Motives and Diophantine
geometry V
July 7, 2006
1
Several conjectures:
(BSD) E/Q elliptic curve. Then
\
δ : E(
Q)→Hf1(Γ, T̂ E)
is surjective.
(Bloch-Kato) X/Q smooth projective variety. Then
(r)
chn,r : K2r−n−1(X)⊗Qp→Hg1(Q, H n(X̄, Qp(r)))
is surjective.
(Fontaine-Mazur) X/Q smooth projective variety. Then
Mixed Motives→Hg1(Γ, H n(X̄, Qp(r))
is surjective.
(Grothendieck) X/Q compact hyperbolic curve,
b ∈ X(Q). Then
X(Q)→H 1(Γ, π̂1(X̄, b))
is surjective.
2
Grothendieck and Deligne expected
Grothendieck’s conjecture ⇒ Mordell conjecture.
In fact,
B-K or F-M ⇒ Mordell conjecture over Q.
Lesson: Importance of abelian techniques
even in the study of non-abelian objects.
3
X/Q: compact hyperbolic curve.
UnB (X(C)): category of unipotent Q-local
systems on X(C).
Unet,p(X̄): category of unipotent Qp-local
systems on X̄ et.
UnDR (XQp ): category of unipotent vector
bundles with flat connections on XQp .
b: a point of X(Q)
x: point in either X(C), X(Q), or X(Qp)
depending on context.
Fiber functors fxB , fxet, fxDR taking values in
VectQ, VectQp .
4
So far defined
U B = UQ(π1(X(C), b)) and
P B (x) = Isom⊗(fbB , fxB ) for x ∈ X(C).
U et = UQp (π̂1(X̄, b)) and
P et(x) = Isom⊗(fbet, fxet) for x ∈ X(Q)
U DR = Aut⊗(fb) and
P DR (x) = Isom⊗(fb, fx) for x ∈ X(Qp).
5
Related by comparison isomorphisms:
P et(x) ' P B (x) ⊗Q Qp
For any embedding Qp,→C,
P DR (x) ⊗ C ' P B (x) ⊗ C
For p a prime of good reduction for X,
P et(x) ⊗ Bcr ' P DR (x) ⊗ Bcr
Also implicitly a De Rham-to-crystalline comparison isomorphism:
P DR (x) ' P cr (x)
endowing P DR with a Frobenius endomorphism.
6
Remark on étale-to De Rham comparison:
If P DR (x) and P et(x) are the coordinate rings
of P DR (x) and P et(x), then
(P et ⊗ Bcr )Γp ' P DR
Left hand side is the value at P et(x) of a
non-abelian Dieudonné functor that sends
U et torsors with Γp-action to U DR -torsors
with Hodge filtration and Frobenius.
7
We will refer to the collection of these unipotent fundamental groups and path torsors
as the motivic fundamental group and the
motivic torsor of paths.
Important for us are the étale torsors with
local and global Galois actions classified by
Hf1(Γ, U et)
and
Hf1(Γp, U et)
as well as the De Rham torsors classified by
U DR /F 0
When we pass to quotients modulo the descending central series, we have corresponding objects
Unet, Pnet(x), UnDR , PnDR (x)
and classifying spaces
Hf1(Γ, Unet), Hf1(Γp, Unet), UnDR /F 0
8
We also described maps
X(Q)→Hf1(Γ, Unet)
x 7→ P et(x)
(with Γ−action)
X(Qp)→Hf1(Γp, Unet)
x 7→ P et(x)
(with Γp-action)
and
X(Qp) 7→ UnDR /F 0
x 7→ P DR (x)
Can be thought of as
x 7→ P M (x),
the motivic torsor of paths.
9
The corresponding map over C takes values
in a complex manifold of the form
DR /F 0
Ln\Un,
C
called the higher Albanese manifolds.
Usual Albanese map corresponds to n = 2.
Note that the map over Qp has no L (periods). Same phenomenon as the global definability of log map.
x 7→ P M (x)
might be called the motivic unipotent Albanese map.
10
Inductive structure:
M →U M →0
0→Z n+1\Z n→Un+1
n
Can use this to compute various dimensions.
For example, if dn := dimZ n+1\Z n, then
have recursive formula
q
Σk|nkdk = (g +
g 2 − 1)n + (g −
q
g 2 − 1)n
and hence,
q
dn ≈ (g +
g 2 − 1)n/n
11
But can also analyze Galois cohomology in
the étale realization.
Hf1(Γ, Unet) ⊂ H 1(ΓT , Unet)
where T is the set of primes of bad reduction
and p, and GT is the Galois group of the
maximal extension of Q with ramification
restricted to T .
We have the sequence:
et )→
0→H 1(ΓT , Z n+1\Z n)→H 1(ΓT , Un+1
δ
→H 1(ΓT , Unet) → H 2(ΓT , Z n+1\Z n)
which is exact in the sense that
et )
H 1(ΓT , Un+1
is a torsor for the vector group
H 1(ΓT , Z n+1\Z n)
over the kernel of δ.
12
Note: All cohomology sets naturally have
the structure of algebraic varieties.
Can use this sequence to estimate the growth
in dimension of H 1(ΓT , Unet) and hence, of
Hf1(ΓT , Unet).
13
Recall diagram:
X(Q)
↓
,→
X(Qp)
↓
&
D
Hf1(Γ, Unet) → Hf1(Γp, Unet) → UnDR /F 0
↓α
Qp
If we show
dimHf1(Γ, Unet) < UnDR /F 0
(∗)n
for some n, then get finiteness of X(Q).
Depends also on local description of algebraic functions on UnDR /F 0 pulled back to
X(Qp) as p-adic iterated integrals.
Reduces to Chabauty’s method when n = 2.
Can prove this kind of statement for hyperbolic curves of genus 0 and genus 1 CM of
rank 1.
Other curves depending on conditions on
GT .
14
Controlling the dimension uses the Euler characteristic formula
dimH 1(ΓT , Z n+1\Z n)−dimH 2(ΓT , Z n+1\Z n)
= dim(Z n+1\Z n)−
where the negative superscript refers to the
(-1) eigenspace of complex conjugation. By
comparison with complex Hodge theory, we
see that the right hand side is
dn/2
for n odd.
15
When the genus is zero,
Z n+1\Z n ' Qp(n)dn
and
H 2(ΓT , Z n+1\Z n) = 0
for n ≥ 2 and Therefore, eventually,
dimH 1(ΓT , Unet) < dimUnDR /F 0
16
The general implications over Q rely on bounds
of the form
dimH 2(ΓT , Z n+1\Z n) ≤ P (n)g n
implied by Bloch-Kato or Fontaine-Mazur.
We have a surjection
H 2(ΓT , H1(X̄, Qp)⊗n)→H 2(ΓT , Z n+1\Z n)→0
and an exact sequence
0→Sh2(H1(X̄, Qp)⊗n)→H 2(ΓT , H1(X̄, Qp)⊗n)→
⊕w∈T H 2(Gw , H1(X̄, Qp)⊗n)
The local groups are bounded by a quantity of the form P (n)g n using Hodge-Tate
decomposition and the monodromy-weight
filtration.
17
Furthermore,
Sh2(H1(X̄, Qp)⊗n) ' (Sh1(H 1(X̄, Qp)⊗n(1)))∗
with the latter group defined by
0→Sh1(H 1(X̄, Qp)⊗n(1))→H 1(ΓT , H 1(X̄, Qp)⊗n(1))
→ ⊕w∈T H 1(Gw , H 1(X̄, Qp)⊗n(1)))
But either Bloch-Kato or Fontaine-Mazur
implies that Sh1
n = 0 for n ≥ 2.
18
Note that we have in place all the ingredients predicted by Weil’s fantasy:
-Vector bundles;
-π1;
-application to arithmetic.
Not yet a ‘π1-proof’ of finiteness. But at
least marginal progress.
19
Remarks on future direction.
Need a non-abelian method of KolyvaginKato.
X/Q genus 1.
Kato produces c ∈ H 1(Γ, H1(X̄, Qp)) such
that the map
H 1(Γ, H 1(X̄, Qp)(1))→
exp∗
1
1
→H (Γp, H (X̄, Qp)(1)) →
F 0H1DR (Xp)
takes
c 7→ LX (1)α
α a global 1-form. Using it to annihilate
points (local-global duality)
x ∈ X(Q) ⊂ X(Qp) ⊂ TeX = H1DR /F 0
gives finiteness of X(Q) if L(1) 6= 0.
20
Should be promoted to a precise study of
map
D
Hf1(Γ, Unet)→Hf1(Gp, Unet) → UnDR /F 0
enabling the construction of a function vanishing on the image.
Foundational work:
Non-abelian dualities in Galois cohomology.
Difficult part:
Production of a non-abelian ‘dual’ global
cohomology class.
21
For example, start with pairing
ExtΓp (U, Qp(1))×H 1(Γp, U )→H 2(Γp, Qp(1)) ' Qp
and try to produce good global elements in
ExtΓp (U, Qp(1)).
Fantasy:
Non-abelian (p-adic) L-functions?
Should take values in a homogenous space
for the fundamental group?
22