Applications of projective geometry

Transcription

1 January 2009

2 Euclid (Elements, Book I)

3 An old result

4 An old result Dans certains pays d Europe, dont la France, le théorème de Thalès désigne un théorème de géométrie qui affirme que, dans un plan, une droite parallèle à l un des côtés d un triangle sectionne ce dernier en un triangle semblable.

5 An old result Dans certains pays d Europe, dont la France, le théorème de Thalès désigne un théorème de géométrie qui affirme que, dans un plan, une droite parallèle à l un des côtés d un triangle sectionne ce dernier en un triangle semblable. Dans d autres langues, notamment en anglais, ce résultat est connu sous le nom de théorème d intersection.

6 An old result Dans certains pays d Europe, dont la France, le théorème de Thalès désigne un théorème de géométrie qui affirme que, dans un plan, une droite parallèle à l un des côtés d un triangle sectionne ce dernier en un triangle semblable. Dans d autres langues, notamment en anglais, ce résultat est connu sous le nom de théorème d intersection. En anglais et allemand, le théorème de Thalès désigne un autre théorème de géométrie qui affirme qu un triangle inscrit dans un cercle et dont un côté est un diamètre est un triangle rectangle.

7 An old result Dans certains pays d Europe, dont la France, le théorème de Thalès désigne un théorème de géométrie qui affirme que, dans un plan, une droite parallèle à l un des côtés d un triangle sectionne ce dernier en un triangle semblable. Dans d autres langues, notamment en anglais, ce résultat est connu sous le nom de théorème d intersection. En anglais et allemand, le théorème de Thalès désigne un autre théorème de géométrie qui affirme qu un triangle inscrit dans un cercle et dont un côté est un diamètre est un triangle rectangle. Le théorème de Thalès suisse exprime par contre le carré de la hauteur dans un triangle rectangle.

8 Multiplication

9 More on projective geometry: axiomatization Steiner 1832: Durch gehörige 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 naturgemäss ineinander greifen, in schönster Ordnung sich in Reihen stellen...

10 More on projective geometry: axiomatization Steiner 1832: Durch gehörige 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 naturgemäss ineinander greifen, in schönster Ordnung sich in Reihen stellen... Staudt 1847: Ich habe in dieser Schrift versucht, die Geometrie der Lage zu einer selbstständigen Wissenschaft zu machen, welche des Messens nicht bedarf.

11 More on projective geometry: axiomatization Steiner 1832: Durch gehörige 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 naturgemäss ineinander greifen, in schönster Ordnung sich in Reihen stellen... Staudt 1847: Ich habe in dieser Schrift versucht, die Geometrie der Lage zu einer selbstständigen Wissenschaft zu machen, welche des Messens nicht bedarf. Klein, Pasch, Pieri,...

12 More on projective geometry: axiomatization Steiner 1832: Durch gehörige 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 naturgemäss ineinander greifen, in schönster Ordnung sich in Reihen stellen... Staudt 1847: Ich habe in dieser Schrift versucht, die Geometrie der Lage zu einer selbstständigen Wissenschaft zu machen, welche des Messens nicht bedarf. Klein, Pasch, Pieri,... Schur: proof of the fundamental theorem of projective geometry from incidence axioms, Desargues and Pappus axiom

13 More on projective geometry: axiomatization Steiner 1832: Durch gehörige 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 naturgemäss ineinander greifen, in schönster Ordnung sich in Reihen stellen... Staudt 1847: Ich habe in dieser Schrift versucht, die Geometrie der Lage zu einer selbstständigen Wissenschaft zu machen, welche des Messens nicht bedarf. Klein, Pasch, Pieri,... Schur: proof of the fundamental theorem of projective geometry from incidence axioms, Desargues and Pappus axiom Hessenberg: Desargues follows from Pappus

14 More on projective geometry: axiomatization Steiner 1832: Durch gehörige 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 naturgemäss ineinander greifen, in schönster Ordnung sich in Reihen stellen... Staudt 1847: Ich habe in dieser Schrift versucht, die Geometrie der Lage zu einer selbstständigen Wissenschaft zu machen, welche des Messens nicht bedarf. Klein, Pasch, Pieri,... Schur: proof of the fundamental theorem of projective geometry 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.

15 More on projective geometry: axiomatization Steiner 1832: Durch gehörige 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 naturgemäss ineinander greifen, in schönster Ordnung sich in Reihen stellen... Staudt 1847: Ich habe in dieser Schrift versucht, die Geometrie der Lage zu einer selbstständigen Wissenschaft zu machen, welche des Messens nicht bedarf. Klein, Pasch, Pieri,... Schur: proof of the fundamental theorem of projective geometry 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.

16 Fano plane

17 Universality theorems

18 Universality theorems Configuration spaces: moduli of finitely many points with specified alignments.

19 Universality theorems Configuration spaces: moduli of finitely many points with specified alignments. Mnëv 1988 Any scheme over Z arises as a configuration space of points in P 2.

20 Universality theorems Configuration spaces: moduli of finitely many points with specified alignments. Mnëv 1988 Any scheme over Z arises as a configuration space of points in P 2. Lafforgue 2002: singularities of certain strata in some moduli spaces arising in the Geometric Langlands Program, e.g., compactifications of PGL n+1 r /PGL r.

21 Universality theorems Configuration spaces: moduli of finitely many points with specified alignments. Mnëv 1988 Any scheme over Z arises as a configuration space of points in P 2. Lafforgue 2002: singularities of certain strata in some moduli spaces arising in the Geometric Langlands Program, e.g., compactifications of PGL n+1 r /PGL r. Vakil: Murphy s law - badly behaved moduli spaces, e.g., Hilbert schemes of smooth curves in projective space, surfaces in P 4, etc.

22 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 such that s, s l; l = l(s, s ) L P4 for every quadruple of pairwise distinct s, s, t, t S one has l(s, s ) l(t, t ) l(s, t) l(s, t ).

23 Axioms A morphism of projective structures ρ : (S, L) (S, L ) is a map of sets ρ : S S preserving lines, i.e., ρ(l) L, for all l L.

24 Axioms A morphism of projective structures ρ : (S, L) (S, L ) is a map of 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 six points and lines in these subspaces as below the intersections are collinear.

25 Fundamental theorem Reconstruction Let (S, L) be a projective structure of dimension n 2 which satisfies Pappus axiom. Then there exists a vector space V over a field k and an isomorphism σ : Pk(V ) S. Moreover, for any two such triples (V, k, σ) and (V, k, σ ) there is an isomorphism V /k V /k compatible with σ, σ and unique up to homothety v λv, λ k.

26 Main example Let k be a field and P n the usual projective space over k of dimension n 2. Then P n (k) carries a projective structure: lines are the usual projective lines P 1 (k) P n (k).

27 Main example Let k be a field and P n the usual projective space over k of dimension n 2. Then P n (k) carries a projective structure: lines are the usual projective lines P 1 (k) P n (k). Let K/k be an extension of fields. Then S := Pk(K) = (K \ 0)/k carries a natural (possibly, infinite-dimensional) projective structure.

28 Main example Let k be a field and P n the usual projective space over k of dimension n 2. Then P n (k) carries a projective structure: lines are the usual projective lines P 1 (k) P n (k). Let K/k be an extension of fields. Then S := Pk(K) = (K \ 0)/k carries a natural (possibly, infinite-dimensional) projective structure. Multiplication in K /k preserves this structure.

29 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 and of projective structures. Then k k and K K.

30 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 with projective structures. Then k k and K is isomorphic to a subfield of K.

31 Pregeometries and geometries

32 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 that a cl(y ) (finite character), (exchange condition) if a cl(y {b}) \ cl(y ), then b cl(y {a}).

33 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 that a cl(y ) (finite character), (exchange condition) if a cl(y {b}) \ cl(y ), then b cl(y {a}). A geometry is a pregeometry such that cl(a) = a, for all a P, and cl( ) =.

34 Examples 1 P = V /k, a vector space over a field k and cl(y ) the k-span of Y P

35 Examples 1 P = V /k, a vector space over a field k and cl(y ) the k-span of Y P 2 P = Pk(V ), the usual projective space over a k

36 Examples 1 P = V /k, a vector space over a field k and cl(y ) the k-span of Y P 2 P = Pk(V ), the usual projective space over a k 3 P = P k (K), a field K containing an algebraically closed subfield k and cl(y ) - the normal closure of k(y ) in K;

37 Examples 1 P = V /k, a vector space over a field k and cl(y ) the k-span of Y P 2 P = Pk(V ), the usual projective space over a k 3 P = P k (K), a field K containing an algebraically closed subfield k and cl(y ) - the normal closure of k(y ) in K; a geometry is obtained after factoring by x y iff cl(x) = cl(y).

38 Combinatorial geometries of field extensions Evans Hrushovski 1991 / Gismatullin 2008 Let k and k be algebraically closed fields, K/k and K /k field extensions of transcendence degree 5 over k, resp. k. Then, every isomorphism of combinatorial geometries P k (K) P k (K ) is induced by an isomorphism of purely inseparable closures K K.

39 K-theory

40 K-theory Let K M i (K) be i-th Milnor K-group of a field K. Recall that K M 1 (K) = K and that there is a canonical surjective homomorphism σ K : K M 1 (K) K M 1 (K) K M 2 (K) whose kernel is generated by symbols (x, 1 x), for x K \ 1.

41 K-theory Let K M i (K) be i-th Milnor K-group of a field K. Recall that K M 1 (K) = K and that there is a canonical surjective homomorphism σ K : K M 1 (K) K M 1 (K) K M 2 (K) whose kernel is generated by symbols (x, 1 x), for x K \ 1. Let K M i (K) := K M i (K)/infinitely divisible, i = 1, 2, be the component generated by nondivisible elements.

42 Reconstructing fields Let K and L be function fields of algebraic varieties of dimension 2 over algebraically closed fields k and l, respectively. Assume that there exist isomorphisms ψ i : K M i (K) K M i (L), i = 1, 2, of abelian groups with a commutative diagram K M 1 (K) KM 1 (K) ψ 1 ψ 1 K M 1 (L) KM 1 (L) σ K K M 2 (K) ψ 2 K M 2 (L). σ L

43 Reconstructing fields Bogomolov-T Then there exists an isomorphism of fields ψ : K L, compatible with ψ 1.

44 Reconstructing fields Assume that there exist isomorphisms ψ i : K M i (K) K M i (L), i = 1, 2, of abelian groups with a commutative diagram K M 1 (K) K M 1 (K) ψ 1 ψ 1 K M 1 (L) K M 1 (L) σ K K M 2 (K) ψ2 K M 2 (L). σ L

45 Reconstructing fields Assume that there exist isomorphisms ψ i : K M i (K) K M i (L), i = 1, 2, of abelian groups with a commutative diagram K M 1 (K) K M 1 (K) ψ 1 ψ 1 K M 1 (L) K M 1 (L) σ K K M 2 (K) ψ2 K M 2 (L). σ L Then there exists a (compatible) isomorphism of fields ψ : K L.

46 K-groups of function fields Let K and L be function fields of transcendence degree 2 over an algebraically closed field k, resp. l. Let be an injective homomorphism. ψ 1 : K M 1 (K) K M 1 (L)

47 K-groups of function fields Let K and L be function fields of transcendence degree 2 over an algebraically closed field k, resp. l. Let ψ 1 : K M 1 (K) K M 1 (L) be an injective homomorphism. Assume that there is a commutative diagram K M 1 (K) K M 1 (K) ψ 1 ψ 1 K M 1 (L) K M 1 (L) σ K K M 2 (K) ψ2 K M 2 (L). σ L

48 K-groups of function fields Let K and L be function fields of transcendence degree 2 over an algebraically closed field k, resp. l. Let ψ 1 : K M 1 (K) K M 1 (L) be an injective homomorphism. Assume that there is a commutative diagram K M 1 (K) K M 1 (K) ψ 1 ψ 1 K M 1 (L) K M 1 (L) σ K K M 2 (K) ψ2 K M 2 (L). σ L Assume that ψ 1 (K /k ) E /l, for 1-dimensional E L.

49 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 ψ 1 r.

50 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 ψ r 1. Outlook: existence of sections of fibrations X B, e.g., uniqueness of the Brauer obstruction to the existence of points.

51 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 ψ r 1. Outlook: existence of sections of fibrations X B, e.g., uniqueness of the Brauer obstruction to the existence of points. birational invariants of quotients V /G, where G is a finite group and V its representation

52 Sketch of proof The ground field: Infinitely divisible elements An element f K = K M 1 (K) is infinitely divisible if and only if f k. In particular, K M 1 (K) = K /k.

53 Sketch of proof 1-dimensional subfields Given a nonconstant f 1 K /k, we have Ker 2 (f 1 ) = E /k, where E = k(f 1 ) K is the normal closure in K of the 1-dimensional field generated by f 1 and Ker 2 (f ) := { g K /k = K M 1 (K) (f, g) = 0 K M 2 (K) }.

54 Sketch of proof Reconstructing lines: Functional equations Assume that x, y K are algebraically independent and that if both x b, y b K then b Z.

55 Sketch of proof Reconstructing lines: Functional equations Assume that x, y K are algebraically independent and that if both x b, y b K then b Z. Let p k(x), q k(y) be such that x, y, p, q are multiplicatively independent in K /k.

56 Sketch of proof Reconstructing lines: Functional equations Assume that x, y K are algebraically independent and that if both x b, y b K then b Z. Let p k(x), q k(y) be such that x, y, p, q are multiplicatively independent in K /k. Assume that there is a nonconstant Π k(x/y) y k(p/q) q. Assume moreover that this Π arises from infinitely many, modulo scalars, elements p, q as above.

57 Sketch of proof Reconstructing lines: Functional equations Assume that x, y K are algebraically independent and that if both x b, y b K then b Z. Let p k(x), q k(y) be such that x, y, p, q are multiplicatively independent in K /k. Assume that there is a nonconstant Π k(x/y) y k(p/q) q. Assume moreover that this Π arises from infinitely many, modulo scalars, elements p, q as above. Then, modulo k, with κ k and δ = ±1. Π = Π κ,δ (x, y) := (x δ κy δ ) δ, (1)

58 Sketch of proof Reconstructing lines: Functional equations The corresponding p and q are given by p κx,1(x) = x + κ x, q κy,1(y) = y + κ y p κx, 1(x) = (x 1 + κ x ) 1, q κx, 1(y) = (y 1 + κ y ) 1 with κ x κ y = κ.

59 Anabelian geometry Grothendieck s Anabelian program The Galois group of a function field determines the field.

60 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.

61 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 G K := Gal( K/K). Let G K be the pro-l-completion of G K, for l char(k) a prime.

62 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 G K := Gal( K/K). Let G K be the pro-l-completion of G K, for l char(k) a prime. Uchida, Tamagawa, Mochizuki, Pop, Königsmann, Zaidi...: reconstruction of function fields from the full G K or G K.

63 Almost abelian anabelian geometry Let G a K := G K /[G K, G K ], G c K := G K /[G K, [G K, G K ]] be the abelianization, resp. its canonical central extension.

64 Almost abelian anabelian geometry Let G a K := G K /[G K, G K ], G c K := G K /[G K, [G K, G K ]] be the abelianization, resp. its canonical central extension. The group G a K is a torsion-free Z l-module of infinite rank.

65 Almost abelian anabelian geometry Let G a K := G K /[G K, G K ], G c K := G K /[G K, [G K, G K ]] be the abelianization, resp. its canonical central extension. The group G a K is a torsion-free Z l-module of infinite rank. Let Σ K be the set of all topologically noncyclic subgroups of G a K that lift to abelian subgroups of G c K.

66 Almost abelian anabelian geometry Let G a K := G K /[G K, G K ], G c K := G K /[G K, [G K, G K ]] be the abelianization, resp. its canonical central extension. The group G a K is a torsion-free Z l-module of infinite rank. Let Σ K be the set of all topologically noncyclic subgroups of G a K that lift to abelian subgroups of G c K. Bogomolov s program The pair (G a K, Σ K ) determines K.

67 Anabelian geometry of surfaces Theorem (Bogomolov-T. 2004) Let K and L be function fields over algebraic closures of finite fields k, l of characteristic l. Assume that K = k(x ) is a function field of a surface X /k and that there exists an isomorphism inducing a bijection of sets ψ : G a K Ga L Σ K = Σ L. Then, for some c Z l, cψ is induced by an isomorphism of purely inseparable closures of K and L.

68 Sketch of proof: Kummer theory The abelianized Galois group G a K is dual to ˆK, the pro-l-completion of K, and one obtains an isomorphism ˆK ˆL.

69 Sketch of proof: Kummer theory The abelianized Galois group G a K is dual to ˆK, the pro-l-completion of K, and one obtains an isomorphism ˆK ˆL. In our setup, we can interpret G a K as homomorphisms K /k Zl(1), arising from G a K /ln γ n (f γ( ln f )/ l n f ).

70 Sketch of proof: Kummer theory The abelianized Galois group G a K is dual to ˆK, the pro-l-completion of K, and one obtains an isomorphism ˆK ˆL. In our setup, we can interpret G a K as homomorphisms K /k Zl(1), arising from G a K /ln γ n (f γ( ln f )/ l n f ). For a subfield E K, the map G a K Ga E is simply restriction to E.

71 Valuations A value group, Γ, is a totally ordered (torsion-free) abelian group.

72 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, Γ ν ) consisting of a value group Γ ν and a map such that ν : K Γ ν, = Γ ν ν : K Γ ν is a surjective homomorphism; ν(κ + κ ) min(ν(κ), ν(κ )) for all κ, κ K; ν(0) =.

73 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, Γ ν ) consisting of a value group Γ ν and a map such that ν : K Γ ν, = Γ ν ν : K Γ ν 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.

74 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, Γ ν ) consisting of a value group Γ ν and a map such that ν : K Γ ν, = Γ ν ν : K Γ ν 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 flag map on K: every finite-dimensional Fp-subspace V K has a flag V = V 1 V 2... such that ν is constant on V j \ V j+1.

75 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, Γ ν ) consisting of a value group Γ ν and a map such that ν : K Γ ν, = Γ ν ν : K Γ ν 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 flag map on K: every finite-dimensional Fp-subspace V K has a flag V = V 1 V 2... such that ν is constant on V j \ V j+1. Conversely, every flag map gives rise to a valuation.

76 Valuations Denote by K ν, o ν, m ν and K ν := o ν /m ν the completion of K with respect to ν, the ring of ν-integers in K, the maximal ideal of o ν and the residue field.

77 Valuations Denote by K ν, o ν, m ν and K ν := o ν /m ν the completion of K with respect to ν, the ring of ν-integers in K, the maximal ideal of o ν and the residue field. Keep in mind the exact sequences: 1 o ν K Γ ν 1 1 (1 + m ν ) o ν K ν 1.

78 Valuations A homomorphism χ : Γ ν Zl(1) gives rise to a homomorphism χ ν : K Zl(1), thus to an element of GK a, an inertia element of ν. These form the inertia subgroup Iν a GK a.

79 Valuations A homomorphism χ : Γ ν Zl(1) gives rise to a homomorphism χ ν : K Zl(1), thus to an element of GK a, an inertia element of ν. These form the inertia subgroup Iν a GK a. The decomposition group Dν a is the image of GK a ν an embedding GK a ν GK a and an isomorphism in GK a. We have D a ν/i a ν G a K ν.

80 A dictionary Let K be a function field over k = Fp. We have GK a = {homomorphisms γ : K Zl(1)} Dν a = {µ GK a µ trivial on (1 + m ν)}, Iν a = {ι GK a ι trivial on o ν}.

81 A dictionary Let K be a function field over k = Fp. We have GK a = {homomorphisms γ : K Zl(1)} Dν a = {µ GK a µ trivial on (1 + m ν)}, Iν a = {ι GK a ι trivial on o ν}. Inertia elements define flag maps on K.

82 Projective geometry of the Galois group Key fact Let γ, γ GK a Z l be two nonproportional elements lifting to commuting elements in GK c. Then, for any nonconstant f K the restrictions of γ, γ to the projective line PF p (Fp f Fp) are proportional (modulo addition of constants).

83 Projective geometry of the Galois group Key fact Let γ, γ GK a Z l be two nonproportional elements lifting to commuting elements in GK c. Then, for any nonconstant f K the restrictions of γ, γ to the projective line PF p (Fp f Fp) are proportional (modulo addition of constants). Consider the map K /k = Pk(K) A 2 (Zl) f (γ(f ), γ (f ))

84 Projective geometry of the Galois group Key fact Let γ, γ GK a Z l be two nonproportional elements lifting to commuting elements in GK c. Then, for any nonconstant f K the restrictions of γ, γ to the projective line PF p (Fp f Fp) are proportional (modulo addition of constants). Consider the map K /k = Pk(K) A 2 (Zl) f (γ(f ), γ (f )) This maps every projective line into an affine line, a collineation.

85 Projective geometry of the Galois group Lemma A map α : P 2 (Fp) Z/2 is a flag map iff the restiction to every P 1 (Fp) P 2 (Fp) is a flag map, i.e., constant on the complement of one point.

86 Projective geometry of the Galois group Lemma A map α : P 2 (Fp) Z/2 is a flag map iff the restiction to every P 1 (Fp) P 2 (Fp) is a flag map, i.e., constant on the complement of one point. Counterexample: the Fano plane (0:1:0) (0:1:1) (1:1:0) (0:0:1) (1:0:1) (1:0:0)

87 Projective geometry of the Galois group Projective/affine geometry considerations produce a flag map in the Zl-linear span of γ, γ. Every noncyclic subgroup of GK a lifting to an abelian subgroup of GK c contains an inertia element ι = ι ν for some valuation ν of K.

88 Projective geometry of the Galois group Projective/affine geometry considerations produce a flag map in the Zl-linear span of γ, γ. Every noncyclic subgroup of GK a lifting to an abelian subgroup of GK c contains an inertia element ι = ι ν for some valuation ν of K. The elements commuting with ι form Dν. a

89 Projective geometry of the Galois group Projective/affine geometry considerations produce a flag map in the Zl-linear span of γ, γ. Every noncyclic subgroup of GK a lifting to an abelian subgroup of GK c contains an inertia element ι = ι ν for some valuation ν of K. The elements commuting with ι form Dν. a The combinatorial structure of the fan Σ K allows to reconstruct the projective structure of Pk(K).

90 What about curves? Let k = Fp and K = k(c). Let G K be the absolute Galois group of K. Let I K := {I a ν}, the set of inertia subgroups Iν a GK a valuations of K (i.e., points of C). of nontrivial divisorial Bogomolov-T Assume that g(c) > 2 and that (G a K, I K ) (G ã K, I K ). Then J J.

91 Curves and their Jacobians Let k be any field and C/k a smooth curve over k of genus g(c) 2, with C(k). For each n N, we have (c 1,..., c n ) (c c n ) C n Sym n (C) J n λ n

92 Curves and their Jacobians Let k be any field and C/k a smooth curve over k of genus g(c) 2, with C(k). For each n N, we have (c 1,..., c n ) (c c n ) C n Sym n (C) J n Choosing c 0 C(k), we may identify J n J. λ n

93 Curves and their Jacobians Let k be any field and C/k a smooth curve over k of genus g(c) 2, with C(k). For each n N, we have (c 1,..., c n ) (c c n ) C n Sym n (C) J n Choosing c 0 C(k), we may identify J n J. Image(λ g 1 ) = Θ J, the Theta divisor λ n

94 Curves and their Jacobians Let k be any field and C/k a smooth curve over k of genus g(c) 2, with C(k). For each n N, we have (c 1,..., c n ) (c c n ) C n Sym n (C) J n Choosing c 0 C(k), we may identify J n J. Image(λ g 1 ) = Θ J, the Theta divisor Torelli: the pair (J, Θ) determines C, up to isomorphism λ n

95 Curves and their Jacobians Let k be any field and C/k a smooth curve over k of genus g(c) 2, with C(k). For each n N, we have (c 1,..., c n ) (c c n ) C n Sym n (C) J n Choosing c 0 C(k), we may identify J n J. Image(λ g 1 ) = Θ J, the Theta divisor Torelli: the pair (J, Θ) determines C, up to isomorphism for n 2g 1, λ n is a P n g -bundle λ n

96 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 l p(ql/zl) 2g.

97 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 l p(ql/zl) 2g. Tate Hom(A, Ã) Z l = Hom Zl [Fr](T l (A), T l (Ã)).

98 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 l p(ql/zl) 2g. Tate Hom(A, Ã) Z l = Hom Zl [Fr](T l (A), T l (Ã)). In particular, A and à are isogenous iff the characteristic polynomials of the Frobenius coincide.

99 Divisibilities Bogomolov-T Let A and à be abelian varieties of dimension g over finite fields k, resp. k. Let k n /k, resp. k n / k, be the unique extensions of degree n. Assume that #A(k n ) #A( k n ) for infinitely many n N.

100 Divisibilities Bogomolov-T Let A and à be abelian varieties of dimension g over finite fields k, resp. k. Let k n /k, resp. k n / k, be the unique extensions of degree n. Assume that #A(k n ) #A( k n ) for infinitely many n N. Then char(k) = char( k) and A and à are isogenous over k.

101 Sketch of proof Let A be an abelian variety over k 1 := Fq. Let {α j } j=1,...,2g be the set of eigenvalues of Frobenius on H 1 et(ā, Ql), for l p, and Γ A C the multiplicative subgroup spanned by α.

102 Sketch of proof Let A be an abelian variety over k 1 := Fq. Let {α j } j=1,...,2g be the set of eigenvalues of Frobenius on H 1 et(ā, Ql), for l p, and Γ A C the multiplicative subgroup spanned by α. The sequence R(n) := #A(k n ) = 2g j=1 (α n j 1). is a simple linear recurrence with roots in Γ = Γ A.

103 Sketch of proof Let A be an abelian variety over k 1 := Fq. Let {α j } j=1,...,2g be the set of eigenvalues of Frobenius on H 1 et(ā, Ql), for l p, and Γ A C the multiplicative subgroup spanned by α. The sequence R(n) := #A(k n ) = 2g j=1 (α n j 1). is a simple linear recurrence with roots in Γ = Γ A. There is an isomorphism of rings { Recurrences with roots in Γ} C[Γ].

104 Sketch of proof: Recurrence sequences Corvaja-Zannier 2002 Let R and R be simple linear recurrences such that 1 R(n), R(ñ) 0, for all n, ñ 0; 2 the subgroup Γ C generated by the roots of R and R is torsion-free; 3 there is a finitely-generated subring A C with R(n)/ R(n) A, for infinitely many n N.

105 Sketch of proof: Recurrence sequences Corvaja-Zannier 2002 Let R and R be simple linear recurrences such that 1 R(n), R(ñ) 0, for all n, ñ 0; 2 the subgroup Γ C generated by the roots of R and R is torsion-free; 3 there is a finitely-generated subring A C with R(n)/ R(n) A, for infinitely many n N. Then Q : N C n R(n)/ R(n) is a simple linear recurrence. In particular, the F Q C[Γ] and F Q F R = F R.

106 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) J 1 ( k) j 1 C( k) φ 0 φ 1 φ s where J( k) J1 j ( k) 1 C( k) φ 0 : isomorphism of abstract abelian groups; φ 1 : isomorphism of homogeneous spaces, compatible with φ 0 ; the restriction φ s : C( k) C( k) of φ 1 is a bijection of sets.

107 Curves and their Jacobians For all #k 0 the group J(k) is generated by C(k).

108 Curves and their Jacobians For all #k 0 the group J(k) is generated by C(k). Let k = k 1 k 2... k n... be the tower of degree 2 extensions.

109 Curves and their Jacobians For all #k 0 the group J(k) is generated by C(k). Let k = k 1 k 2... k n... be the tower of degree 2 extensions. To characterize J(k n ) it suffices to characterize C(k n ).

110 Curves and their Jacobians For all #k 0 the group J(k) is generated by C(k). Let k = k 1 k 2... k n... be the tower of degree 2 extensions. To characterize J(k n ) it suffices to characterize C(k n ). Let C be a nonhyperelliptic curve of genus g(c) 3.

111 Curves and their Jacobians For all #k 0 the group J(k) is generated by C(k). Let k = k 1 k 2... k n... be the tower of degree 2 extensions. To characterize J(k n ) it suffices to characterize C(k n ). Let C be a nonhyperelliptic curve of genus g(c) 3. Inductive characterization of J(k n ), n N J(k n ) is generated by c C( k) such that there exists a point c C( k) with c + c J(k n 1 ).

112 Curves and their Jacobians: Torelli Theorem Let (C, J) ( C, J) be an isomorphism of pairs. Then J is isogenous to J.

113 Curves and their Jacobians: Torelli Theorem Let (C, J) ( C, J) be an isomorphism of pairs. Then J is isogenous to J. Proof. 1 Choose k 1, k 1 (sufficiently large) such that φ(j(k 1 )) J( k 1 )

114 Curves and their Jacobians: Torelli Theorem Let (C, J) ( C, J) be an isomorphism of pairs. Then J is isogenous to J. Proof. 1 Choose k 1, k 1 (sufficiently large) such that φ(j(k 1 )) J( k 1 ) 2 Define C(k n ), resp. C( k n ), intrinsically, as above.

115 Curves and their Jacobians: Torelli Theorem Let (C, J) ( C, J) be an isomorphism of pairs. Then J is isogenous to J. Proof. 1 Choose k 1, k 1 (sufficiently large) such that φ(j(k 1 )) J( k 1 ) 2 Define C(k n ), resp. C( k n ), intrinsically, as above. 3 Have φ(j(k n )) J( k n ), for all n N.

116 Curves and their Jacobians: Torelli Theorem Let (C, J) ( C, J) be an isomorphism of pairs. Then J is isogenous to J. Proof. 1 Choose k 1, k 1 (sufficiently large) such that φ(j(k 1 )) J( k 1 ) 2 Define C(k n ), resp. C( k n ), intrinsically, as above. 3 Have φ(j(k n )) J( k n ), for all n N. 4 #J(k n ) # J( k n )

117 Curves and their Jacobians: Torelli Theorem Let (C, J) ( C, J) be an isomorphism of pairs. Then J is isogenous to J. Proof. 1 Choose k 1, k 1 (sufficiently large) such that φ(j(k 1 )) J( k 1 ) 2 Define C(k n ), resp. C( k n ), intrinsically, as above. 3 Have φ(j(k n )) J( k n ), for all n N. 4 #J(k n ) # J( k n ) 5 Apply the result about divisibility of recurrence sequences.