Diophantine Geometry and Non-Abelian Reciprocity Laws

Transcription

1 Diophantine Geometry and Non-Abelian Reciprocity Laws Minhyong Kim Oxford, July, 2014

2 Diophantine Geometry: Abelian Case The Hasse-Minkowski theorem says that ax 2 + by 2 = c has a solution in a number field F and only if it has a solution in F v for all v. There are straightforward algorithms for determining this. For example, we need only check for v = and v 2abc, and there, a solution exists if and only if (a, b) v (b, c) v (c, a) v (c, 1) v = 1.

3 Diophantine Geometry: Main Local-to-Global Problem Locate X (F ) X (A F ) = X (F v ) v The question is How do the global points sit inside the local points? In fact, there is a classical answer of satisfactory sort for conic equations.

4 Diophantine Geometry: Main Local-to-Global Problem In that case, assume for simplicity that there is a rational point (and that the points at infinity are rational), so that X G m. Then X (F ) = F, X (F v ) = F v. Problem becomes that of locating F A F.

5 Diophantine Geometry: Abelian Class Field Theory We have the Artin reciprocity map Rec = v Rec v : A F G ab F. Here, and G ab F = Gal(F ab /F ), F ab is the maximal abelian algebraic extension of F.

6 Diophantine Geometry: Abelian Class Field Theory Artin s reciprocity law: The map is zero. F A F Rec G ab F Key point is that the reciprocity law becomes a result of Diophantine geometry. That is, the reciprocity map gives a defining equation for G m (F ).

7 Diophantine Geometry: Non-Abelian Reciprocity? We would like to generalize this to other equations by way of a non-abelian reciprocity law Start with a rather general Diophantine equation X for which we would like to understand X (F ) via X (F ) X (A F ) RecNA some target with base-point in such way that becomes an equation for X (F ). Rec NA = base-point

8 Diophantine Geometry: Non-Abelian Reciprocity To rephrase: we would like to Construct class field theory with coefficients in a general variety X generalizing CFT with coefficients in G m. Will describe a version that works for smooth hyperbolic curves.

9 Diophantine Geometry: Non-Abelian Reciprocity (Joint with Jonathan Pridham) Notation: F : number field. G F = Gal( F /F ). G v = Gal( F v /F v ) for a place v of F. S: finite set of places of F. A F : Adeles of F A S F : S-integral adeles of F. GF S = Gal(F S /F ), where F S is the maximal extension of F unramified outside S.

10 Diophantine Geometry: Non-Abelian Reciprocity X : a smooth curve over F with genus at least two; b X (F ) (sometimes tangential). = π 1 ( X, b) : Pro-finite étale fundamental group of X = X Spec(F ) Spec( F ) with base-point b. [n] Lower central series with [1] =. n = / [n+1]. T n = [n] / [n+1].

11 Diophantine Geometry: Non-Abelian Reciprocity We then have a nilpotent class field theory with coefficients in X made up of a filtration X (A F ) = X (A F ) 1 X (A F ) 2 X (A F ) 3 and a sequence of maps rec n : X (A F ) n G n (X ) to a sequence G n (X ) of profinite abelian groups in such a way that X (A F ) n+1 = rec 1 n (0).

12 Diophantine Geometry: Non-Abelian Reciprocity X (A F ) 3 = rec 1 2 (0) X (A F ) 2 = rec 1 1 (0) X (A F ) 1 = X (A F ) rec 3 rec 2 rec 1 G 3 (X ) G 2 (X ) G 1 (X ) rec n is defined not on all of X (A F ), but only on the kernel (the inverse image of 0) of all the previous rec i.

13 Diophantine Geometry: Non-Abelian Reciprocity The G n (X ) are defined as G n (X ) := Hom[H 1 (G F, D(T n )), Q/Z] where D(T n ) = lim m Hom(T n, µ m ). When X = G m, then G n (X ) = 0 for n 2 and G 1 = Hom[H 1 (G F, D(Ẑ(1))), Q/Z] = Hom[H 1 (G F, Q/Z), Q/Z] = G ab F.

14 Diophantine Geometry: Non-Abelian Reciprocity The reciprocity maps are defined using the local period maps j v : X (F v ) H 1 (G v, ); x [π 1 ( X ; b, x)]. Because the homotopy classes of étale paths π 1 ( X ; b, x) form a torsor for with compatible action of G v, we get a corresponding class in non-abelian cohomology of G v with coefficients in.

15 Diophantine Geometry: Non-Abelian Reciprocity These assemble to a map j loc : X (A F ) H 1 (G v, ), which comes in levels j loc n : X (A F ) H 1 (G v, n ).

16 Diophantine Geometry: Non-Abelian Reciprocity The first reciprocity map is just defined using where x X (A F ) d 1 (j loc 1 (x)), d 1 : S M H 1 (G v, M 1 ) S M H 1 (G v, D( M 1 )) loc H 1 (G S M F, M 1 ), is obtained from Tate duality and the dual of localization. One needs first to work with a pro-m quotient for a finite set of primes M and S M = S M. Here, S M H 1 (G v, M 1 ) = v S M H 1 (G v, M 1 ) v / S M H 1 (G v /I v, M n ).

17 Diophantine Geometry: Non-Abelian Reciprocity To define the higher reciprocity maps, we use the exact sequences 0 H 1 c (G S M F, T M n+1) H 1 z (G S M F, M n+1) H 1 z (G S M F, n ) δ n+1 Hc 2 (G S M F, Tn+1) M for non-abelian cohomology with support and Poitou-Tate duality d n+1 : Hc 2 (G S M F, Tn+1) M H 1 (G S M F, D(Tn+1)) M.

18 Diophantine Geometry: Non-Abelian Reciprocity Essentially, rec M n+1 = d n+1 δ n+1 loc 1 j n. x X (A F ) n+1 jn loc S M H 1 (G v, M n ) loc 1 H 1 j loc n (x)(g S M F, M n ) δ n+1 Hc 2 (G S M F, Tn+1) M dn+1 H 1 (G S M F, D(Tn+1)) M. We take a limit over M to get the reciprocity maps. (Need some care when 2 M.)

19 Diophantine Geometry: Non-Abelian Reciprocity Put X (A F ) = n=1x (A F ) n. Theorem (Non-abelian reciprocity) X (F ) X (A F ).

20 Diophantine Geometry: Non-Abelian Reciprocity Remark: When F = Q and p is a prime of good reduction, suppose there is a finite set T of places such that H 1 (G S F, p n) v T H 1 (G v, p n) is injective. Then the reciprocity law implies finiteness of X (F ).

21 Diophantine Geometry: Non-Abelian Reciprocity With the formalism in place, the proof of the reciprocity law is almost obvious. If x X (A F ) comes from a global point x g X (F ), then there will be a class j g n (x g ) H 1 j n(x) (G S M F, M n ) for every n corresponding to the global torsor π et,m 1 ( X ; b, x g ). That is, jn g (x g ) = loc 1 (jn loc (x)) and for every n. δ n+1 (j g n (x g )) = 0

22 A non-abelian conjecture of Birch and Swinnerton-Dyer type Let Pr p : X (A F ) X (F v ) be the projection to the v-adic component of the adeles. Conjecture: Let X /Q be a projective smooth curve of genus at least 2. Then for any prime p of good reduction, we have Pr p (X (Q)) = Pr p (X (A Q ) ).

23 A non-abelian conjecture of Birch and Swinnerton-Dyer type Can consider more generally integral points on affine hyperbolic X as well. Conjecture: Let X be an affine smooth curve with non-abelian fundamental group and S a finite set of primes. Then for any prime p / S of good reduction, we have Pr p (X (Z[1/S])) = Pr p (X (A S Q ) ). Should allow us to compute X (Q) X (Q p ) or X (Z[1/S]) X (Z p ).

24 A non-abelian conjecture of Birch and Swinnerton-Dyer type Whenever we have an element we get a function k n H 1 (G S, Hom(T M n, Q p (1))), X (A S Q ) n rec n H 1 (G S, D(T M n )) kn Q p that kills X (Z[1/S]) X (A S Q ) n. Need an explicit reciprocity law that describes the image X (Z p ) n := Pr p (X (A S Q ) n).

25 A non-abelian conjecture of Birch and Swinnerton-Dyer type: examples [Joint work with Jennifer Balakrishnan, Ishai Dan-Cohen, and Stefan Wewers] Let X = P 1 \ {0, 1, }. Then X (Z) = φ. X (Z p ) 2 = {z log(z) = 0, log(1 z) = 0}. Must have z = ζ n and 1 z = ζ m, and hence, z = ζ 6 or z = ζ 1 6.

26 A non-abelian conjecture of Birch and Swinnerton-Dyer type: examples Thus, if p = 3 or p 2 mod 3, we have X (Z p ) 2 = φ = X (Z), so the conjecture holds already at level 2. When p 1 mod 3 and we must go to a higher level. X (Z) = φ {ζ 6, ζ 1 6 } = X (Z p) 2

27 A non-abelian conjecture of Birch and Swinnerton-Dyer type: examples Let be the dilogarithm. Then Li 2 (z) = n z n n 2 X (Z p ) 3 = {z log(z) = 0, log(1 z) = 0, Li 2 (z) = 0}. and the conjecture is true for X (Z) if Li 2 (ζ 6 ) 0. Can check this numerically for all 2 < p < 10 5.

28 A non-abelian conjecture of Birch and Swinnerton-Dyer type: examples Let X = E \ O where E is a semi-stable elliptic curve of rank 0 and X(E)(p) <. log(z) = (b is a tangential base-point.) Then z b (dx/y). X (Z p ) 2 = {z X (Z p ) log(z) = 0} = E(Z p )[tor] \ O. For small p, it happens frequently that E(Z)[tor] = E(Z p )[tor] and hence that X (Z) = X (Z p ) 2. But of course, this fails as p grows.

29 A non-abelian conjecture of Birch and Swinnerton-Dyer type: examples Must then examine the inclusion X (Z) X (Z p ) 3. Let D 2 (z) = z b (dx/y)(xdx/y).

30 A non-abelian conjecture of Birch and Swinnerton-Dyer type: examples Let S be the set of primes of bad reduction. For each l S, let N l = ord l ( E ), where E is the minimal discriminant. Define a set W l := {(n(n l n)/2n l ) log l 0 n < N l }, and for each w = (w l ) l S W := l S W l, define w = l S w l.

31 A non-abelian conjecture of Birch and Swinnerton-Dyer type: examples Theorem Suppose E has rank zero and that X E [p ] <. With assumptions as above where X (Z p ) 3 = w W Ψ(w), Ψ(w) := {z X (Z p ) log(z) = 0, D 2 (z) = w }. Of course, X (Z) X (Z p ) 3, but depending on the reduction of E, the latter could be made up of a large number of Ψ(w), creating potential for some discrepancy.

32 A non-abelian conjecture of Birch and Swinnerton-Dyer type: examples The curve has integral points y 2 + xy = x 3 x x (675, ±108). We find X (Z) = {z log(z) = 0, D 2 (z) = 0} = X (Z p ) 3 for all p such that 5 p 79. Note that D 2 (675, ±108) = 0 is already non-obvious. (A non-abelian reciprocity law.)

33 A non-abelian conjecture of Birch and Swinnerton-Dyer type: examples In fact, so far, we have checked X (Z) = X (Z p ) 3 for the prime p = 5 and 256 semi-stable elliptic curves of rank zero.

34 A non-abelian conjecture of Birch and Swinnerton-Dyer type: examples Cremona label number of w -values 1122m m m a d a h h h d j j b1 108

35 A non-abelian conjecture of Birch and Swinnerton-Dyer type: examples Cremona label number of w -values 3774f g b h b j j c e e e e d1 216

36 A non-abelian conjecture of Birch and Swinnerton-Dyer type: examples Hence, for example, for the curve 1122m2, y 2 + xy = x x there are potentially 384 of the Ψ(w) s that make up X (Z p ) 3. Of these, all but 4 end up being empty, while the points in those Ψ(w) consist exactly of the integral points (752, 17800), (752, 17048), (2864, ), (2864, ).

37 A non-abelian conjecture of Birch and Swinnerton-Dyer type: examples Another kind of test is to fix a few curves and let p grow. For example, for the curve ( 378b3 ) we found that for 5 p 97. y 2 + xy = x 3 x x , X (Z p ) 3 = X (Z) = {(19, 9), (19, 10)}

38 A non-abelian conjecture of Birch and Swinnerton-Dyer type: examples As one might expect, as p gets large, X (Z p ) 2 becomes significantly larger than X (Z). For p = 97, we have X (Z 97 ) 2 = 89. However, imposing the additional constraint defining X (Z 97 ) 3 exactly cuts out the integral points.

39 A non-abelian conjecture of Birch and Swinnerton-Dyer type: difficulties The case of X = E \ O where E is of rank 1 and Tamagawa number 1. Assume there is a point y X (Z) of infinite order and put c = D 2 (y)/ log 2 (y). Then X (Z p ) 2 = X (Z p ) 1 = X (Z p ) and X (Z p ) 3 is the zero set of D 2 (z) c log 2 (z). Can also write this as D 2 (z) log 2 (z) = c.

40 A non-abelian conjecture of Birch and Swinnerton-Dyer type: difficulties has integral points X : y 2 + y = x 3 x; p = 7 P = (0, 0), 2P = (1, 0), 3P = ( 1, 1), 4P = (2, 3), 6P = (6, 14). We find D 2 (P) log 2 (P) = D 2(2P) log 2 (2P) = D 2(3P) log 2 (3P) = D 2(4P) log 2 (4P) = D 2(6P) log 2 (6P) = O(7 5 ). However, in this case, we have X (Z) X (Z p ) 3 in most case, so need to compute X (Z p ) 4 X (Z p ) 3. (Work in progress by Balakrishnan and Netan Dogra.)

41 A non-abelian conjecture of Birch and Swinnerton-Dyer type: difficulties Analogously, investigating S-integral analogues for affine curves, say 2-integral points in X = P 1 \ {0, 1, }. Then while X (Z p ) 3 is the zero set of X (Z[1/2]) = {2, 1/2, 1}, 2Li 2 (z) + log(z) log(1 z). So far, checked equality for p = 3, 5, 7. For larger p, appear to have X (Z[1/2]) X (Z p ) 3 indicating the need to look at X (Z p ) 4. (Currently being studied by Dan-Cohen, Wewers, and Brown.)

42 Non-abelian reciprocity: a brief comparison Usual (Langlands) reciprocity: L(M) = L(π) where M is a motive and π is an algebraic automorphic representation on GL n (A F ). The relevance to arithemic comes from conjectures that say L(N M) encodes RHom(N, M). So in some sense, L functions classify motives. However, in classical (non-linear) Diophantine geometry, we are interested in schemes, not motives, in particular, actual maps between schemes. Hence, a need for a nonlinear reciprocity of some sort.

43 Non-abelian reciprocity: a brief comparison X /F as above, n, T n = n / n+1, etc. Langlands reciprocity ρ H 1 (G F, GL(T 1 )) functions on GL(H DR 1 (F ))\GL(H DR 1 (X )(A F )). π 1 reciprocity k H 1 (G F, T n ) functions on X (A F ) via functions on H 1 (G F, U n )\ H 1 (G v, U n ).