...:reconstruction of function fields from the full GK or GK .
Applications of projective geometry
Anabelian geometry
Grothendieck’s Anabelian program
The Galois group of a function field determines the field.
Two group operations, + and ·, are encoded in one group.
Let K be a field with absolute Galois group GK := Gal(K/K ).
Let GK be the pro-`-completion of GK , for ` 6= char(K ) a prime.
Uchida, Tamagawa, Mochizuki, Pop, Konigsmann, Zaidi ...:reconstruction of function fields from the full GK or GK .
Applications of projective geometry
Almost abelian anabelian geometry
LetGa
K := GK/[GK ,GK ], GcK := GK/[GK , [GK ,GK ]]
be the abelianization, resp. its canonical central extension.
Thegroup Ga
K is a torsion-free Z`-module of infinite rank.
Let ΣK be the set of all topologically noncyclic subgroups of GaK
that lift to abelian subgroups of GcK .
Bogomolov’s program
The pair (GaK ,ΣK ) determines K .
Applications of projective geometry
Almost abelian anabelian geometry
LetGa
K := GK/[GK ,GK ], GcK := GK/[GK , [GK ,GK ]]
be the abelianization, resp. its canonical central extension. Thegroup Ga
K is a torsion-free Z`-module of infinite rank.
Let ΣK be the set of all topologically noncyclic subgroups of GaK
that lift to abelian subgroups of GcK .
Bogomolov’s program
The pair (GaK ,ΣK ) determines K .
Applications of projective geometry
Almost abelian anabelian geometry
LetGa
K := GK/[GK ,GK ], GcK := GK/[GK , [GK ,GK ]]
be the abelianization, resp. its canonical central extension. Thegroup Ga
K is a torsion-free Z`-module of infinite rank.
Let ΣK be the set of all topologically noncyclic subgroups of GaK
that lift to abelian subgroups of GcK .
Bogomolov’s program
The pair (GaK ,ΣK ) determines K .
Applications of projective geometry
Almost abelian anabelian geometry
LetGa
K := GK/[GK ,GK ], GcK := GK/[GK , [GK ,GK ]]
be the abelianization, resp. its canonical central extension. Thegroup Ga
K is a torsion-free Z`-module of infinite rank.
Let ΣK be the set of all topologically noncyclic subgroups of GaK
that lift to abelian subgroups of GcK .
Bogomolov’s program
The pair (GaK ,ΣK ) determines K .
Applications of projective geometry
Anabelian geometry of surfaces
Theorem (Bogomolov-T. 2004)
Let K and L be function fields over algebraic closures of finite fieldsk , l of characteristic 6= `. Assume that K = k(X ) is a functionfield of a surface X/k and that there exists an isomorphism
ψ : GaK ' Ga
L
inducing a bijection of sets
ΣK = ΣL.
Then, for some c ∈ Z∗` , cψ is induced by an isomorphism of purelyinseparable closures of K and L.
Applications of projective geometry
Sketch of proof: Kummer theory
The abelianized Galois group GaK is dual to K ∗, the
pro-`-completion of K ∗, and one obtains an isomorphism
K ∗ ' L∗.
In our setup, we can interpret GaK as homomorphisms
K ∗/k∗→Z`(1),
arising from
GaK/`
n 3 γn 7→(
f 7→ γ(`n√
f )/`n√
f).
For a subfield E ⊂ K , the map GaK→Ga
E is simply restriction to E .
Applications of projective geometry
Sketch of proof: Kummer theory
The abelianized Galois group GaK is dual to K ∗, the
pro-`-completion of K ∗, and one obtains an isomorphism
K ∗ ' L∗.
In our setup, we can interpret GaK as homomorphisms
K ∗/k∗→Z`(1),
arising from
GaK/`
n 3 γn 7→(
f 7→ γ(`n√
f )/`n√
f).
For a subfield E ⊂ K , the map GaK→Ga
E is simply restriction to E .
Applications of projective geometry
Sketch of proof: Kummer theory
The abelianized Galois group GaK is dual to K ∗, the
pro-`-completion of K ∗, and one obtains an isomorphism
K ∗ ' L∗.
In our setup, we can interpret GaK as homomorphisms
K ∗/k∗→Z`(1),
arising from
GaK/`
n 3 γn 7→(
f 7→ γ(`n√
f )/`n√
f).
For a subfield E ⊂ K , the map GaK→Ga
E is simply restriction to E .
Applications of projective geometry
Valuations
A value group, Γ, is a totally ordered (torsion-free) abelian group.
A (nonarchimedean) valuation on a field K is a pair ν = (ν, Γν)consisting of a value group Γν and a map
ν : K→Γν,∞ = Γν ∪∞
such that
ν : K ∗→Γν is a surjective homomorphism;
ν(κ+ κ′) ≥ min(ν(κ), ν(κ′)) for all κ, κ′ ∈ K ;
ν(0) =∞.
Note that Fp admits only the trivial valuation. A valuation is a flagmap on K : every finite-dimensional Fp-subspace V ⊂ K has a flagV = V1 ⊃ V2 . . . such that ν is constant on Vj \ Vj+1. Conversely,every flag map gives rise to a valuation.
Applications of projective geometry
Valuations
A value group, Γ, is a totally ordered (torsion-free) abelian group.A (nonarchimedean) valuation on a field K is a pair ν = (ν, Γν)consisting of a value group Γν and a map
ν : K→Γν,∞ = Γν ∪∞
such that
ν : K ∗→Γν is a surjective homomorphism;
ν(κ+ κ′) ≥ min(ν(κ), ν(κ′)) for all κ, κ′ ∈ K ;
ν(0) =∞.
Note that Fp admits only the trivial valuation. A valuation is a flagmap on K : every finite-dimensional Fp-subspace V ⊂ K has a flagV = V1 ⊃ V2 . . . such that ν is constant on Vj \ Vj+1. Conversely,every flag map gives rise to a valuation.
Applications of projective geometry
Valuations
A value group, Γ, is a totally ordered (torsion-free) abelian group.A (nonarchimedean) valuation on a field K is a pair ν = (ν, Γν)consisting of a value group Γν and a map
ν : K→Γν,∞ = Γν ∪∞
such that
ν : K ∗→Γν is a surjective homomorphism;
ν(κ+ κ′) ≥ min(ν(κ), ν(κ′)) for all κ, κ′ ∈ K ;
ν(0) =∞.
Note that Fp admits only the trivial valuation.
A valuation is a flagmap on K : every finite-dimensional Fp-subspace V ⊂ K has a flagV = V1 ⊃ V2 . . . such that ν is constant on Vj \ Vj+1. Conversely,every flag map gives rise to a valuation.
Applications of projective geometry
Valuations
A value group, Γ, is a totally ordered (torsion-free) abelian group.A (nonarchimedean) valuation on a field K is a pair ν = (ν, Γν)consisting of a value group Γν and a map
ν : K→Γν,∞ = Γν ∪∞
such that
ν : K ∗→Γν is a surjective homomorphism;
ν(κ+ κ′) ≥ min(ν(κ), ν(κ′)) for all κ, κ′ ∈ K ;
ν(0) =∞.
Note that Fp admits only the trivial valuation. A valuation is a flagmap on K : every finite-dimensional Fp-subspace V ⊂ K has a flagV = V1 ⊃ V2 . . . such that ν is constant on Vj \ Vj+1.
Conversely,every flag map gives rise to a valuation.
Applications of projective geometry
Valuations
A value group, Γ, is a totally ordered (torsion-free) abelian group.A (nonarchimedean) valuation on a field K is a pair ν = (ν, Γν)consisting of a value group Γν and a map
ν : K→Γν,∞ = Γν ∪∞
such that
ν : K ∗→Γν is a surjective homomorphism;
ν(κ+ κ′) ≥ min(ν(κ), ν(κ′)) for all κ, κ′ ∈ K ;
ν(0) =∞.
Note that Fp admits only the trivial valuation. A valuation is a flagmap on K : every finite-dimensional Fp-subspace V ⊂ K has a flagV = V1 ⊃ V2 . . . such that ν is constant on Vj \ Vj+1. Conversely,every flag map gives rise to a valuation.
Applications of projective geometry
Valuations
Denote by Kν , oν ,mν and Kν := oν/mν the completion of K withrespect to ν, the ring of ν-integers in K , the maximal ideal of oνand the residue field.
Keep in mind the exact sequences:
1→o∗ν→K ∗ → Γν→1
1→(1 + mν)→o∗ν→K∗ν→1.
Applications of projective geometry
Valuations
Denote by Kν , oν ,mν and Kν := oν/mν the completion of K withrespect to ν, the ring of ν-integers in K , the maximal ideal of oνand the residue field. Keep in mind the exact sequences:
1→o∗ν→K ∗ → Γν→1
1→(1 + mν)→o∗ν→K∗ν→1.
Applications of projective geometry
Valuations
A homomorphism χ : Γν → Z`(1) gives rise to a homomorphism
χ ◦ ν : K ∗ → Z`(1),
thus to an element of GaK , an inertia element of ν. These form the
inertia subgroup Iaν ⊂ Ga
K .
The decomposition group Daν is the image of Ga
Kνin Ga
K . We havean embedding Ga
Kν↪→ Ga
K and an isomorphism
Daν/Ia
ν ' GaKν .
Applications of projective geometry
Valuations
A homomorphism χ : Γν → Z`(1) gives rise to a homomorphism
χ ◦ ν : K ∗ → Z`(1),
thus to an element of GaK , an inertia element of ν. These form the
inertia subgroup Iaν ⊂ Ga
K .
The decomposition group Daν is the image of Ga
Kνin Ga
K . We havean embedding Ga
Kν↪→ Ga
K and an isomorphism
Daν/Ia
ν ' GaKν .
Applications of projective geometry
A dictionary
Let K be a function field over k = Fp. We have
GaK = {homomorphisms γ : K ∗→Z`(1)}Daν = {µ ∈ Ga
K |µ trivial on (1 + mν)},Iaν = {ι ∈ Ga
K | ι trivial on o∗ν}.
Inertia elements define flag maps on K .
Applications of projective geometry
A dictionary
Let K be a function field over k = Fp. We have
GaK = {homomorphisms γ : K ∗→Z`(1)}Daν = {µ ∈ Ga
K |µ trivial on (1 + mν)},Iaν = {ι ∈ Ga
K | ι trivial on o∗ν}.
Inertia elements define flag maps on K .
Applications of projective geometry
Projective geometry of the Galois group
Key fact
Let γ, γ′ ∈ GaK ' Z∞` be two nonproportional elements lifting to
commuting elements in GcK . Then, for any nonconstant f ∈ K ∗ the
restrictions of γ, γ′ to the projective line PFp (Fp ⊕ f Fp) areproportional (modulo addition of constants).
Consider the map
K ∗/k∗ = Pk(K ) → A2(Z`)
f 7→ (γ(f ), γ′(f ))
This maps every projective line into an affine line, a collineation.
Applications of projective geometry
Projective geometry of the Galois group
Key fact
Let γ, γ′ ∈ GaK ' Z∞` be two nonproportional elements lifting to
commuting elements in GcK . Then, for any nonconstant f ∈ K ∗ the
restrictions of γ, γ′ to the projective line PFp (Fp ⊕ f Fp) areproportional (modulo addition of constants).
Consider the map
K ∗/k∗ = Pk(K ) → A2(Z`)
f 7→ (γ(f ), γ′(f ))
This maps every projective line into an affine line, a collineation.
Applications of projective geometry
Projective geometry of the Galois group
Key fact
Let γ, γ′ ∈ GaK ' Z∞` be two nonproportional elements lifting to
commuting elements in GcK . Then, for any nonconstant f ∈ K ∗ the
restrictions of γ, γ′ to the projective line PFp (Fp ⊕ f Fp) areproportional (modulo addition of constants).
Consider the map
K ∗/k∗ = Pk(K ) → A2(Z`)
f 7→ (γ(f ), γ′(f ))
This maps every projective line into an affine line, a collineation.
Applications of projective geometry
Projective geometry of the Galois group
Lemma
A map α : P2(Fp)→Z/2 is a flag map iff the restiction to everyP1(Fp) ⊂ P2(Fp) is a flag map, i.e., constant on the complementof one point.
Counterexample: the Fano plane
(0:1:0)
(1:0:0)(1:0:1)(0:0:1)
(0:1:1) (1:1:0)
Applications of projective geometry
Projective geometry of the Galois group
Lemma
A map α : P2(Fp)→Z/2 is a flag map iff the restiction to everyP1(Fp) ⊂ P2(Fp) is a flag map, i.e., constant on the complementof one point.
Counterexample: the Fano plane
(0:1:0)
(1:0:0)(1:0:1)(0:0:1)
(0:1:1) (1:1:0)
Applications of projective geometry
Projective geometry of the Galois group
Projective/affine geometry considerations produce a flag map inthe Z`-linear span of γ, γ′.
Every noncyclic subgroup of GaK lifting to an abelian subgroup of
GcK contains an inertia element ι = ιν for some valuation ν of K .
The elements “commuting” with ι form Daν .
The combinatorial structure of the fan ΣK allows to reconstructthe projective structure of Pk(K ).
Applications of projective geometry
Projective geometry of the Galois group
Projective/affine geometry considerations produce a flag map inthe Z`-linear span of γ, γ′.
Every noncyclic subgroup of GaK lifting to an abelian subgroup of
GcK contains an inertia element ι = ιν for some valuation ν of K .
The elements “commuting” with ι form Daν .
The combinatorial structure of the fan ΣK allows to reconstructthe projective structure of Pk(K ).
Applications of projective geometry
Projective geometry of the Galois group
Projective/affine geometry considerations produce a flag map inthe Z`-linear span of γ, γ′.
Every noncyclic subgroup of GaK lifting to an abelian subgroup of
GcK contains an inertia element ι = ιν for some valuation ν of K .
The elements “commuting” with ι form Daν .
The combinatorial structure of the fan ΣK allows to reconstructthe projective structure of Pk(K ).
Applications of projective geometry
What about curves?
Let k = Fp and K = k(C ). Let GK be the absolute Galois groupof K . Let
IK := {Iaν},
the set of inertia subgroups Iaν ⊂ G a
K of nontrivial divisorialvaluations of K (i.e., points of C ).
Bogomolov-T. 2008
Assume that g(C ) > 2 and that
(G aK , IK ) ' (G a
K, IK ).
ThenJ ∼ J.
Applications of projective geometry
Curves and their Jacobians
Let k be any field and C/k a smooth curve over k of genusg(C ) ≥ 2, with C (k) 6= ∅. For each n ∈ N, we have
(c1, . . . , cn) // (c1 + · · ·+ cn)
Cn // Symn(C )
λn
��Jn
Choosing c0 ∈ C (k), we may identify Jn ' J.
Image(λg−1) = Θ ⊂ J, the Theta divisor
Torelli: the pair (J,Θ) determines C , up to isomorphism
for n ≥ 2g − 1, λn is a Pn−g-bundle
Applications of projective geometry
Curves and their Jacobians
Let k be any field and C/k a smooth curve over k of genusg(C ) ≥ 2, with C (k) 6= ∅. For each n ∈ N, we have
(c1, . . . , cn) // (c1 + · · ·+ cn)
Cn // Symn(C )
λn
��Jn
Choosing c0 ∈ C (k), we may identify Jn ' J.
Image(λg−1) = Θ ⊂ J, the Theta divisor
Torelli: the pair (J,Θ) determines C , up to isomorphism
for n ≥ 2g − 1, λn is a Pn−g-bundle
Applications of projective geometry
Curves and their Jacobians
Let k be any field and C/k a smooth curve over k of genusg(C ) ≥ 2, with C (k) 6= ∅. For each n ∈ N, we have
(c1, . . . , cn) // (c1 + · · ·+ cn)
Cn // Symn(C )
λn
��Jn
Choosing c0 ∈ C (k), we may identify Jn ' J.
Image(λg−1) = Θ ⊂ J, the Theta divisor
Torelli: the pair (J,Θ) determines C , up to isomorphism
for n ≥ 2g − 1, λn is a Pn−g-bundle
Applications of projective geometry
Curves and their Jacobians
Let k be any field and C/k a smooth curve over k of genusg(C ) ≥ 2, with C (k) 6= ∅. For each n ∈ N, we have
(c1, . . . , cn) // (c1 + · · ·+ cn)
Cn // Symn(C )
λn
��Jn
Choosing c0 ∈ C (k), we may identify Jn ' J.
Image(λg−1) = Θ ⊂ J, the Theta divisor
Torelli: the pair (J,Θ) determines C , up to isomorphism
for n ≥ 2g − 1, λn is a Pn−g-bundle
Applications of projective geometry
Curves and their Jacobians
Let k be any field and C/k a smooth curve over k of genusg(C ) ≥ 2, with C (k) 6= ∅. For each n ∈ N, we have
(c1, . . . , cn) // (c1 + · · ·+ cn)
Cn // Symn(C )
λn
��Jn
Choosing c0 ∈ C (k), we may identify Jn ' J.
Image(λg−1) = Θ ⊂ J, the Theta divisor
Torelli: the pair (J,Θ) determines C , up to isomorphism
for n ≥ 2g − 1, λn is a Pn−g-bundle
Applications of projective geometry
Abelian varieties over finite fields
Let A be an abelian variety of dimension g over a finite field k .Recall that
A(k) = p-part⊕⊕6=p
(Q`/Z`)2g.
Tate
Hom(A, A)⊗ Z` = HomZ`[Fr](T`(A),T`(A)).
In particular, A and A are isogenous iff the characteristicpolynomials of the Frobenius coincide.
Applications of projective geometry
Abelian varieties over finite fields
Let A be an abelian variety of dimension g over a finite field k .Recall that
A(k) = p-part⊕⊕6=p
(Q`/Z`)2g.
Tate
Hom(A, A)⊗ Z` = HomZ`[Fr](T`(A),T`(A)).
In particular, A and A are isogenous iff the characteristicpolynomials of the Frobenius coincide.
Applications of projective geometry
Abelian varieties over finite fields
Let A be an abelian variety of dimension g over a finite field k .Recall that
A(k) = p-part⊕⊕6=p
(Q`/Z`)2g.
Tate
Hom(A, A)⊗ Z` = HomZ`[Fr](T`(A),T`(A)).
In particular, A and A are isogenous iff the characteristicpolynomials of the Frobenius coincide.
Applications of projective geometry
Divisibilities
Bogomolov-T. 2008
Let A and A be abelian varieties of dimension g over finite fields k ,resp. k . Let kn/k , resp. kn/k , be the unique extensions of degreen. Assume that
#A(kn) | #A(kn)
for infinitely many n ∈ N.
Then char(k) = char(k) and A and Aare isogenous over k .
Applications of projective geometry
Divisibilities
Bogomolov-T. 2008
Let A and A be abelian varieties of dimension g over finite fields k ,resp. k . Let kn/k , resp. kn/k , be the unique extensions of degreen. Assume that
#A(kn) | #A(kn)
for infinitely many n ∈ N. Then char(k) = char(k) and A and Aare isogenous over k .
Applications of projective geometry
Sketch of proof
Let A be an abelian variety over k1 := Fq. Let {αj}j=1,...,2g be theset of eigenvalues of Frobenius on H1
et(A,Q`), for ` 6= p, andΓA ⊂ C∗ the multiplicative subgroup spanned by α.
The sequence
R(n) := #A(kn) =
2g∏j=1
(αnj − 1).
is a simple linear recurrence with roots in Γ = ΓA.
There is an isomorphism of rings
{ Recurrences with roots in Γ} ⇔ C[Γ].
Applications of projective geometry
Sketch of proof
Let A be an abelian variety over k1 := Fq. Let {αj}j=1,...,2g be theset of eigenvalues of Frobenius on H1
et(A,Q`), for ` 6= p, andΓA ⊂ C∗ the multiplicative subgroup spanned by α.The sequence
R(n) := #A(kn) =
2g∏j=1
(αnj − 1).
is a simple linear recurrence with roots in Γ = ΓA.
There is an isomorphism of rings
{ Recurrences with roots in Γ} ⇔ C[Γ].
Applications of projective geometry
Sketch of proof
Let A be an abelian variety over k1 := Fq. Let {αj}j=1,...,2g be theset of eigenvalues of Frobenius on H1
et(A,Q`), for ` 6= p, andΓA ⊂ C∗ the multiplicative subgroup spanned by α.The sequence
R(n) := #A(kn) =
2g∏j=1
(αnj − 1).
is a simple linear recurrence with roots in Γ = ΓA.
There is an isomorphism of rings
{ Recurrences with roots in Γ} ⇔ C[Γ].
Applications of projective geometry
Sketch of proof: Recurrence sequences
Corvaja-Zannier 2002
Let R and R be simple linear recurrences such that
1 R(n), R(n) 6= 0, for all n, n� 0;
2 the subgroup Γ ⊂ C∗ generated by the roots of R and R istorsion-free;
3 there is a finitely-generated subring A ⊂ C withR(n)/R(n) ∈ A, for infinitely many n ∈ N.
ThenQ : N → C
n 7→ R(n)/R(n)
is a simple linear recurrence. In particular, the FQ ∈ C[Γ] and
FQ · FR = FR .
Applications of projective geometry
Sketch of proof: Recurrence sequences
Corvaja-Zannier 2002
Let R and R be simple linear recurrences such that
1 R(n), R(n) 6= 0, for all n, n� 0;
2 the subgroup Γ ⊂ C∗ generated by the roots of R and R istorsion-free;
3 there is a finitely-generated subring A ⊂ C withR(n)/R(n) ∈ A, for infinitely many n ∈ N.
ThenQ : N → C
n 7→ R(n)/R(n)
is a simple linear recurrence. In particular, the FQ ∈ C[Γ] and
FQ · FR = FR .
Applications of projective geometry
Curves and their Jacobians
Let C be another smooth projective curve and J its Jacobian.Isomorphism of pairs:
φ : (C , J)→(C , J)
J(k)
φ0
��
J1(k)
φ1
��
C (k)j1oo
φs
��J(k) J1(k) C (k)
j1oo
where
φ0: isomorphism of abstract abelian groups;
φ1: isomorphism of homogeneous spaces, compatible with φ0;
the restriction φs : C (k)→C (k) of φ1 is a bijection of sets.
Applications of projective geometry
Curves and their Jacobians
For all #k � 0 the group J(k) is generated by C (k).
Let
k = k1 ⊂ k2 ⊂ . . . ⊂ kn ⊂ . . .
be the tower of degree 2 extensions. To characterize J(kn) itsuffices to characterize C (kn). Let C be a nonhyperelliptic curve ofgenus g(C ) ≥ 3.
Inductive characterization of J(kn), n ∈ N
J(kn) is generated by c ∈ C (k) such that there exists a pointc ′ ∈ C (k) with
c + c ′ ∈ J(kn−1).
Applications of projective geometry
Curves and their Jacobians
For all #k � 0 the group J(k) is generated by C (k). Let
k = k1 ⊂ k2 ⊂ . . . ⊂ kn ⊂ . . .
be the tower of degree 2 extensions.
To characterize J(kn) itsuffices to characterize C (kn). Let C be a nonhyperelliptic curve ofgenus g(C ) ≥ 3.
Inductive characterization of J(kn), n ∈ N
J(kn) is generated by c ∈ C (k) such that there exists a pointc ′ ∈ C (k) with
c + c ′ ∈ J(kn−1).
Applications of projective geometry
Curves and their Jacobians
For all #k � 0 the group J(k) is generated by C (k). Let
k = k1 ⊂ k2 ⊂ . . . ⊂ kn ⊂ . . .
be the tower of degree 2 extensions. To characterize J(kn) itsuffices to characterize C (kn).
Let C be a nonhyperelliptic curve ofgenus g(C ) ≥ 3.
Inductive characterization of J(kn), n ∈ N
J(kn) is generated by c ∈ C (k) such that there exists a pointc ′ ∈ C (k) with
c + c ′ ∈ J(kn−1).
Applications of projective geometry
Curves and their Jacobians
For all #k � 0 the group J(k) is generated by C (k). Let
k = k1 ⊂ k2 ⊂ . . . ⊂ kn ⊂ . . .
be the tower of degree 2 extensions. To characterize J(kn) itsuffices to characterize C (kn). Let C be a nonhyperelliptic curve ofgenus g(C ) ≥ 3.
Inductive characterization of J(kn), n ∈ N
J(kn) is generated by c ∈ C (k) such that there exists a pointc ′ ∈ C (k) with
c + c ′ ∈ J(kn−1).
Applications of projective geometry
Curves and their Jacobians
For all #k � 0 the group J(k) is generated by C (k). Let
k = k1 ⊂ k2 ⊂ . . . ⊂ kn ⊂ . . .
be the tower of degree 2 extensions. To characterize J(kn) itsuffices to characterize C (kn). Let C be a nonhyperelliptic curve ofgenus g(C ) ≥ 3.
Inductive characterization of J(kn), n ∈ N
J(kn) is generated by c ∈ C (k) such that there exists a pointc ′ ∈ C (k) with
c + c ′ ∈ J(kn−1).
Applications of projective geometry
Curves and their Jacobians: Torelli
Theorem
Let (C , J)→ (C , J) be an isomorphism of pairs. Then J isisogenous to J.
Proof.
1 Choose k1, k1 (sufficiently large) such that φ(J(k1)) ⊂ J(k1)
2 Define C (kn), resp. C (kn), intrinsically, as above.
3 Have φ(J(kn)) ⊂ J(kn), for all n ∈ N.
4 #J(kn) | #J(kn)
5 Apply the result about divisibility of recurrence sequences.
Applications of projective geometry
Curves and their Jacobians: Torelli
Theorem
Let (C , J)→ (C , J) be an isomorphism of pairs. Then J isisogenous to J.
Proof.
1 Choose k1, k1 (sufficiently large) such that φ(J(k1)) ⊂ J(k1)
2 Define C (kn), resp. C (kn), intrinsically, as above.
3 Have φ(J(kn)) ⊂ J(kn), for all n ∈ N.
4 #J(kn) | #J(kn)
5 Apply the result about divisibility of recurrence sequences.
Applications of projective geometry
Curves and their Jacobians: Torelli
Theorem
Let (C , J)→ (C , J) be an isomorphism of pairs. Then J isisogenous to J.
Proof.
1 Choose k1, k1 (sufficiently large) such that φ(J(k1)) ⊂ J(k1)
2 Define C (kn), resp. C (kn), intrinsically, as above.
3 Have φ(J(kn)) ⊂ J(kn), for all n ∈ N.
4 #J(kn) | #J(kn)
5 Apply the result about divisibility of recurrence sequences.
Applications of projective geometry
Curves and their Jacobians: Torelli
Theorem
Let (C , J)→ (C , J) be an isomorphism of pairs. Then J isisogenous to J.
Proof.
1 Choose k1, k1 (sufficiently large) such that φ(J(k1)) ⊂ J(k1)
2 Define C (kn), resp. C (kn), intrinsically, as above.
3 Have φ(J(kn)) ⊂ J(kn), for all n ∈ N.
4 #J(kn) | #J(kn)
5 Apply the result about divisibility of recurrence sequences.
Applications of projective geometry
Curves and their Jacobians: Torelli
Theorem
Let (C , J)→ (C , J) be an isomorphism of pairs. Then J isisogenous to J.
Proof.
1 Choose k1, k1 (sufficiently large) such that φ(J(k1)) ⊂ J(k1)
2 Define C (kn), resp. C (kn), intrinsically, as above.
3 Have φ(J(kn)) ⊂ J(kn), for all n ∈ N.
4 #J(kn) | #J(kn)
5 Apply the result about divisibility of recurrence sequences.
Applications of projective geometry
Curves and their Jacobians: Torelli
Theorem
Let (C , J)→ (C , J) be an isomorphism of pairs. Then J isisogenous to J.
Proof.
1 Choose k1, k1 (sufficiently large) such that φ(J(k1)) ⊂ J(k1)
2 Define C (kn), resp. C (kn), intrinsically, as above.
3 Have φ(J(kn)) ⊂ J(kn), for all n ∈ N.
4 #J(kn) | #J(kn)
5 Apply the result about divisibility of recurrence sequences.
Applications of projective geometry