NON-ABELIAN RECIPROCITY, PERIOD MAPS, AND
NON-ABELIAN RECIPROCITY, PERIOD MAPS, AND DIOPHANTINE GEOMETRY MINHYONG KIM
1. Some p-adic analytic functions We start with the evaluation in Q2 a convergent power series ∞ X 2n =0 n2 n=1
The value of zero, not at all obvious, was observed by Coleman about 30 years ago[7]. In fact, the value is zero even when the series is not convergent, that is, in Qp for all p. The general statement is that `2 (2) = 0, where `2 is the p-adic dilogarithm defined by iterated Coleman integration: Z z ∞ X zn dt dt `2 (z) := = t 1 − t n=1 n2 b Here, b is the tangential base point at 0 defined by dt(b) = 1 with respect to the coordinate t on P1 \ {0, 1, ∞}. There are many approaches to Coleman integration, but one efficient (albeit abstract) way [4] to define the iterated integral as the upper right hand corner of the matrix Rz Rz 1 b dt/t b (dt/t)(dt/(1 − t)) R z Mbz = 0 1 dt/(1 − t) b 0 0 1 where Mbz is the holonomy of the rank 3 unipotent connection on P1 \ {0, 1, ∞} given by the connection form 0 dt/t 0 0 dt/(1 − t) − 0 0 0 0 The holonomy is computed locally by solving the differential equations d`1 = dt/(1 − t); d`2 = `1 dt/t. with initial condition at a point. But it is a somewhat deep fact that given a unipotent connection (V, ∇) and two points x, y ∈ P1 \ {0, 1, ∞}(Qp ) 1991 Mathematics Subject Classification. 14G10, 11G40, 81T45 . M.K. Supported by grant EP/M024830/1 from the EPSRC. 1
2
MINHYONG KIM
(possibly tangental), there is a canonical isomorphism Mxy (V, ∇) : V (]¯ x[)∇=0
'
- V (]¯ y [)∇=0
determined by the property that it’s compatible with morphisms between connections and Frobenius pull-backs. One gets thereby a holonomy transformation between any two points. More generally, the k-logarithm Z z `k (z) := (dt/t)(dt/t) · · · (dt/t)(dt/(1 − t)) b
is defined as the upper right hand corner (k + 1) × (k + 1) connection form 0 dt/t 0 0 0 0 dt/t 0 0 0 0 dt/t − .. . 0 0 0 0 0 0 0 0
of the holonomy matrix arising from the ... ... ...
0 0 0
. . . dt/t ... 0 ... 0
0 0 0
0 dt/(1 − t) 0
Using the geometry of P1 \ {0, 1, ∞}, Coleman derived the following functional equations: −1 `k (z) + (−1)k `k (z −1 ) = logk (z); k! D2 (z) = −D2 (z −1 ); D2 (z) = −D2 (1 − z); where D2 (z) = `2 (z) + (1/2) log(z) log(1 − z). (The upper right hand corner of the log of the holonomy matrix.) From this, we get D2 (−1) = −D2 (1/(−1)) = 0, and D2 (2) = −D2 (1 − 2) = 0. But D2 (2) = `2 (2) + (1/2) log(2) log(−1) = `2 (2), giving us the desired vanishing. We note also that D2 (1/2) = −D2 (2) = 0. The beginning of our investigation is the observation that {−1, 2, 1/2} are exactly the 2-integral points of P1 \ {0, 1, ∞}. It is possible thereby to give a global proof of the vanishing [10] using, in some sense, the arithmetic geometry of Spec(Z) \ {2, p, ∞}. We will expand on this in detail later, but for now, we add also the following result of Ishai Dan-Cohen and Stefan Wewers [8].
NON-ABELIAN RECIPROCITY, PERIOD MAPS, AND DIOPHANTINE GEOMETRY
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Let D4 (z) = ζ(3)`4 (z) + (8/7)[log3 2/24 + `4 (1/2)/ log 2] log(z)`3 (z) +[(4/21)(log3 2/24 + `4 (1/2)/ log 2) + ζ(3)/24] log3 (z) log(1 − z). Even though the expression is somewhat complicated, note that it is of the form ζ(3)`4 (z) + A log(z)`3 (z) + B log3 (z) log(1 − z), which is homogeneous of degree 4 in a suitable sense. With this definition, it turns out that [P1 \ {0, 1, ∞}](Z[1/2]) ⊂ {D2 (z) = 0, D4 (z) = 0} and numerical computations for p ≤ 29 indicate equality. A key theme is that these equations are examples of non-abelian explicit reciprocity laws. 1 In fact, to describe √ P \ {0, 1, ∞}(Z[1/2]), we definitely need the extra equation since, for example, 5 ∈ Z11 , and √ √ √ 3± 5 1± 5 −1 ± 5 ) = D2 ( ) = D2 ( ) = 0. D2 ( 2 2 2 2. The local-to-global problem in Diophantine geometry We go on to describe the general context for these formulae [11]. Given a number field F and X/F a smooth variety (with an integral model), the key local-to-global problem is to locate 0 Y X(F ) ⊂ X(AF ) = X(Fv ) v
That is, we would like to know How do the global points sit inside the local points? There is a classical answer for X = Gm , in which case X(F ) = F ∗ ,
X(Fv ) = Fv∗ .
Thus, the problem becomes that of locating F ∗ ⊂ A× F. We have the Artin reciprocity map Y rec = recv : A× F
- Gab F ,
v
The Artin-Takagi reciprocity law says that the composed map F∗
⊂
- A× F
rec
- Gab F
is zero. That is, the reciprocity map gives a defining equation for Gm (F ) ⊂ Gm (AF ). We would like to generalize this to other equations by way of a non-abelian reciprocity law. Is it possible to start with a rather general variety X and understand X(F ) via X(F )
⊂
- X(AF )
recN A
-
some target with base-point 0
4
MINHYONG KIM
in such way that recN A = 0. becomes an equation for X(F )? 3. Non-abelian reciprocity Here is a brief list of essentially standard notation to be used for the remainder of this note. F : number field. GF = Gal(F¯ /F ). Gv = Gal(F¯v /Fv ) for a place v of F . S: finite set of places of F . AF : finite Adeles of F ASF : finite S-integral adeles of F . GSF = Gal(F S /F ), where F S is the maximal extension of F unramified outside S. Q
: product over non-Archimedean places in S. H 1 (Gv , A): product over non-Archimedean places in S and ‘unramified cohomology’ outside of S. X: a smooth variety over F . b ∈ X(F ) (sometimes tangential). QSS
Now for some more sophisticated objects: We let ¯ b)(2) , ∆ = π1 (X, the pro-finite prime-to-2, étale fundamental group of ¯ =X× ¯ X Spec(F ) Spec(F ) with base-point b. Denote with superscripts ∆[n] , [1] the lower central series with ∆ = ∆. Subscripts will be used for the corresponding quotients: ∆n = ∆/∆[n+1] . Tn = ∆[n] /∆[n+1] . We also consider ∆M , (∆n )M , TnM , the pro-M quotients for various finite sets of prime M . We will be assuming that the following condition is satisfied (which is true for curves): For each n and M sufficiently large, TnM is torsion-free. This implies H 1 (GSF , TnM )
0 Y loc H 1 (Gv , TnM )
is injective. With this, we get a non-abelian class field theory with coefficients in the nilpotent completion of X. This consists of a filtration X(AF ) = X(AF )1 ⊃ X(AF )21 ⊃ X(AF )2 ⊃ X(AF )32 ⊃ X(AF )3 ⊃ X(AF )43 ⊃ · · ·
NON-ABELIAN RECIPROCITY, PERIOD MAPS, AND DIOPHANTINE GEOMETRY
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and a sequence of maps recn : X(AF )n
- Gn (X)
- Gn+1 (X) recn+1 : X(AF )n+1 n n n n+1 to a sequence Gn (X), Gn (X) of profinite abelian groups that are organised in such a way that X(AF )n+1 = rec−1 n n (0) and X(AF )n+1 = (recn+1 )−1 (0). n It is also useful to display the maps as follows: 2 −1 · · · rec−1 (0) ⊂ rec−1 2 (0) ⊂ (rec1 ) 1 (0) ⊂
||
||
X(AF )
||
||
· · · X(AF )32 ⊂ X(AF )2 ⊂ X(AF )21 ⊂ X(AF )1 rec32
rec21
rec2
? · · · G32 (X)
? G2 (X)
rec1
? G21 (X)
? G1 (X)
The targets Gn (X) are defined as Gn (X) := ∨
1
H (GF , D(Tn )) := lim Hom[H 1 (GF , D(TnM )), Q/Z] ←− M
where D(TnM ) = lim Hom(Tn , µm ). −→ m
For the intermediate targets, M Gn+1 (X) := lim X1 (D(Tn+1 ))∨ n ←−
H 1 (GF , D(Tn+1 ))∨ .
M
When X = Gm , then Gn (X) = 0 for n ≥ 2, Gn+1 (X) = 0 n for all n, and ˆ (2) )), Q/Z] G1 = Hom[H 1 (GF , D(Z(1) = Hom[H 1 (GF , [Q/Z](2) ), Q/Z] = [G(2) ]ab F . In this case, rec1 reduces to the prime-to-2 part of the usual reciprocity map. The reciprocity maps are defined using the local non-abelian étale period maps j v : X(Fv ) - H 1 (Gv , ∆); (2)
¯ b, x)]. x 7→ [π1 (X;
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MINHYONG KIM
Because the homotopy classes of étale paths (2)
¯ b, x) π1 (X; form a torsor for ∆ with compatible action of Gv , we get a corresponding class in non-abelian cohomology of Gv with coefficients in ∆. These assemble to a map j loc : X(AF )
-
jnloc : X(AF )
-
0 Y
H 1 (Gv , ∆),
which comes in levels 0 Y
H 1 (Gv , ∆n ).
We also have pro-M versions jnloc : X(AF )
-
jnloc : X(ASF )
-
0 Y
H 1 (Gv , ∆M n )
and integral versions S Y
H 1 (Gv , ∆M n ).
To give a flavour of the definition of the reciprocity maps, we will just define pro-M versions on X(ASF ) and assume that locS Y 1 H 1 (GSF , TnM ) H (Gv , TnM ) S
are injective. In general, one needs first to work with a pro-M quotient for a finite set of primes M and S ⊃ M , then take a limit over S and M . The first reciprocity map is just defined using x ∈ X(AF ) 7→ d1 (j1loc (x)), where D1 :
Y
H 1 (Gv , ∆M 1 )
-
Y
S
∨ H 1 (Gv , D(∆M 1 ))
∗ loc∨ H 1 (GSF , D(∆M 1 )) ,
S
is obtained from Tate duality and the dual of localisation. To define the higher reciprocity maps, we use the exact sequences 0
M - H 1 (GSF , Tn+1 )
- H 1 (GSF , ∆M n+1 )
pn+1 n
- H 1 (GSF , ∆n )
δn+1
M - H 2 (GSF , Tn+1 ) for non-abelian cohomology and Poitou-Tate duality stating that Y Dn+1 M M - H 1 (GS , D(Tn+1 H 1 (GSF , Tn+1 ) H 1 (Gv , TnM ) ))∨ S
is exact. We proceed as follows: rec21 (x) = P T ◦ δ2 ◦ loc−1 (j1 (x)) ∈ X1S (D(T2M ))∨ and rec2 (x) = D2 (loc((p21 )−1 (loc−1 (j1 (x)))) − j2 (x)) ∈ H 1 (GSF , D(T2M ))∨ .
NON-ABELIAN RECIPROCITY, PERIOD MAPS, AND DIOPHANTINE GEOMETRY
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To understand these formulae, we use the diagram Q H 1 (GSF , T2M ) ⊂- S H 1 (Gv , T2M )3 k2
? ? Q 1 M ⊂[p21 ]−1 (loc−1 (j1 )) ∈H 1 (GSF , ∆M ) H (G v , ∆2 ) 3 j2 2 S p21
loc−1 (j1 ) ∈
? ? Q H 1 (GSF , T1M ) ⊂- S H 1 (Gv , T1M ) 3 j1
That is, we have δ2 ◦ loc−1 (j1 (x)) ∈ X2S (T2M ) which maps to rec21 (x) via P T : X2S (T2M ) ' X1S (D(T2M ))∨
H 1 (GS , D(T2M ))∨ .
When this vanishes, we can lift loc−1 (j1 (x)) to (p21 )−1 (loc−1 (j1 (x)) and localise. The difference between this global lift and the second level of the local period map gives Y k2 (x) := loc[[p21 ]−1 (loc−1 (j1 (x)))] − j2 (x) ∈ H 1 (Gv , T2M ) 1
S S (GF , D(T2M ))∨
7→ rec2 (x) := D2 (k2 (x)) ∈ H Poitou-Tate duality implies that this element is independent of the lift. It is clear how to continue to higher levels. Now put X(AF )∞ = ∩∞ n=1 X(AF )n . Theorem 3.1 (Non-abelian reciprocity). X(F ) ⊂ X(AF )∞ . Let F = Q, X ia compact curve of genus at least two, p is a prime of good reduction, and S a finite set of places containing p and all primes of bad reduction. Suppose there is a finite set T ⊃ S of places such that Y H 1 (Gv , ∆pn ) H 1 (GSF , ∆pn ) v∈T
is injective for all sufficiently large n. Then the reciprocity law implies finiteness of X(F ). The proof of the reciprocity law is easy, being implied by the existence of a global period map on global points, which causes all obstruction to vanish: - X(AF ) X(F ) g jn
loc jn
? ? Q loc 1 M H 1 (GSF , ∆M ) H (G v , ∆n ) n S
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MINHYONG KIM
H 1 (GSF , ∆M n+1 ) 1
g
j n+
X(F )
? - H 1 (GS , ∆M n ) F
g jn
4. A non-abelian conjecture of Shafarevich-Tate type This is an elaboration of the discussion in [11] and [1]. The relation to computation of points arises via the projection P rv : X(AF ) - X(Fv ) to the v-adic component of the adeles for some fixed place v. Define X(Fv )n := P rv (X(AF )n ) and X(Fv )n+1 := P rv (X(AF )n+1 ). n n Thus, we get a filtation on v-adic points amenable to computation: X(Fv ) = X(Fv )1 ⊃ X(Fv )21 ⊃ X(Fv )2 ⊃ · · · ⊃ X(Fv )∞ ⊃ X(F ). Conjecture: Let X/Q be a projective smooth curve of genus at least 2. Then for any prime p of good reduction, we have X(Qp )∞ = X(Q). Can consider more generally S-integral points on affine hyperbolic X as well where we get an induced filtration X(ASF ) ⊃ X(ASF )21 ⊃ X(ASF )2 ⊃ X(ASF )32 ⊃ · · · . By projecting to X(OFv ) for v ∈ / S, get a flitration X(OFv ) ⊃ X(OFv )2S,1 ⊃ X(OFv )S,2 ⊃ X(OFv )3S,2 ⊃ · · · . and X(OFv )S,∞ = ∩n X(OFv )S,n . Conjecture: Let X/Q 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 X(Z[1/S]) = X(Zp )S,∞ . These give us conjectural methods to ‘compute’ X(Q) ⊂ X(Qp ) or X(Z[1/S]) ⊂ X(Zp ). In principle, functions annihilating global points should come from cohomology classes kn ∈ H 1 (GSQ , Hom(TnM , Qp (1))).
NON-ABELIAN RECIPROCITY, PERIOD MAPS, AND DIOPHANTINE GEOMETRY
9
Such an element gives a function X(AQ )n
recn
- H 1 (GSF , D(TnM ))∨
kn
- Qp
that kills X(Q) ⊂ X(AQ )n . However, to make this effective, we need an explicit reciprocity law that describes the image X(Qp )n . All computations all rely on the theory [9, 12] of U (X, b), ¯ with Galois action, and the diagram the Qp -pro-unipotent fundamental group of X - X(Qp ) X(Q) j g jn
p jn
? Hf1 (GSQ , Un )
D n R
? D ' - H 1 (Gp , Un ) - UnDR /F 0 f
locp n
That is, U is the Tannakian fundamental group of Qp -unipotent local systems on ¯ and the H 1 are all algebraic schemes. The key point is that the map X X(Qp )
j DR
- U DR /F 0
can be computed explicitly using iterated integrals, and X(Q) ⊂ X(Qp )n ⊂ [jnDR ]−1 [Im(D ◦ locpn )]. 5. Some explicit reciprocity 1. [1, 8] Let X = P1 \ {0, 1, ∞}. Then X(Z[1/2]) = {2, −1, 1/2}. X(Zp ){2},2 ⊂ ∪n,m {z | log(z) = n log(2), log(1 − z) = m log(2)}. X(Zp ){2},3 ⊂ [∪m,n {z | log(z) = n log(2), log(1 − z) = m log(2)}] ∩ {D2 (z) = 0}. Probably, X(Zp ){2},4 = X(Zp ){2},3 . Also, X(Zp ){2},5 ⊂ [∪m,n {z | log(z) = n log(2), log(1 − z) = m log(2)}] ∩{D2 (z) = 0} ∩ {D4 (z) = 0}. Numerically, this appears to be equal to {2, −1, 1/2}. 2. [1] Let X = E \ O where E is a semi-stable elliptic curve of rank 0 and |X(E)(p)| < ∞.
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MINHYONG KIM
Z log(z) =
z
(dx/y). b
(b is a tangential base-point.) Then X(Zp )2 = {z ∈ X(Zp ) | log(z) = 0} = E(Zp )[tor] \ O. Now examine the inclusion X(Z) ⊂ X(Zp )3 . Let Z
z
D2 (z) =
(dx/y)(xdx/y). b
Let T be the set of primes of bad reduction. For each l ∈ T , let Nl = ordl (∆E ), where ∆E is the minimal discriminant. Define a set Wl := {(n(Nl − n)/2Nl ) log l | 0 ≤ n < Nl }, Q and for each w = (wl )l∈S ∈ W := l∈S Wl , define kwk =
X
wl .
l∈S
Theorem 5.1. Suppose E has rank zero and that XE [p∞ ] < ∞. With assumptions as above X(Zp )3 ⊂ ∪w∈W Ψ(w), where Ψ(w) := {z ∈ X(Zp ) | log(z) = 0, D2 (z) = kwk}. Of course, X(Z) ⊂ X(Zp )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. In fact, so far, we have checked X(Z) = X(Zp )3 for the prime p = 5 and 256 semi-stable elliptic curves of rank zero. A small list is in the following table:
NON-ABELIAN RECIPROCITY, PERIOD MAPS, AND DIOPHANTINE GEOMETRY
Cremona label 1122m1 1122m2 1122m4 1254a2 1302d2 1506a2 1806h1 2442h1 2442h2 2706d2 2982j1 2982j2 3054b1
11
number of ||w||-values 128 384 84 140 96 112 120 78 84 120 160 140 108
Hence, for example, for the curve 1122m2, y 2 + xy = x3 − 41608x − 90515392 there are potentially 384 of the Ψ(w)’s that make up X(Zp )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, −154024), (2864, 151160). 6. The cursed curve [Jennifer Balakrishnan, Netan Dogra, Stefan Mueller-Stach, Jan Tuitman, Jan Vonk] Xs+ (N ) = X(N )/Cs+ (N ), where X(N ) is the compactification of the moduli space of pairs (E, φ : E[N ] ' (Z/N )2 ), and Cs+ (N ) ⊂ GL2 (Z/N ) is the normaliser of a split Cartan subgroup. Bilu, Parent, and Rebolledo [5, 6] had shown that Xs+ (p)(Q) consists entirely of cusps and CM points for all primes p > 7, p 6= 13. They called p = 13 the ‘cursed level’. Theorem 6.1. [2] Xs+ (13)(Q) = Xs+ (13)(Q17 )3 . This set consists of 7 rational points, which are the cusp and 6 CM points. This concludes an important chapter of a question of Serre: Find an absolute constant A such that GQ
- Aut(E[p])
is surjective for all non-CM elliptic curves E/Q and primes p > A.
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MINHYONG KIM
The proof relies on a careful computation of the lower horizontal map, which is algebraic: - X(Qp ) X(Q) j g jn
D n R
p jn
? Hf1 (GT , Un )
? D ' - H 1 (Gp , Un ) - UnDR /F 0 f
locp n
As mentioned, the defining equations for Im(locpn ) pull back to analytic defining equation for rational points. In fact, in this case, there is a pushout: - U 2 /U 3 - U2 - U1 - 0 0
0
? - Qp (1)
? - W2
? - U1
- 0
induced by a polarisation U 2 /U 3 ' ∧2 U1 /Qp (1)
- Qp (1)
orthogonal to the Weil pairing. Recall that U1 ' Vp = Tp JX ⊗ Qp . X(Q)
- X(Qp )
? ? Hf1 (GT , W2 ) - Hf1 (Gp , W2 ) - W2DR /F 0 The target can be identified with a space of mixed extensions:
E ⊃ E1 ⊃ E2, such that E2 ' Qp (1), E 1 /E 2 ' Vp , E/E 1 ' Qp . Thus, they are mixtures of extensions 0 - Qp (1) - E 1
- Vp
- 0
and 0 - Vp - E/E 2 - Qp - 0, coming up in Nekovar’s theory of height pairings Hf1 (V ) × Hf1 (V ) - Qp . We give a general idea of how this works with a simpler example: X =E\0 where E/Q is an elliptic curve of rank 1 with square-free minimal discriminant. We have the p-adic quadratic height [14] h : E(Q)
- Qp ,
NON-ABELIAN RECIPROCITY, PERIOD MAPS, AND DIOPHANTINE GEOMETRY
13
Thus, if y ∈ E(Q) is non-torsion, then cE := h(y)/ log2 (y) is independent of y. But log is an analytic function on E(Qp ), while h has a decomposition X h = hp + hv , v6=p
with
Z hp (z) =
z
αβ + CE , b
where CE = (a21 + 4a2 )/12 − Eis2 (E, α)/12, α is an integral invariant differential, and β = xα. But if z is integral, then h(z) = hp (z). Thus, the equation h(z)/ log2 (z) = cE = h(y)/ log2 (y) becomes
Z
z
αβ + (CE − cE ) log2 (z) = 0,
b
a defining equation for integral points. The case of Xs+ (13) is a substantially more complicated version of this argument using a relation between the functions h(z), logi (z) logj (z) for 1 ≤ i ≤ j ≤ 3. 7. Critical points When X is proper, so that X(Q) = X(Z) (for some integral model, which is left implicit), actually interested in Y Y Im(H 1 (GT , U )) ∩ Hf1 (Gv , U ) ⊂ H 1 (Gv , U ), v∈T
v∈T
where Hf1 (Gv , U ) ⊂ H 1 (Gv , U ) is a subvariety defined by some integral or Hodge-theoretic conditions. We should use the structure of this intersection more actively. In order to apply symplectic techniques, replace U by T ∗ (1)U := (LieU )∗ (1) o U. Then arithmetic duality theorems [13] imply that Y H 1 (Gv , T ∗ (1)U ) v∈T
is a symplectic variety and Im(H 1 (GT , T ∗ (1)U )),
Y v∈T
Hf1 (Gv , T ∗ (1)U )
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MINHYONG KIM
are Lagrangian subvarieties. Thus, the derived intersection Y DT (X) := Im(H 1 (GT , T ∗ (1)U )) ∩ Hf1 (Gv , T ∗ (1)U ) v∈T
has a [−1]-shifted symplectic structure. This should be the critical set of a function [3]. We have the diagram, - j −1 (DT (X)) ⊂ - Q X(Qv ) X(Z) T
jg
v∈T
jT
? Hf1 (GT , T ∗ (1)U )
locT
jT
? - DT (X)
⊂
- Q
v∈T
? H 1 (Gv , T ∗ (1)Un )
whereby the critical equation for the intersection pulls back to a defining equation for X(Z). It is also suggestive to take a limit over T : - j −1 (D(X)) ⊂ - X(AQ ) X(Z) jg
? Hf1 (G, T ∗ (1)U )
j loc
? - D(X)
j
⊂
? - Q0 H 1 (Gv , T ∗ (1)Un )
The point is of this diagram is that it reduces the problem of computing solutions to Diophantine equations to that of finding the critical set of a functional. References [1] Balakrishnan, Jennifer, Dan-Cohen, Ishai, Kim, Minhyong, Wewers, Stefan A non-abelian conjecture of Birch and Swinnerton-Dyer type for hyperbolic curves. arXiv:1209.0640 [2] Balakrishnan, Jennifer S., Dogra, Netan, Müller, J. Steffen, Tuitman, Jan, Vonk, Jan, Explicit Chabauty-Kim for the Split Cartan Modular Curve of Level 13. arXiv:1711.05846 [math.NT] [3] Ben-Bassat, Oren, Brav, Christopher, Bussi, Vittoria, Joyce, Dominic, A ’Darboux Theorem’ for shifted symplectic structures on derived Artin stacks, with applications. arXiv:1312.0090 [math.AG] [4] Besser, Amnon Coleman integration using the Tannakian formalism. Math. Ann. 322 (2002), no. 1, 19–48. [5] Bilu, Yuri; Parent, Pierre Serre’s uniformity problem in the split Cartan case. Ann. of Math. (2) 173 (2011), no. 1, 569–584. [6] Bilu, Yuri; Parent, Pierre; Rebolledo, Marusia Rational points on X0+ (pr ). Ann. Inst. Fourier (Grenoble) 63 (2013), no. 3, 957–984. [7] Coleman, Robert F. Dilogarithms, regulators and p-adic L-functions. Invent. Math. 69 (1982), no. 2, 171–208. [8] Dan-Cohen, Ishai, Wewers, Stefan Mixed Tate motives and the unit equation. arXiv:1311.7008 [9] Deligne, Pierre Le groupe fondamental de la droite projective moins trois points. Galois groups over Q (Berkeley, CA, 1987), 79–297, Math. Sci. Res. Inst. Publ., 16, Springer, New York, 1989. [10] Kim, Minhyong The motivic fundamental group of P1 \ {0, 1, ∞} and the theorem of Siegel. Invent. Math. 161 (2005), no. 3, 629–656. [11] Kim, Minhyong Diophantine geometry and non-abelian reciprocity I. Elliptic curves, modular forms and Iwasawa theory, 311-334, Springer Proc. Math. Stat., 188, Springer, Cham., 2016. [12] Kim, Minhyong The unipotent Albanese map and Selmer varieties for curves. Publ. Res. Inst. Math. Sci. 45 (2009), no. 1, 89–133.
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[13] Milne, James Arithmetic Duality Theorems. 2nd ed., available from www.jamesmilne.org (2006). [14] Mazur, Barry; Stein, William; Tate, John Computation of p-adic heights and log convergence. Doc. Math. 2006, Extra Vol., 577–614. M.K: Mathematical Institute, University of Oxford, and the Korea Institute for Advanced Study
1. Some p-adic analytic functions We start with the evaluation in Q2 a convergent power series ∞ X 2n =0 n2 n=1
The value of zero, not at all obvious, was observed by Coleman about 30 years ago[7]. In fact, the value is zero even when the series is not convergent, that is, in Qp for all p. The general statement is that `2 (2) = 0, where `2 is the p-adic dilogarithm defined by iterated Coleman integration: Z z ∞ X zn dt dt `2 (z) := = t 1 − t n=1 n2 b Here, b is the tangential base point at 0 defined by dt(b) = 1 with respect to the coordinate t on P1 \ {0, 1, ∞}. There are many approaches to Coleman integration, but one efficient (albeit abstract) way [4] to define the iterated integral as the upper right hand corner of the matrix Rz Rz 1 b dt/t b (dt/t)(dt/(1 − t)) R z Mbz = 0 1 dt/(1 − t) b 0 0 1 where Mbz is the holonomy of the rank 3 unipotent connection on P1 \ {0, 1, ∞} given by the connection form 0 dt/t 0 0 dt/(1 − t) − 0 0 0 0 The holonomy is computed locally by solving the differential equations d`1 = dt/(1 − t); d`2 = `1 dt/t. with initial condition at a point. But it is a somewhat deep fact that given a unipotent connection (V, ∇) and two points x, y ∈ P1 \ {0, 1, ∞}(Qp ) 1991 Mathematics Subject Classification. 14G10, 11G40, 81T45 . M.K. Supported by grant EP/M024830/1 from the EPSRC. 1
2
MINHYONG KIM
(possibly tangental), there is a canonical isomorphism Mxy (V, ∇) : V (]¯ x[)∇=0
'
- V (]¯ y [)∇=0
determined by the property that it’s compatible with morphisms between connections and Frobenius pull-backs. One gets thereby a holonomy transformation between any two points. More generally, the k-logarithm Z z `k (z) := (dt/t)(dt/t) · · · (dt/t)(dt/(1 − t)) b
is defined as the upper right hand corner (k + 1) × (k + 1) connection form 0 dt/t 0 0 0 0 dt/t 0 0 0 0 dt/t − .. . 0 0 0 0 0 0 0 0
of the holonomy matrix arising from the ... ... ...
0 0 0
. . . dt/t ... 0 ... 0
0 0 0
0 dt/(1 − t) 0
Using the geometry of P1 \ {0, 1, ∞}, Coleman derived the following functional equations: −1 `k (z) + (−1)k `k (z −1 ) = logk (z); k! D2 (z) = −D2 (z −1 ); D2 (z) = −D2 (1 − z); where D2 (z) = `2 (z) + (1/2) log(z) log(1 − z). (The upper right hand corner of the log of the holonomy matrix.) From this, we get D2 (−1) = −D2 (1/(−1)) = 0, and D2 (2) = −D2 (1 − 2) = 0. But D2 (2) = `2 (2) + (1/2) log(2) log(−1) = `2 (2), giving us the desired vanishing. We note also that D2 (1/2) = −D2 (2) = 0. The beginning of our investigation is the observation that {−1, 2, 1/2} are exactly the 2-integral points of P1 \ {0, 1, ∞}. It is possible thereby to give a global proof of the vanishing [10] using, in some sense, the arithmetic geometry of Spec(Z) \ {2, p, ∞}. We will expand on this in detail later, but for now, we add also the following result of Ishai Dan-Cohen and Stefan Wewers [8].
NON-ABELIAN RECIPROCITY, PERIOD MAPS, AND DIOPHANTINE GEOMETRY
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Let D4 (z) = ζ(3)`4 (z) + (8/7)[log3 2/24 + `4 (1/2)/ log 2] log(z)`3 (z) +[(4/21)(log3 2/24 + `4 (1/2)/ log 2) + ζ(3)/24] log3 (z) log(1 − z). Even though the expression is somewhat complicated, note that it is of the form ζ(3)`4 (z) + A log(z)`3 (z) + B log3 (z) log(1 − z), which is homogeneous of degree 4 in a suitable sense. With this definition, it turns out that [P1 \ {0, 1, ∞}](Z[1/2]) ⊂ {D2 (z) = 0, D4 (z) = 0} and numerical computations for p ≤ 29 indicate equality. A key theme is that these equations are examples of non-abelian explicit reciprocity laws. 1 In fact, to describe √ P \ {0, 1, ∞}(Z[1/2]), we definitely need the extra equation since, for example, 5 ∈ Z11 , and √ √ √ 3± 5 1± 5 −1 ± 5 ) = D2 ( ) = D2 ( ) = 0. D2 ( 2 2 2 2. The local-to-global problem in Diophantine geometry We go on to describe the general context for these formulae [11]. Given a number field F and X/F a smooth variety (with an integral model), the key local-to-global problem is to locate 0 Y X(F ) ⊂ X(AF ) = X(Fv ) v
That is, we would like to know How do the global points sit inside the local points? There is a classical answer for X = Gm , in which case X(F ) = F ∗ ,
X(Fv ) = Fv∗ .
Thus, the problem becomes that of locating F ∗ ⊂ A× F. We have the Artin reciprocity map Y rec = recv : A× F
- Gab F ,
v
The Artin-Takagi reciprocity law says that the composed map F∗
⊂
- A× F
rec
- Gab F
is zero. That is, the reciprocity map gives a defining equation for Gm (F ) ⊂ Gm (AF ). We would like to generalize this to other equations by way of a non-abelian reciprocity law. Is it possible to start with a rather general variety X and understand X(F ) via X(F )
⊂
- X(AF )
recN A
-
some target with base-point 0
4
MINHYONG KIM
in such way that recN A = 0. becomes an equation for X(F )? 3. Non-abelian reciprocity Here is a brief list of essentially standard notation to be used for the remainder of this note. F : number field. GF = Gal(F¯ /F ). Gv = Gal(F¯v /Fv ) for a place v of F . S: finite set of places of F . AF : finite Adeles of F ASF : finite S-integral adeles of F . GSF = Gal(F S /F ), where F S is the maximal extension of F unramified outside S. Q
: product over non-Archimedean places in S. H 1 (Gv , A): product over non-Archimedean places in S and ‘unramified cohomology’ outside of S. X: a smooth variety over F . b ∈ X(F ) (sometimes tangential). QSS
Now for some more sophisticated objects: We let ¯ b)(2) , ∆ = π1 (X, the pro-finite prime-to-2, étale fundamental group of ¯ =X× ¯ X Spec(F ) Spec(F ) with base-point b. Denote with superscripts ∆[n] , [1] the lower central series with ∆ = ∆. Subscripts will be used for the corresponding quotients: ∆n = ∆/∆[n+1] . Tn = ∆[n] /∆[n+1] . We also consider ∆M , (∆n )M , TnM , the pro-M quotients for various finite sets of prime M . We will be assuming that the following condition is satisfied (which is true for curves): For each n and M sufficiently large, TnM is torsion-free. This implies H 1 (GSF , TnM )
0 Y loc H 1 (Gv , TnM )
is injective. With this, we get a non-abelian class field theory with coefficients in the nilpotent completion of X. This consists of a filtration X(AF ) = X(AF )1 ⊃ X(AF )21 ⊃ X(AF )2 ⊃ X(AF )32 ⊃ X(AF )3 ⊃ X(AF )43 ⊃ · · ·
NON-ABELIAN RECIPROCITY, PERIOD MAPS, AND DIOPHANTINE GEOMETRY
5
and a sequence of maps recn : X(AF )n
- Gn (X)
- Gn+1 (X) recn+1 : X(AF )n+1 n n n n+1 to a sequence Gn (X), Gn (X) of profinite abelian groups that are organised in such a way that X(AF )n+1 = rec−1 n n (0) and X(AF )n+1 = (recn+1 )−1 (0). n It is also useful to display the maps as follows: 2 −1 · · · rec−1 (0) ⊂ rec−1 2 (0) ⊂ (rec1 ) 1 (0) ⊂
||
||
X(AF )
||
||
· · · X(AF )32 ⊂ X(AF )2 ⊂ X(AF )21 ⊂ X(AF )1 rec32
rec21
rec2
? · · · G32 (X)
? G2 (X)
rec1
? G21 (X)
? G1 (X)
The targets Gn (X) are defined as Gn (X) := ∨
1
H (GF , D(Tn )) := lim Hom[H 1 (GF , D(TnM )), Q/Z] ←− M
where D(TnM ) = lim Hom(Tn , µm ). −→ m
For the intermediate targets, M Gn+1 (X) := lim X1 (D(Tn+1 ))∨ n ←−
H 1 (GF , D(Tn+1 ))∨ .
M
When X = Gm , then Gn (X) = 0 for n ≥ 2, Gn+1 (X) = 0 n for all n, and ˆ (2) )), Q/Z] G1 = Hom[H 1 (GF , D(Z(1) = Hom[H 1 (GF , [Q/Z](2) ), Q/Z] = [G(2) ]ab F . In this case, rec1 reduces to the prime-to-2 part of the usual reciprocity map. The reciprocity maps are defined using the local non-abelian étale period maps j v : X(Fv ) - H 1 (Gv , ∆); (2)
¯ b, x)]. x 7→ [π1 (X;
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MINHYONG KIM
Because the homotopy classes of étale paths (2)
¯ b, x) π1 (X; form a torsor for ∆ with compatible action of Gv , we get a corresponding class in non-abelian cohomology of Gv with coefficients in ∆. These assemble to a map j loc : X(AF )
-
jnloc : X(AF )
-
0 Y
H 1 (Gv , ∆),
which comes in levels 0 Y
H 1 (Gv , ∆n ).
We also have pro-M versions jnloc : X(AF )
-
jnloc : X(ASF )
-
0 Y
H 1 (Gv , ∆M n )
and integral versions S Y
H 1 (Gv , ∆M n ).
To give a flavour of the definition of the reciprocity maps, we will just define pro-M versions on X(ASF ) and assume that locS Y 1 H 1 (GSF , TnM ) H (Gv , TnM ) S
are injective. In general, one needs first to work with a pro-M quotient for a finite set of primes M and S ⊃ M , then take a limit over S and M . The first reciprocity map is just defined using x ∈ X(AF ) 7→ d1 (j1loc (x)), where D1 :
Y
H 1 (Gv , ∆M 1 )
-
Y
S
∨ H 1 (Gv , D(∆M 1 ))
∗ loc∨ H 1 (GSF , D(∆M 1 )) ,
S
is obtained from Tate duality and the dual of localisation. To define the higher reciprocity maps, we use the exact sequences 0
M - H 1 (GSF , Tn+1 )
- H 1 (GSF , ∆M n+1 )
pn+1 n
- H 1 (GSF , ∆n )
δn+1
M - H 2 (GSF , Tn+1 ) for non-abelian cohomology and Poitou-Tate duality stating that Y Dn+1 M M - H 1 (GS , D(Tn+1 H 1 (GSF , Tn+1 ) H 1 (Gv , TnM ) ))∨ S
is exact. We proceed as follows: rec21 (x) = P T ◦ δ2 ◦ loc−1 (j1 (x)) ∈ X1S (D(T2M ))∨ and rec2 (x) = D2 (loc((p21 )−1 (loc−1 (j1 (x)))) − j2 (x)) ∈ H 1 (GSF , D(T2M ))∨ .
NON-ABELIAN RECIPROCITY, PERIOD MAPS, AND DIOPHANTINE GEOMETRY
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To understand these formulae, we use the diagram Q H 1 (GSF , T2M ) ⊂- S H 1 (Gv , T2M )3 k2
? ? Q 1 M ⊂[p21 ]−1 (loc−1 (j1 )) ∈H 1 (GSF , ∆M ) H (G v , ∆2 ) 3 j2 2 S p21
loc−1 (j1 ) ∈
? ? Q H 1 (GSF , T1M ) ⊂- S H 1 (Gv , T1M ) 3 j1
That is, we have δ2 ◦ loc−1 (j1 (x)) ∈ X2S (T2M ) which maps to rec21 (x) via P T : X2S (T2M ) ' X1S (D(T2M ))∨
H 1 (GS , D(T2M ))∨ .
When this vanishes, we can lift loc−1 (j1 (x)) to (p21 )−1 (loc−1 (j1 (x)) and localise. The difference between this global lift and the second level of the local period map gives Y k2 (x) := loc[[p21 ]−1 (loc−1 (j1 (x)))] − j2 (x) ∈ H 1 (Gv , T2M ) 1
S S (GF , D(T2M ))∨
7→ rec2 (x) := D2 (k2 (x)) ∈ H Poitou-Tate duality implies that this element is independent of the lift. It is clear how to continue to higher levels. Now put X(AF )∞ = ∩∞ n=1 X(AF )n . Theorem 3.1 (Non-abelian reciprocity). X(F ) ⊂ X(AF )∞ . Let F = Q, X ia compact curve of genus at least two, p is a prime of good reduction, and S a finite set of places containing p and all primes of bad reduction. Suppose there is a finite set T ⊃ S of places such that Y H 1 (Gv , ∆pn ) H 1 (GSF , ∆pn ) v∈T
is injective for all sufficiently large n. Then the reciprocity law implies finiteness of X(F ). The proof of the reciprocity law is easy, being implied by the existence of a global period map on global points, which causes all obstruction to vanish: - X(AF ) X(F ) g jn
loc jn
? ? Q loc 1 M H 1 (GSF , ∆M ) H (G v , ∆n ) n S
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MINHYONG KIM
H 1 (GSF , ∆M n+1 ) 1
g
j n+
X(F )
? - H 1 (GS , ∆M n ) F
g jn
4. A non-abelian conjecture of Shafarevich-Tate type This is an elaboration of the discussion in [11] and [1]. The relation to computation of points arises via the projection P rv : X(AF ) - X(Fv ) to the v-adic component of the adeles for some fixed place v. Define X(Fv )n := P rv (X(AF )n ) and X(Fv )n+1 := P rv (X(AF )n+1 ). n n Thus, we get a filtation on v-adic points amenable to computation: X(Fv ) = X(Fv )1 ⊃ X(Fv )21 ⊃ X(Fv )2 ⊃ · · · ⊃ X(Fv )∞ ⊃ X(F ). Conjecture: Let X/Q be a projective smooth curve of genus at least 2. Then for any prime p of good reduction, we have X(Qp )∞ = X(Q). Can consider more generally S-integral points on affine hyperbolic X as well where we get an induced filtration X(ASF ) ⊃ X(ASF )21 ⊃ X(ASF )2 ⊃ X(ASF )32 ⊃ · · · . By projecting to X(OFv ) for v ∈ / S, get a flitration X(OFv ) ⊃ X(OFv )2S,1 ⊃ X(OFv )S,2 ⊃ X(OFv )3S,2 ⊃ · · · . and X(OFv )S,∞ = ∩n X(OFv )S,n . Conjecture: Let X/Q 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 X(Z[1/S]) = X(Zp )S,∞ . These give us conjectural methods to ‘compute’ X(Q) ⊂ X(Qp ) or X(Z[1/S]) ⊂ X(Zp ). In principle, functions annihilating global points should come from cohomology classes kn ∈ H 1 (GSQ , Hom(TnM , Qp (1))).
NON-ABELIAN RECIPROCITY, PERIOD MAPS, AND DIOPHANTINE GEOMETRY
9
Such an element gives a function X(AQ )n
recn
- H 1 (GSF , D(TnM ))∨
kn
- Qp
that kills X(Q) ⊂ X(AQ )n . However, to make this effective, we need an explicit reciprocity law that describes the image X(Qp )n . All computations all rely on the theory [9, 12] of U (X, b), ¯ with Galois action, and the diagram the Qp -pro-unipotent fundamental group of X - X(Qp ) X(Q) j g jn
p jn
? Hf1 (GSQ , Un )
D n R
? D ' - H 1 (Gp , Un ) - UnDR /F 0 f
locp n
That is, U is the Tannakian fundamental group of Qp -unipotent local systems on ¯ and the H 1 are all algebraic schemes. The key point is that the map X X(Qp )
j DR
- U DR /F 0
can be computed explicitly using iterated integrals, and X(Q) ⊂ X(Qp )n ⊂ [jnDR ]−1 [Im(D ◦ locpn )]. 5. Some explicit reciprocity 1. [1, 8] Let X = P1 \ {0, 1, ∞}. Then X(Z[1/2]) = {2, −1, 1/2}. X(Zp ){2},2 ⊂ ∪n,m {z | log(z) = n log(2), log(1 − z) = m log(2)}. X(Zp ){2},3 ⊂ [∪m,n {z | log(z) = n log(2), log(1 − z) = m log(2)}] ∩ {D2 (z) = 0}. Probably, X(Zp ){2},4 = X(Zp ){2},3 . Also, X(Zp ){2},5 ⊂ [∪m,n {z | log(z) = n log(2), log(1 − z) = m log(2)}] ∩{D2 (z) = 0} ∩ {D4 (z) = 0}. Numerically, this appears to be equal to {2, −1, 1/2}. 2. [1] Let X = E \ O where E is a semi-stable elliptic curve of rank 0 and |X(E)(p)| < ∞.
10
MINHYONG KIM
Z log(z) =
z
(dx/y). b
(b is a tangential base-point.) Then X(Zp )2 = {z ∈ X(Zp ) | log(z) = 0} = E(Zp )[tor] \ O. Now examine the inclusion X(Z) ⊂ X(Zp )3 . Let Z
z
D2 (z) =
(dx/y)(xdx/y). b
Let T be the set of primes of bad reduction. For each l ∈ T , let Nl = ordl (∆E ), where ∆E is the minimal discriminant. Define a set Wl := {(n(Nl − n)/2Nl ) log l | 0 ≤ n < Nl }, Q and for each w = (wl )l∈S ∈ W := l∈S Wl , define kwk =
X
wl .
l∈S
Theorem 5.1. Suppose E has rank zero and that XE [p∞ ] < ∞. With assumptions as above X(Zp )3 ⊂ ∪w∈W Ψ(w), where Ψ(w) := {z ∈ X(Zp ) | log(z) = 0, D2 (z) = kwk}. Of course, X(Z) ⊂ X(Zp )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. In fact, so far, we have checked X(Z) = X(Zp )3 for the prime p = 5 and 256 semi-stable elliptic curves of rank zero. A small list is in the following table:
NON-ABELIAN RECIPROCITY, PERIOD MAPS, AND DIOPHANTINE GEOMETRY
Cremona label 1122m1 1122m2 1122m4 1254a2 1302d2 1506a2 1806h1 2442h1 2442h2 2706d2 2982j1 2982j2 3054b1
11
number of ||w||-values 128 384 84 140 96 112 120 78 84 120 160 140 108
Hence, for example, for the curve 1122m2, y 2 + xy = x3 − 41608x − 90515392 there are potentially 384 of the Ψ(w)’s that make up X(Zp )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, −154024), (2864, 151160). 6. The cursed curve [Jennifer Balakrishnan, Netan Dogra, Stefan Mueller-Stach, Jan Tuitman, Jan Vonk] Xs+ (N ) = X(N )/Cs+ (N ), where X(N ) is the compactification of the moduli space of pairs (E, φ : E[N ] ' (Z/N )2 ), and Cs+ (N ) ⊂ GL2 (Z/N ) is the normaliser of a split Cartan subgroup. Bilu, Parent, and Rebolledo [5, 6] had shown that Xs+ (p)(Q) consists entirely of cusps and CM points for all primes p > 7, p 6= 13. They called p = 13 the ‘cursed level’. Theorem 6.1. [2] Xs+ (13)(Q) = Xs+ (13)(Q17 )3 . This set consists of 7 rational points, which are the cusp and 6 CM points. This concludes an important chapter of a question of Serre: Find an absolute constant A such that GQ
- Aut(E[p])
is surjective for all non-CM elliptic curves E/Q and primes p > A.
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MINHYONG KIM
The proof relies on a careful computation of the lower horizontal map, which is algebraic: - X(Qp ) X(Q) j g jn
D n R
p jn
? Hf1 (GT , Un )
? D ' - H 1 (Gp , Un ) - UnDR /F 0 f
locp n
As mentioned, the defining equations for Im(locpn ) pull back to analytic defining equation for rational points. In fact, in this case, there is a pushout: - U 2 /U 3 - U2 - U1 - 0 0
0
? - Qp (1)
? - W2
? - U1
- 0
induced by a polarisation U 2 /U 3 ' ∧2 U1 /Qp (1)
- Qp (1)
orthogonal to the Weil pairing. Recall that U1 ' Vp = Tp JX ⊗ Qp . X(Q)
- X(Qp )
? ? Hf1 (GT , W2 ) - Hf1 (Gp , W2 ) - W2DR /F 0 The target can be identified with a space of mixed extensions:
E ⊃ E1 ⊃ E2, such that E2 ' Qp (1), E 1 /E 2 ' Vp , E/E 1 ' Qp . Thus, they are mixtures of extensions 0 - Qp (1) - E 1
- Vp
- 0
and 0 - Vp - E/E 2 - Qp - 0, coming up in Nekovar’s theory of height pairings Hf1 (V ) × Hf1 (V ) - Qp . We give a general idea of how this works with a simpler example: X =E\0 where E/Q is an elliptic curve of rank 1 with square-free minimal discriminant. We have the p-adic quadratic height [14] h : E(Q)
- Qp ,
NON-ABELIAN RECIPROCITY, PERIOD MAPS, AND DIOPHANTINE GEOMETRY
13
Thus, if y ∈ E(Q) is non-torsion, then cE := h(y)/ log2 (y) is independent of y. But log is an analytic function on E(Qp ), while h has a decomposition X h = hp + hv , v6=p
with
Z hp (z) =
z
αβ + CE , b
where CE = (a21 + 4a2 )/12 − Eis2 (E, α)/12, α is an integral invariant differential, and β = xα. But if z is integral, then h(z) = hp (z). Thus, the equation h(z)/ log2 (z) = cE = h(y)/ log2 (y) becomes
Z
z
αβ + (CE − cE ) log2 (z) = 0,
b
a defining equation for integral points. The case of Xs+ (13) is a substantially more complicated version of this argument using a relation between the functions h(z), logi (z) logj (z) for 1 ≤ i ≤ j ≤ 3. 7. Critical points When X is proper, so that X(Q) = X(Z) (for some integral model, which is left implicit), actually interested in Y Y Im(H 1 (GT , U )) ∩ Hf1 (Gv , U ) ⊂ H 1 (Gv , U ), v∈T
v∈T
where Hf1 (Gv , U ) ⊂ H 1 (Gv , U ) is a subvariety defined by some integral or Hodge-theoretic conditions. We should use the structure of this intersection more actively. In order to apply symplectic techniques, replace U by T ∗ (1)U := (LieU )∗ (1) o U. Then arithmetic duality theorems [13] imply that Y H 1 (Gv , T ∗ (1)U ) v∈T
is a symplectic variety and Im(H 1 (GT , T ∗ (1)U )),
Y v∈T
Hf1 (Gv , T ∗ (1)U )
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MINHYONG KIM
are Lagrangian subvarieties. Thus, the derived intersection Y DT (X) := Im(H 1 (GT , T ∗ (1)U )) ∩ Hf1 (Gv , T ∗ (1)U ) v∈T
has a [−1]-shifted symplectic structure. This should be the critical set of a function [3]. We have the diagram, - j −1 (DT (X)) ⊂ - Q X(Qv ) X(Z) T
jg
v∈T
jT
? Hf1 (GT , T ∗ (1)U )
locT
jT
? - DT (X)
⊂
- Q
v∈T
? H 1 (Gv , T ∗ (1)Un )
whereby the critical equation for the intersection pulls back to a defining equation for X(Z). It is also suggestive to take a limit over T : - j −1 (D(X)) ⊂ - X(AQ ) X(Z) jg
? Hf1 (G, T ∗ (1)U )
j loc
? - D(X)
j
⊂
? - Q0 H 1 (Gv , T ∗ (1)Un )
The point is of this diagram is that it reduces the problem of computing solutions to Diophantine equations to that of finding the critical set of a functional. References [1] Balakrishnan, Jennifer, Dan-Cohen, Ishai, Kim, Minhyong, Wewers, Stefan A non-abelian conjecture of Birch and Swinnerton-Dyer type for hyperbolic curves. arXiv:1209.0640 [2] Balakrishnan, Jennifer S., Dogra, Netan, Müller, J. Steffen, Tuitman, Jan, Vonk, Jan, Explicit Chabauty-Kim for the Split Cartan Modular Curve of Level 13. arXiv:1711.05846 [math.NT] [3] Ben-Bassat, Oren, Brav, Christopher, Bussi, Vittoria, Joyce, Dominic, A ’Darboux Theorem’ for shifted symplectic structures on derived Artin stacks, with applications. arXiv:1312.0090 [math.AG] [4] Besser, Amnon Coleman integration using the Tannakian formalism. Math. Ann. 322 (2002), no. 1, 19–48. [5] Bilu, Yuri; Parent, Pierre Serre’s uniformity problem in the split Cartan case. Ann. of Math. (2) 173 (2011), no. 1, 569–584. [6] Bilu, Yuri; Parent, Pierre; Rebolledo, Marusia Rational points on X0+ (pr ). Ann. Inst. Fourier (Grenoble) 63 (2013), no. 3, 957–984. [7] Coleman, Robert F. Dilogarithms, regulators and p-adic L-functions. Invent. Math. 69 (1982), no. 2, 171–208. [8] Dan-Cohen, Ishai, Wewers, Stefan Mixed Tate motives and the unit equation. arXiv:1311.7008 [9] Deligne, Pierre Le groupe fondamental de la droite projective moins trois points. Galois groups over Q (Berkeley, CA, 1987), 79–297, Math. Sci. Res. Inst. Publ., 16, Springer, New York, 1989. [10] Kim, Minhyong The motivic fundamental group of P1 \ {0, 1, ∞} and the theorem of Siegel. Invent. Math. 161 (2005), no. 3, 629–656. [11] Kim, Minhyong Diophantine geometry and non-abelian reciprocity I. Elliptic curves, modular forms and Iwasawa theory, 311-334, Springer Proc. Math. Stat., 188, Springer, Cham., 2016. [12] Kim, Minhyong The unipotent Albanese map and Selmer varieties for curves. Publ. Res. Inst. Math. Sci. 45 (2009), no. 1, 89–133.
NON-ABELIAN RECIPROCITY, PERIOD MAPS, AND DIOPHANTINE GEOMETRY
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[13] Milne, James Arithmetic Duality Theorems. 2nd ed., available from www.jamesmilne.org (2006). [14] Mazur, Barry; Stein, William; Tate, John Computation of p-adic heights and log convergence. Doc. Math. 2006, Extra Vol., 577–614. M.K: Mathematical Institute, University of Oxford, and the Korea Institute for Advanced Study
