Diophantine geometry and non-abelian duality

Transcription

1 Diophantine geometry and non-abelian duality Minhyong Kim Leiden, May, 2011

2 Diophantine geometry and abelian duality E: elliptic curve over a number field F. Kummer theory: E(F) Z p H 1 f (G,T p(e)) conjectured to be an isomorphism. Should allow us, in principle, to compute E(F). Furthermore, size of H 1 f (G,T p(e)) should be controlled by an L-function.

3 Diophantine geometry and abelian duality In the theorem L(E/Q,1) 0 E(Q) <, key point is that the image of loc p : Hf 1 (G,T p(e)) Hf 1 (G p,t p (E)) is annihilated using Poitou-Tate duality by a class c H 1 (G,T p (E)) whose image in H 1 (G p,t p (E))/Hf 1 (G p,t p (E)) is non-torsion.

4 Diophantine geometry and abelian duality An explicit local reciprocity law then translates this into an analytic function on E(Q p ) that annihilates E(Q). Exp : H 1 (G p,t p (E)) F 1 H 1 DR (E/Q p); c L(E,1) Ω(E) α where α is an invariant differential form on E.

5 Diophantine geometry and abelian duality E(Q) E(Q p ) L(E,1) Ω(E) ( ) 0 α H 1 f (G,T p(e)) H 1 f (G p,t p (E)) c Q p

6 Non-abelian analogue? Wish to investigate an extension of this phenomenon to hyperbolic curves. That is, curves of -genus zero minus at least three points; -genus one minus at least one point; -genus at least two.

7 Notation F: Number field. S 0 : finite set of primes of F. R := O F [1/S 0 ], the ring of S integers in F. p: odd prime not divisible by primes in S 0 ; v: a prime of F above p with F v = Q p.. G := Gal( F/F); G v = Gal( F v /F v ). G S := Gal(F S /F), where F S is the maximal extension of F unramified outside S = S 0 {v p}. X: smooth curve over Spec(R) with good compactification. (Might be compact itself.) X: generic fiber of X, assumed to be hyperbolic. b X(R), possibly tangential.

8 Grothendieck s section conjecture Suppose X is compact. Then the map ĵ : X(R) H 1 (G,ˆπ 1 ( X,b)); x [ˆπ 1 ( X;b,x)] is a bijection.

9 Unipotent descent tower X(R).. H 1 f (G,U 4) j 4 j 3 j 1 j 2 Hf 1 (G,U 3) Hf 1 (G,U 2) Hf 1 (G,U 1)

10 Unipotent descent tower U = ˆπ 1 ( X,b) Q p, is the Q p -pro-unipotent étale fundamental group of X = X Spec(F) Spec( F) with base-point b. The universal pro-unipotent pro-algebraic group over Q p equipped with a map from ˆπ 1 ( X,b).

11 Unipotent descent tower U = ˆπ 1 ( X,b) Q p, is the Q p -pro-unipotent étale fundamental group of X = X Spec(F) Spec( F) with base-point b. The universal pro-unipotent pro-algebraic group over Q p equipped with a map from ˆπ 1 ( X,b). The map j : x X(R) [P(x)] H 1 f (G,U), associates to a point x, the U-torsor P(x) := ˆπ 1 ( X;b,x) ˆπ1 ( X,b) U of Q p -unipotent étale paths from b to x.

12 Unipotent descent tower U n := U n+1 \U, where U n is the lower central series with U 1 = U. So U 1 = U ab = T p J X Q p.

13 Unipotent descent tower U n := U n+1 \U, where U n is the lower central series with U 1 = U. So U 1 = U ab = T p J X Q p. All these objects have natural actions of G so that P(x) defines a class in H 1 f (G,U), the continuous non-abelian cohomology of G with coefficients in U satisfying local Selmer conditions, the Selmer variety of X, which must be controlled in order to control the points of X.

14 Algebraic localization X(R) X(R v ) j j v Hf 1 (G,U locv n) Hf 1 (G v,u n )

15 Algebraic localization X(R) X(R v ) j j v Hf 1 (G,U locv n) Hf 1 (G v,u n ) Goal: Compute the image of loc v.

16 Algebraic localization One essential fact is that the local map X(R v ) j v H 1 f (G v,u n ) can be computed via a diagram X(R v ) j v Hf 1 (G v,u n ) j DR U DR n /F 0 A N where U DR n /F 0 is a homogeneous space for the De Rham-crystalline fundamental group, and the map j DR can be described explicitly using p-adic iterated integrals.

17 Non-abelian method of Chabauty Meanwhile, the localization map is an algebraic map of varieties over Q p making it feasible, in principle, to discuss its computation.

18 Non-abelian method of Chabauty Meanwhile, the localization map is an algebraic map of varieties over Q p making it feasible, in principle, to discuss its computation. Knowledge of will lead to knowledge of Im(loc v ) H 1 f (G v,u n ) X(R) [j v ] 1 (Im(loc v )) X(R v ). For example, when Im(loc v ) is not Zariski dense, immediately deduce finiteness of X(R).

19 Non-abelian method of Chabauty This deduction is captured by the diagram X(R) X(R v ) such that ψ j et v j et v Hf 1 (G,U locv n) Hf 1 (G v,u n ) kills X(R). ψ 0 Q p

20 Non-abelian method of Chabauty Can use this to give a new proof of finiteness of points in some cases: F = Q and the Jacobian of X has potential CM. (joint with John Coates) F = Q and X, elliptic curve minus one point. F totally real and X of genus zero.

21 Non-abelian method of Chabauty Can use this to give a new proof of finiteness of points in some cases: F = Q and the Jacobian of X has potential CM. (joint with John Coates) F = Q and X, elliptic curve minus one point. F totally real and X of genus zero. In each of these cases, non-vanishing of a p-adic L-function seems to play a key role.

22 Non-abelian method of Chabauty By analogy with the abelian case: Non-vanishing of L-function control of Selmer group finiteness of points;

23 Non-abelian method of Chabauty By analogy with the abelian case: Non-vanishing of L-function control of Selmer group finiteness of points; one has Non-vanishing of L-function control of Selmer variety finiteness of points.

24 Non-abelian method of Chabauty By analogy with the abelian case: Non-vanishing of L-function control of Selmer group finiteness of points; one has Non-vanishing of L-function control of Selmer variety finiteness of points. But would like to construct ψ in some canonical fashion.

25 Motivation: Effective computation of points? The goal is to find an effectively compute m(x,v) such that min{d v (x,y) x y X(R) X(R v )} > m(x,v).

26 Motivation: Effective computation of points? The goal is to find an effectively compute m(x,v) such that min{d v (x,y) x y X(R) X(R v )} > m(x,v). Key point: Computation of m(x,v) + section conjecture computation of X(R).

27 Motivation: Effective computation of points? The goal is to find an effectively compute m(x,v) such that min{d v (x,y) x y X(R) X(R v )} > m(x,v). Key point: Computation of m(x,v) + section conjecture computation of X(R). Remark: Section conjecture can also be used to effectively determine the existence of a point (A. Pal, M. Stoll).

28 Non-abelian duality?

29 Non-abelian duality? Hope that Im(loc v ) might be computed using a sort of non-abelian Poitou-Tate duality.

30 Non-abelian duality? Hope that Im(loc v ) might be computed using a sort of non-abelian Poitou-Tate duality. In the abelian case, we know that Poitou-Tate duality is the basic tool for computing the global image inside local cohomology:

31 Non-abelian duality? Hope that Im(loc v ) might be computed using a sort of non-abelian Poitou-Tate duality. In the abelian case, we know that Poitou-Tate duality is the basic tool for computing the global image inside local cohomology: Say c H 1 (G,V (1)) has local component c w = 0 for all w v and c v 0. Then Im(H 1 (G,V)) H 1 (G v,v) lies in the hyperplane ( ) c v = 0. Would like a non-abelian analogue.

32 Non-abelian duality? Hope that Im(loc v ) might be computed using a sort of non-abelian Poitou-Tate duality. In the abelian case, we know that Poitou-Tate duality is the basic tool for computing the global image inside local cohomology: Say c H 1 (G,V (1)) has local component c w = 0 for all w v and c v 0. Then Im(H 1 (G,V)) H 1 (G v,v) lies in the hyperplane ( ) c v = 0. Would like a non-abelian analogue. Difficulty is that duality for Galois cohomology with coefficients in various non-abelian groups can be interpreted as a sort of non-abelian class field theory.

33 Non-abelian duality: example E/Q: elliptic curve with ranke(q) = 1, trivial Tamagawa numbers, and X(E)[p ] < for a prime p of good reduction. X =: E \{0} given as a minimal Weierstrass model: y 2 + a 1 xy + a 3 y = x 3 + a 2 x 2 + a 4 x + a 6. So X(Z) E(Z) = E(Q).

34 Non-abelian duality: example Let α = dx/(2y + a 1 x + a 3 ), β = xdx/(2y + a 1 x + a 3 ). Get analytic functions on X(Q p ), log α (z) = z b D 2 (z) = α; log β (z) = z b αβ. Here, b is a tangential base-point at 0, and the integral is (iterated) Coleman integration. Locally, the integrals are just anti-derivatives of the forms, while for the iteration, z dd 2 = ( β)α. b z b β;

35 Non-abelian duality: example Theorem Suppose there is a point y X(Z) of infinite order in E(Q). Then the subset X(Z) X(Z p ) lies in the zero set of the analytic function ψ(z) := D 2 (z) D 2(y) ( y b α)2( z b α) 2.

36 Non-abelian duality: example Theorem Suppose there is a point y X(Z) of infinite order in E(Q). Then the subset X(Z) X(Z p ) lies in the zero set of the analytic function ψ(z) := D 2 (z) D 2(y) ( y b α)2( z b α) 2. A fragment of non-abelian duality and explicit reciprocity.

37 Non-abelian duality: example Function ψ is actually a composition X(Z p ) H 1 f (G p,u 2 ) φ Q p U2 DR /F 0 where φ is constructed using secondary cohomology products and has the property that ψ φ(loc p (H 1 f (G,U 2))) = 0.

38 Non-abelian duality: example X(Z) H 1 f (G,U 2) X(Z p ) Hf 1 (G p,u 2 ) φ Q p U2 DR /F 0 ψ

39 Non-abelian duality: example U 2 V Q p (1) where V = T p (E) Q p, with group law (X,a)(Y,b) = (X + Y,a+b+(1/2) < X,Y >). A function a = (a 1,a 2 ) : G p U 2 is a cocycle if and only if da 1 = 0; da 2 = (1/2)[a 1,a 1 ].

40 Non-abelian duality: example For a = (a 1,a 2 ) H 1 f (G p,u 2 ), we define φ(a 1,a 2 ) := [b,a 1 ]+logχ p ( 2a 2 ) H 2 (G p,q p (1)) Q p,

41 Non-abelian duality: example For a = (a 1,a 2 ) H 1 f (G p,u 2 ), we define φ(a 1,a 2 ) := [b,a 1 ]+logχ p ( 2a 2 ) H 2 (G p,q p (1)) Q p, where logχ p : G p Q p is the logarithm of the p-adic cyclotomic character and b : G V is a solution to the equation db = logχ p a 1.

42 Non-abelian duality: example The annihilation comes from the standard exact sequence 0 H 2 (G,Q p (1)) v H 2 (G v,q p (1)) Q p 0. That is, our assumptions imply that the class [π 1 ( X;b,x)] 2 for x X(Z) is trivial at all places l p. On the other hand φ(loc p ([π 1 ( X;b,x)] 2 )) = loc p (φ glob ([π 1 ( X;b,x)] 2 )).

43 Non-abelian duality: example With respect to the coordinates the image H 1 f (G p,u 2 ) U DR 2 /F 0 A 2 = {(s,t)} is described by the equation loc p (H 1 f (G,U 2)) H 1 f (G p,u 2 ) t D 2(y) ( y b α)2 s2 = 0.

44 Non-abelian duality: abstract framework Let L = n N L n be graded Lie algebra over field k. The map D : L L such that D L n = n is a derivation, i.e., an element of H 1 (L,L). Can be viewed as an element of H 2 (L L,k), that is, a central extension of L L: 0 k E L L 0.

45 Non-abelian duality: abstract framework Explicitly described as follows: [(a,α,x),(b,β,y)] = (α(d(y)) β(d(x)),ad X (β) ad Y (α),[x,y]). When L = L 1 and D = I, then this gives a standard Heisenberg extension.

46 Non-abelian duality: abstract framework Explicitly described as follows: [(a,α,x),(b,β,y)] = (α(d(y)) β(d(x)),ad X (β) ad Y (α),[x,y]). When L = L 1 and D = I, then this gives a standard Heisenberg extension. When k = Q p and we are given an action of G or G v, can twist to 0 Q p (1) E L (1) L 0. Also have a corresponding group extension (L = Lie(U)) 0 Q p (1) E L (1) U 0.

47 Non-abelian duality: abstract framework From this, we get a boundary map H 1 (G v,l (1) U) δ H 2 (G v,q p (1)) Q p. This boundary map should form the basis of (unipotent) non-abelian duality.

48 Non-abelian duality: abstract framework From this, we get a boundary map H 1 (G v,l (1) U) δ H 2 (G v,q p (1)) Q p. This boundary map should form the basis of (unipotent) non-abelian duality. H 1 (G v,l (1)) H 1 (G v,l (1) U) δ Q p H 1 (G v,u)

49 Non-abelian duality: difficulties 1. How to get functions on H 1 f (G v,u)?

50 Non-abelian duality: difficulties 1. How to get functions on H 1 f (G v,u)? 2. When U is a unipotent fundamental group, L is not graded in way that s compatible with the Galois action.

51 Non-abelian duality: weighted completions This second difficulty is partially resolved by Hain s theory of weights completions.

52 Non-abelian duality: weighted completions This second difficulty is partially resolved by Hain s theory of weights completions. Let R be the Zariski closure of the image of G S Aut(H 1 ( X,Q p )).

53 Non-abelian duality: weighted completions This second difficulty is partially resolved by Hain s theory of weights completions. Let R be the Zariski closure of the image of G S Aut(H 1 ( X,Q p )). Then R contains the center G m of Aut(H 1 ( X,Q p )).

54 Non-abelian duality: weighted completions Consider the universal pro-algebraic extension 0 T G S R 0 equipped with a lift G S G S R such that T is pro-unipotent and the action of G m on H 1 (T) has negative weights.

55 Non-abelian duality: weighted completions Then H 1 (G S,U) H 1 (G S,U). Easy to see by comparing the splittings of the two rows in 1 U U G S G S 1 = 1 U U G S G S 1

56 Non-abelian duality: weighted completions Furthermore, the exact sequence 0 T G S R 0 splits to give R G S that maps isomorphically to R, and G S T R. In particular, there is a lifted one-parameter subgroup G m G S, which gives a grading on all G S modules. (Actually, the G m -lifting determines R.)

57 Non-abelian duality: weighted completions Corollary Let N = LieT. The lifting G m G S determines a grading on [N (1) L (1)] L N.

58 Non-abelian duality: weighted completions Corollary Let N = LieT. The lifting G m G S determines a grading on [N (1) L (1)] L N. One can use this to construct, in turn, central extensions of [N (1) L (1)] L N;

59 Non-abelian duality: weighted completions Corollary Let N = LieT. The lifting G m G S determines a grading on [N (1) L (1)] L N. One can use this to construct, in turn, central extensions of [N (1) L (1)] L N; [N (1) L (1)] U T;

60 Non-abelian duality: weighted completions Corollary Let N = LieT. The lifting G m G S determines a grading on [N (1) L (1)] L N. One can use this to construct, in turn, central extensions of [N (1) L (1)] L N; and [N (1) L (1)] U T; [N (1) L (1)] U G S ;

61 Non-abelian duality: weighted completions Corollary Let N = LieT. The lifting G m G S determines a grading on [N (1) L (1)] L N. One can use this to construct, in turn, central extensions of [N (1) L (1)] L N; and [N (1) L (1)] U T; [N (1) L (1)] U G S ; which can then be pulled back to [N (1) L (1)] U G S.

62 Non-abelian duality: weighted completions Proposition The G m lift determines a central extension 0 Q p (1) E [N (1) L (1)] U G S 0 giving rise to a boundary map H 1 (G S,[N (1) L (1)] U) H 2 (G S,Q p (1)).

63 Non-abelian duality: weighted completions The central extension can be pulled back to each G w for w S, to give boundary maps H 1 (G w,n (1) L (1)) H 1 (G w,[n (1) L (1)] U) δw Q p H 1 (G w,u)

64 Non-abelian duality: remarks 1. Applying the constructions to the finite level quotients, we get maps and H 1 (G w,[n n(1) L n(1)] U n ) δw,n Q p H 1 (G S,[N n(1) L n(1)] U n ) δ n H 2 (G S,Q p (1)).

65 Non-abelian duality: remarks 1. Applying the constructions to the finite level quotients, we get maps and H 1 (G w,[n n(1) L n(1)] U n ) δw,n Q p H 1 (G S,[N n(1) L n(1)] U n ) δ n H 2 (G S,Q p (1)). 2. These maps are compatible in the following sense: δ n restricted to H 1 (G S,[N n 1 (1) L n 1 (1)] U n) is the composition H 1 (G S,[N n 1 (1) L n 1 (1)] U n) H 1 (G S,[N n 1 (1) L n 1 (1)] U n 1) δ n 1 H 2 (G S,Q p (1)); and the same for the local versions.

66 Non-abelian duality: remarks H 1 (G w,[nn 1 (1) L n 1 (1)] U n) H 1 (G w,[nn(1) L n(1)] U n ) H 1 (G w,[nn 1 (1) L n 1 (1)] U n 1) Q p

67 Non-abelian duality: remarks 3. These boundary maps are quite non-trivial. For example, considering the central subgroup when the boundary map on is restricted to L n n = Un n := Un+1 \U n U n, H 1 (G w,[n n(1) L n(1)] U n ) H 1 (G w,[n n (1) L n (1)] Ln n ),

68 Non-abelian duality: remarks 3. These boundary maps are quite non-trivial. For example, considering the central subgroup when the boundary map on is restricted to L n n = Un n := Un+1 \U n U n, H 1 (G w,[n n(1) L n(1)] U n ) H 1 (G w,[n n (1) L n (1)] Ln n ), then it factors through H 1 (G w,n n (1) (Ln n ) (1) L n n ).

69 Non-abelian duality: remarks On the subspace H 1 (G w,(l n n ) (1) L n n ) H1 (G w,n n (1) (Ln n ) (1) L n n ), the induced map is usual Tate duality multiplied by n.

70 Non-abelian duality: remarks On the subspace H 1 (G w,(l n n ) (1) L n n ) H1 (G w,n n (1) (Ln n ) (1) L n n ), the induced map is usual Tate duality multiplied by n. H 1 (G w,(l n n ) (1) L n n ) H 1 (G w,[nn (1) L n (1)] Ln n ) H 1 (G w,nn (1) (Ln n ) (1) L n n ) H 1 (G w,[nn (1) L n (1)] U n) δ w,n Q p

71 Non-abelian duality: a reciprocity law Theorem The image of in is annihilated by H 1 (G S,[N n(1) L n(1)] U n ) H 1 (G w,[nn (1) L n (1)] U n) w S δ w. w