A non-abelian conjecture of Birch and Swinnerton-Dyer type

Transcription

1 A non-abelian conjecture of Birch and Swinnerton-Dyer type Minhyong Kim Bordeaux, July, 2012

2 Dedicated to Martin Taylor on the occasion of his 60th birthday.

3 Diophantine geometry: general remarks

4 Diophantine geometry: general remarks Diophantine geometry is the study of maps Y X between schemes of finite type over Z (integral points) or Q (rational points).

5 Diophantine geometry: general remarks (1) Galois theory: Spec(L) Spec(K) Spec(F )

6 Diophantine geometry: general remarks (1) Galois theory: Spec(L) (2) Class field theory: Spec(K) Spec(F ) Spec(O K ) Spec(O F ).

7 Diophantine geometry: general remarks (3) Conjectures and theorems of Shafarevich type: There are no abelian schemes f : A Spec(Z). (Fontaine-Abrashkin) There are at most finitely many smooth proper curves f : X Spec(O F,S ) of genus g 2. (Faltings)

8 Diophantine geometry: general remarks (4) Hasse-Minkowski theory: 0 H 2 (K, G m ) v H 2 (K v, G m ) Q/Z 0.

9 Diophantine geometry: general remarks (4) Hasse-Minkowski theory: 0 H 2 (K, G m ) v H 2 (K v, G m ) Q/Z 0. (5) The conjecture of Birch and Swinnerton Dyer: E(Z) Q p Hf 1 (G Q, H 1 (Ē, Q p )) E(Z) < L(E, 1) 0.

10 Diophantine geometry: linearization

11 Diophantine geometry: linearization People believe there is a good unified framework for linearized Diophantine geometry:

12 Diophantine geometry: linearization People believe there is a good unified framework for linearized Diophantine geometry: The theory of motives and the conjectures of Beilinson, Bloch, and Kato (generalizing BSD).

13 Diophantine geometry: linearization People believe there is a good unified framework for linearized Diophantine geometry: The theory of motives and the conjectures of Beilinson, Bloch, and Kato (generalizing BSD). After replacing varieties X and Y by motives H(X ) and H(Y ), it is (sometimes) believed that there should be uniform way to understand RHom Mot (Y, X ) inside the category Mot hom of homological motives.

14 Diophantine geometry: linearization For example, when X and Y are smooth projective varieties over F it is conjectured that Hom Mot (Y, X ) Q p Hom GF [H et (Ȳ, Q p ), H et ( X, Q p )].

15 Diophantine geometry: linearization For example, when X and Y are smooth projective varieties over F it is conjectured that and Hom Mot (Y, X ) Q p Hom GF [H et (Ȳ, Q p ), H et ( X, Q p )]. Ext 1 Mot (Y, X ) Q p Ext 1 G F,f (Het (Ȳ, Q p ), H et ( X, Q p )),

16 Diophantine geometry: linearization Also conjectured are numerical criteria:

17 Diophantine geometry: linearization Also conjectured are numerical criteria: dimext 1 Mot (Y, X ) dimhom Mot(Y, X ) = ord s=1 L(H(X ) H(Y ), s)

18 Diophantine geometry: linearization Also conjectured are numerical criteria: dimext 1 Mot (Y, X ) dimhom Mot(Y, X ) = ord s=1 L(H(X ) H(Y ), s) All maps in this theory are treated in a uniform manner, without regard, for example, to the relative dimensions of the objects.

19 Diophantine geometry: problems of linearization

20 Diophantine geometry: problems of linearization In linearized Diophantine geometry, X will pick up many more virtual Y -points, making it difficult to recover the original maps of interest.

21 Diophantine geometry: problems of linearization In linearized Diophantine geometry, X will pick up many more virtual Y -points, making it difficult to recover the original maps of interest. When Y = F itself, then given a K point x X (K) for a finite field extension K of F, the formal linear combination σ(x), where σ(x) runs over the Galois conjugates of x, will define a map in M from Spec(F ) to X.

22 Homotopical Diophantine geometry

23 Homotopical Diophantine geometry Programme to recover non-linear information by studying π(x ) instead of H(X ).

24 Homotopical Diophantine geometry Programme to recover non-linear information by studying π(x ) instead of H(X ). Precise nature of π(x ), a homotopy type, is still undetermined.

25 Homotopical Diophantine geometry Programme to recover non-linear information by studying π(x ) instead of H(X ). Precise nature of π(x ), a homotopy type, is still undetermined. Grothendieck proposed the profinite étale π 1 for a certain class of schemes including algebraic number fields and curves of genus 2.

26 Homotopical Diophantine geometry Programme to recover non-linear information by studying π(x ) instead of H(X ). Precise nature of π(x ), a homotopy type, is still undetermined. Grothendieck proposed the profinite étale π 1 for a certain class of schemes including algebraic number fields and curves of genus 2. Toen and Vezzosi propose schematic homotopy types.

27 Homotopical Diophantine geometry Programme to recover non-linear information by studying π(x ) instead of H(X ). Precise nature of π(x ), a homotopy type, is still undetermined. Grothendieck proposed the profinite étale π 1 for a certain class of schemes including algebraic number fields and curves of genus 2. Toen and Vezzosi propose schematic homotopy types. Deligne and Ihara emphazised the importance of rational homotopy types that have a motivic nature.

28 Homotopical Diophantine geometry Programme to recover non-linear information by studying π(x ) instead of H(X ). Precise nature of π(x ), a homotopy type, is still undetermined. Grothendieck proposed the profinite étale π 1 for a certain class of schemes including algebraic number fields and curves of genus 2. Toen and Vezzosi propose schematic homotopy types. Deligne and Ihara emphazised the importance of rational homotopy types that have a motivic nature. Hope (fantasy?) is to construct non-linear refinements of all aspects of the theory of motives.

29 Homotopical Diophantine geometry A somewhat more accessible fantasy is to extend the ideas surrounding the conjecture of Birch and Swinnerton-Dyer to hyperbolic curves, e.g.,

30 Homotopical Diophantine geometry A somewhat more accessible fantasy is to extend the ideas surrounding the conjecture of Birch and Swinnerton-Dyer to hyperbolic curves, e.g., [X (Q) < ] [non-vanishing of L-values].

31 Homotopical Diophantine geometry A somewhat more accessible fantasy is to extend the ideas surrounding the conjecture of Birch and Swinnerton-Dyer to hyperbolic curves, e.g., [X (Q) < ] [non-vanishing of L-values]. The focus of this lecture is extending the portion E(Z) Q p H 1 f (G Q, H 1 (Ē, Q p )) to the hyperbolic setting, where the right-hand side gets replaced by non-abelian cohomology with coefficients in a fundamental group.

32 type

33 type X /Z is one of the following: P 1 \ {0, 1, }; The regular minimal model of an curve of genus one with at least one point removed; The regular minimal model of a compact smooth curve of genus 2; The regular minimal model of a compact smooth curve of genus 2 with some points removed.

34 type X /Z is one of the following: P 1 \ {0, 1, }; The regular minimal model of an curve of genus one with at least one point removed; The regular minimal model of a compact smooth curve of genus 2; The regular minimal model of a compact smooth curve of genus 2 with some points removed. Denote by b either an integral point of X ; or an integral tangent vector at a missing point.

35 type X /Z is one of the following: P 1 \ {0, 1, }; The regular minimal model of an curve of genus one with at least one point removed; The regular minimal model of a compact smooth curve of genus 2; The regular minimal model of a compact smooth curve of genus 2 with some points removed. Denote by b either an integral point of X ; or an integral tangent vector at a missing point. Let p be an odd prime of good reduction and T a finite set of primes containing all primes of bad reduction and p. X = X Spec(Z) Spec( Q).

36 type Denote by U = π Qp 1 ( X, b) the Q p -pro-unipotent étale fundamental group of X with base-point b. That is, the universal pro-unipotent pro-algebraic group over Q p with a continuous homomorphism π et 1 ( X, b) U(Q p ).

37 type Denote by U = π Qp 1 ( X, b) the Q p -pro-unipotent étale fundamental group of X with base-point b. That is, the universal pro-unipotent pro-algebraic group over Q p with a continuous homomorphism π et 1 ( X, b) U(Q p ). U 1 = U, U n+1 = [U, U n ], U n = U/U n+1. U 4 U 3 U 2 U 1 Each U n is a unipotent Q p -algebraic group.

38 A conjecture of Birch and Swinnerton-Dyer type We have U 1 = H 1 ( X, Q p ) = V p J X and in the affine case, while in the compact case. U 2 /U 3 2 U 1 U 2 /U 3 2 U 1 /Q p (1)

39 A conjecture of Birch and Swinnerton-Dyer type We have U 1 = H 1 ( X, Q p ) = V p J X and in the affine case, while in the compact case. U 2 /U 3 2 U 1 U 2 /U 3 2 U 1 /Q p (1) When X = P 1 \ {0, 1, }, get an exact sequence 1 Q p (2) U 2 Q p (1) Q p (1) 1.

40 A conjecture of Birch and Swinnerton-Dyer type We have U 1 = H 1 ( X, Q p ) = V p J X and in the affine case, while in the compact case. U 2 /U 3 2 U 1 U 2 /U 3 2 U 1 /Q p (1) When X = P 1 \ {0, 1, }, get an exact sequence 1 Q p (2) U 2 Q p (1) Q p (1) 1. When X = E \ O, get an exact sequence 1 Q p (1) U 2 V p E 1.

41 type For any other point x X (Z), we have P(x) := π Qp 1 ( X ; b, x) = [π et 1 ( X ; b, x) U]/π et 1 ( X, b), the homotopy classes of Q p -pro-unipotent étale paths from b to x. This is a torsor for U, the push-out of the π1 et( X, b)-torsor π1 et( X ; b, x) via the map π et 1 ( X, b) U.

42 type For any other point x X (Z), we have P(x) := π Qp 1 ( X ; b, x) = [π et 1 ( X ; b, x) U]/π et 1 ( X, b), the homotopy classes of Q p -pro-unipotent étale paths from b to x. This is a torsor for U, the push-out of the π1 et( X, b)-torsor π1 et( X ; b, x) via the map π et 1 ( X, b) U. Both U and P(x) admit compatible actions of G = G Q that factor through G T = Gal(Q T /Q). The induced action action of G p = Gal( Q p /Q p ) is also crystalline.

43 type Get a global Q p -pro-unipotent period map: X (Z) j H 1 f (G T, U); x [P(x)]; to a moduli scheme Hf 1 (G T, U) of torsors for U that are unramified outside T and crystalline at p.

44 type Get a global Q p -pro-unipotent period map: X (Z) j H 1 f (G T, U); x [P(x)]; to a moduli scheme Hf 1 (G T, U) of torsors for U that are unramified outside T and crystalline at p. Similarly, for each prime v, get local period maps X (Z v ) j v H 1 (G v, U). such that X (Z p ) j p H 1 f (G p, U) H 1 (G p, U).

45 type.. H 1 f (G, U 3) j 3 j 2 Hf 1 (G, U 2) X (Z) j 1 Hf 1 (G, U 1)

46 type Compatible with localization X (Z) X (Z v ) j Hf 1 (G T, U n ) where each loc v is an algebraic map. j v loc v H 1 (G v, U n )

47 type Define H 1 Z (U n) := v p loc 1 v [Im(j v )] H 1 f (G T, U n ) Thee are the torsors that are locally path torsors for each place v p and crystalline at p.

48 type Define H 1 Z (U n) := v p loc 1 v [Im(j v )] H 1 f (G T, U n ) Thee are the torsors that are locally path torsors for each place v p and crystalline at p. Remarks: Im(j v ) is finite for all v p (actually, im(j v ) = 0 for v / T ). Thus, H 1 Z (U n) is a closed subcheme.

49 type Define H 1 Z (U n) := v p loc 1 v [Im(j v )] H 1 f (G T, U n ) Thee are the torsors that are locally path torsors for each place v p and crystalline at p. Remarks: Im(j v ) is finite for all v p (actually, im(j v ) = 0 for v / T ). Thus, H 1 Z (U n) is a closed subcheme. H 1 f (G p, U n ) = j p (X (Z p )) H 1 (G p, U n ).

50 type X (Z) X (Z p ) j j p HZ 1 (U locp n) Hf 1 (G p, U n )

51 type X (Z) X (Z p ) j j p Define HZ 1 (U locp n) Hf 1 (G p, U n ) X (Z p ) n := jp 1 (loc p [HZ 1 (U n)]), the p-adic points that are cohomologically global of level n.

52 type From the diagrams H 1 f (G p, U n+1 ) loc p H 1 Z (U n+1 ) j p X (Z p ) j p Hf 1 (G p, U n ) loc p HZ 1 (U n)

53 type From the diagrams H 1 f (G p, U n+1 ) loc p H 1 Z (U n+1 ) j p X (Z p ) j p Hf 1 (G p, U n ) loc p HZ 1 (U n) we see that X (Z p )

54 type From the diagrams H 1 f (G p, U n+1 ) loc p H 1 Z (U n+1 ) j p X (Z p ) j p Hf 1 (G p, U n ) loc p HZ 1 (U n) we see that X (Z p ) X (Z p ) 1

55 type From the diagrams H 1 f (G p, U n+1 ) loc p H 1 Z (U n+1 ) j p X (Z p ) j p Hf 1 (G p, U n ) loc p HZ 1 (U n) we see that X (Z p ) X (Z p ) 1 X (Z p ) 2

56 type From the diagrams H 1 f (G p, U n+1 ) loc p H 1 Z (U n+1 ) j p X (Z p ) j p Hf 1 (G p, U n ) loc p HZ 1 (U n) we see that X (Z p ) X (Z p ) 1 X (Z p ) 2

57 type From the diagrams H 1 f (G p, U n+1 ) loc p H 1 Z (U n+1 ) j p X (Z p ) j p Hf 1 (G p, U n ) loc p HZ 1 (U n) we see that X (Z p ) X (Z p ) 1 X (Z p ) 2 X (Z p ) n

58 type From the diagrams H 1 f (G p, U n+1 ) loc p H 1 Z (U n+1 ) j p X (Z p ) j p Hf 1 (G p, U n ) loc p HZ 1 (U n) we see that X (Z p ) X (Z p ) 1 X (Z p ) 2 X (Z p ) n X (Z p ) n+1

59 type From the diagrams H 1 f (G p, U n+1 ) loc p H 1 Z (U n+1 ) j p X (Z p ) j p Hf 1 (G p, U n ) loc p HZ 1 (U n) we see that X (Z p ) X (Z p ) 1 X (Z p ) 2 X (Z p ) n X (Z p ) n+1 X (Z).

60 type Conjecture: for n >> 0. X (Z p ) n = X (Z)

61 type Conjecture: for n >> 0. X (Z p ) n = X (Z) Key point is that X (Z p ) n should be computable, in principle.

62 type Easy part: p-adic Hodge theory gives us a description of X (Z p ) j p H 1 f (G p, U n ) A rn as j p (z) = (j w p (z)) where each coordinate j w p (z) is a p-adic analytic function defined by iterated Coleman integrals z b α 1 α 2 α n of differential forms on X.

63 type X (Z) X (Z p ) j j p (j w p ) HZ 1 (U locp n) Hf 1 (G p, U n ) A rn

64 type Difficult part is compute the defining ideal L(n) for the global image at level n: loc p [HZ 1 (U n)] Hf 1 (G p, U n ) = A rn. from which we wish to get the set X (Z p ) n as zeros of f j p, for f L(n).

65 type As soon as L(n) 0, we get finiteness of X (Z p ) n, and hence, of X (Z).

66 type As soon as L(n) 0, we get finiteness of X (Z p ) n, and hence, of X (Z). Standard motivic conjectures (Fontaine-Mazur-Jannsen, Bloch-Kato) imply that for n large, j L(n + 1) j L(n) and in fact contains elements that are algebraically independent of the elements in j L(n).

67 type As soon as L(n) 0, we get finiteness of X (Z p ) n, and hence, of X (Z). Standard motivic conjectures (Fontaine-Mazur-Jannsen, Bloch-Kato) imply that for n large, j L(n + 1) j L(n) and in fact contains elements that are algebraically independent of the elements in j L(n). So something in the common zero set for all n should be there for a good reason.

68 type: Examples [Joint work in progress with Jennifer Balakrishnan, Ishai Dan-Cohen, and Stefan Wewers]

69 type: Examples [Joint work in progress with Jennifer Balakrishnan, Ishai Dan-Cohen, and Stefan Wewers] Let X = P 1 \ {0, 1, }.

70 type: Examples [Joint work in progress with Jennifer Balakrishnan, Ishai Dan-Cohen, and Stefan Wewers] Let X = P 1 \ {0, 1, }. We have and so there is an exact sequence U 1 = Q p (1) Q p (1) U 2 /U 3 = Q p (2), 0 Q p (2) U 2 Q p (1) Q p (1) 0.

71 type: Examples The diagram X (Z) X (Z p ) j j p HZ 1 (U locp 1) Hf 1 (G p, U 1 )

72 type: Examples The diagram X (Z) X (Z p ) j j p becomes HZ 1 (U locp 1) Hf 1 (G p, U 1 ) φ X (Z p ) j j p loc p 0 A 2

73 type: Examples The map j p takes the form z (log(z), log(1 z)), so that X (Z p ) 1 is the common zero set of log(z) and log(1 z).

74 type: Examples The map j p takes the form z (log(z), log(1 z)), so that X (Z p ) 1 is the common zero set of log(z) and log(1 z). Must have z = ζ n and 1 z = ζ m, and hence, z = ζ 6.

75 type: Examples The map j p takes the form z (log(z), log(1 z)), so that X (Z p ) 1 is the common zero set of log(z) and log(1 z). Must have z = ζ n and 1 z = ζ m, and hence, z = ζ 6. Thus, we have if p = 3 or p 2 mod 3. X (Z p ) 1 = φ = X (Z)

76 type: Examples The map j p takes the form z (log(z), log(1 z)), so that X (Z p ) 1 is the common zero set of log(z) and log(1 z). Must have z = ζ n and 1 z = ζ m, and hence, z = ζ 6. Thus, we have if p = 3 or p 2 mod 3. When p 1 mod 3 X (Z p ) 1 = φ = X (Z) X (Z) = φ {ζ 6 } = X (Z p ) 1 and we must go to a higher level.

77 type: Examples We have and is where H 1 f (G p, U 2 ) = A 3 j p : X (Z p ) A 3 j p (z) = (log(z), log(1 z), Li 2 (z)), Li 2 (z) = n z n n 2 is the dilogarithm.

78 type: Examples Meanwhile, H 1 Z (U 2) = 0. (We have H 1 (G T, Q p (2)) = 0 by Soulé vanishing.)

79 type: Examples Meanwhile, H 1 Z (U 2) = 0. (We have H 1 (G T, Q p (2)) = 0 by Soulé vanishing.) X (Z p ) 2 = {z log(z) = 0, log(1 z) = 0, Li 2 (z) = 0}.

80 type: Examples Meanwhile, H 1 Z (U 2) = 0. (We have H 1 (G T, Q p (2)) = 0 by Soulé vanishing.) X (Z p ) 2 = {z log(z) = 0, log(1 z) = 0, Li 2 (z) = 0}. Li 2 (ζ 6 ) 0?

81 type: Examples Meanwhile, H 1 Z (U 2) = 0. (We have H 1 (G T, Q p (2)) = 0 by Soulé vanishing.) X (Z p ) 2 = {z log(z) = 0, log(1 z) = 0, Li 2 (z) = 0}. Li 2 (ζ 6 ) 0? Can check this numerically for all p < 500.

82 type: Examples Let X = E \ O where E is an elliptic curve of rank 0 and Tamagawa number 1.

83 type: Examples Let X = E \ O where E is an elliptic curve of rank 0 and Tamagawa number 1. Then (assuming BSD), H 1 Z (U 1) = 0,

84 type: Examples Let X = E \ O where E is an elliptic curve of rank 0 and Tamagawa number 1. Then (assuming BSD), and we get H 1 Z (U 1) = 0, X (Z) X (Z p ) j j p loc p 0 Hf 1 (G p, U 1 ) = A 1

85 type: Examples The map is given by where j p : X (Z p ) Hf 1 G p, U 1 ) = A 1 z log(z) log(z) = z b (dx/y).

86 type: Examples Thus, X (Z p ) 1 = E(Z p )[tor] \ O.

87 type: Examples Thus, X (Z p ) 1 = 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 ) 1. But of course, this fails as p grows.

88 type: Examples So then, examine j p : X (Z p ) H 1 f (U 2) which is given by where z (log(z), D 2 (z)), D 2 (z) = z b (dx/y)(xdx/y).

89 type: Examples So then, examine j p : X (Z p ) H 1 f (U 2) which is given by where z (log(z), D 2 (z)), D 2 (z) = z b (dx/y)(xdx/y). Under our assumptions, still have H 1 Z (U 2) = 0, so that X (Z p ) 2 is the common zero set of log(z) and D 2 (z).

90 type: Examples So then, examine j p : X (Z p ) H 1 f (U 2) which is given by where z (log(z), D 2 (z)), D 2 (z) = z b (dx/y)(xdx/y). Under our assumptions, still have H 1 Z (U 2) = 0, so that X (Z p ) 2 is the common zero set of log(z) and D 2 (z). Here, X (Z) = X (Z p ) 2 appears to be very common.

91 type: Examples The curve has integral points y 2 + xy = x 3 x x (675, ±108).

92 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 ) 2 for all p such that 5 p 79.

93 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 ) 2 for all p such that 5 p 79. Note that D 2 (675, ±108) = 0 is already non-obvious.

94 type: Examples Higher genus:

95 type: Examples Higher genus: is identified with j p : X (Z p ) H 1 f (G p, U 1 ) log : X (Z p ) J(Z p ) T e J,

96 type: Examples Higher genus: is identified with j p : X (Z p ) H 1 f (G p, U 1 ) log : X (Z p ) J(Z p ) T e J, So X (Z p ) J(Z p )[tor] X (Z p ) 1. and we have equality whenever HZ 1 (U 1) = 0.

97 type: Examples Using this, we get X (Z) = X (Z p ) 1 for x 5 + y 5 = z 5 whenever ζ 5 / Q p and for x 7 + y 7 = z 7 when ζ 7 / Q p.

98 type: Examples In general, for (l 5), x l + y l = z l X (Z) = X (Z p ) J(Z p )[tor] whenever ζ l / Q p by the theorem of Wiles and Coleman-Tamagawa-Tzermias.

99 type: Examples In general, for (l 5), x l + y l = z l X (Z) = X (Z p ) J(Z p )[tor] whenever ζ l / Q p by the theorem of Wiles and Coleman-Tamagawa-Tzermias. But the rank of the Jacobian is positive as soon as l > 7. The discrepancy between X (Z) and X (Z p ) 1 is captured by X (Z) = X (Z p ) J X (Z) = X (Z p ) J(Z p )[tor] X (Z p ) 1 = X (Z p ) J X (Z) + J X (Z p )[tor].

100 type: Examples If d 0, the affine hyperelliptic curve y 2 = d 3 x 6 27d 2 x 4 648dx , has integral points (0, ±108).

101 type: Examples If d 0, the affine hyperelliptic curve y 2 = d 3 x 6 27d 2 x 4 648dx , has integral points (0, ±108). For p such that d / Q p, we often have X (Z) = X (Z p ) 2, e.g. d = 2, p = 5, 11, 13, 19, 29, 53.

102 type: Examples Begun to examine the case of X = E \ O where E is of rank 1 and Tamagawa number 1.

103 type: Examples Begun to examine 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. Then is computed to be where loc p : H 1 Z (U 2) H 1 p(g p, U 2 ) = A 2 A 1 A 2 ; t (t, ct 2 ), c = D 2 (y)/ log 2 (y). So image is defined by x 2 cx 2 1 = 0.

104 type: Examples Meanwhile, is Thus, is the zero set of Can also write this as X (Z p ) H 1 f (G p, U 2 ) z (log(z), D 2 (z)). X (Z p ) 2 D 2 (z) c log 2 (z). D 2 (z) log 2 (z) = c.

105 type: Examples 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).

106 type: Examples 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 ).

107 type: Examples Meanwhile, D 2 (5P) log 2 (5P) = O(7 7 ) D 2 (7P) log 2 (7P) = O(7 5 ) D 2 (8P) log 2 (8P) = O(7 7 ) D 2 (9P) log 2 (9P) = O(7 5 )

108 type: Examples Analogously, investigating S-integral analogues for affine curves, say 2-integral points in X = P 1 \ {0, 1, }. There, the diagram X (Z) X (Z p ) j j p becomes HZ 1 (U locp 2) Hf 1 (G p, U 2 )

109 type: Examples {2, 1/2, 1} X (Z p ) j A 2 where (Dan-Cohen, Wewers) j p loc p A 3 Recall that loc p (x, y) = ((log 2)x, (log 2)y, (1/2)(log 2 2)xy). j p (z) = (log(z), log(1 z), Li 2 (z)).

110 type: Examples Therefore, X (Z p ) 2 is the zero set of 2Li 2 (z) + log(z) log(1 z).

111 type: Examples Therefore, X (Z p ) 2 is the zero set of 2Li 2 (z) + log(z) log(1 z). So far, we have checked that this is exactly {2, 1/2, 1} for p = 3, 5, 7.

112 type: Examples Therefore, X (Z p ) 2 is the zero set of 2Li 2 (z) + log(z) log(1 z). So far, we have checked that this is exactly {2, 1/2, 1} for p = 3, 5, 7. For larger p, appear to have X (Z[1/2]) X (Z p ) 2 indicating we need to look at X (Z p ) 3.

113 type: Plan

114 type: Plan Develop a systematic method to compute loc p : H 1 Z (U n) H 1 f (G p, U n ).

115 type: Plan Develop a systematic method to compute loc p : H 1 Z (U n) H 1 f (G p, U n ). Make the conjecture falsifiable.

116 type: Plan Develop a systematic method to compute loc p : H 1 Z (U n) H 1 f (G p, U n ). Make the conjecture falsifiable. Investigate possibility that there is a precise formula for X (Z) hidden in the conjecture, perhaps using a another interpretation of the defining ideal L(n) of loc p (H 1 Z )(U n) H 1 f (G p, U n ).

117