Motives and Diophantine geometry V July 7, 2006

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Several conjectures: (BSD) E/Q elliptic curve. Then

\ δ : E( Q)→Hf1(Γ, TˆE) is surjective. (Bloch-Kato) X/Q smooth projective variety. Then (r)

¯ Qp(r))) chn,r : K2r−n−1(X)⊗Qp→Hg1(Q, H n(X, is surjective. (Fontaine-Mazur) X/Q smooth projective variety. Then ¯ Qp(r)) Mixed Motives→Hg1(Γ, H n(X, is surjective. (Grothendieck) X/Q compact hyperbolic curve, b ∈ X(Q). Then ¯ b)) X(Q)→H 1(Γ, π ˆ1(X, 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.

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X/Q: compact hyperbolic curve. UnB (X(C)): category of unipotent Q-local systems on X(C). ¯ Unet,p(X): category of unipotent Qp-local ¯ et. systems on X 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 .

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So far defined U B = UQ(π1(X(C), b)) and P B (x) = Isom⊗(fbB , fxB ) for x ∈ X(C). ¯ b)) and π1(X, U et = UQp (ˆ 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).

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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.

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Remark on ´ etale-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 Dieudonn´ e functor that sends U et torsors with Γp-action to U DR -torsors with Hodge filtration and Frobenius.

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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 ´ etale 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.

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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.

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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

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But can also analyze Galois cohomology in the ´ etale 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).

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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.

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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

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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 ¯ Qp)⊗n)→H 2(ΓT , Z n+1\Z n)→0 H 2(ΓT , H1(X, and an exact sequence ¯ Qp)⊗n)→H 2(ΓT , H1(X, ¯ Qp)⊗n)→ 0→Sh2(H1(X, ¯ Qp)⊗n) ⊕w∈T H 2(Gw , H1(X, The local groups are bounded by a quantity of the form P (n)g n using Hodge-Tate decomposition and the monodromy-weight filtration.

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Furthermore, ¯ Qp)⊗n) ' (Sh1(H 1(X, ¯ Qp)⊗n(1)))∗ Sh2(H1(X, with the latter group defined by

¯ Qp)⊗n(1))→H 1(ΓT , H 1(X, ¯ Qp)⊗n(1)) 0→Sh1(H 1(X, ¯ Qp)⊗n(1))) → ⊕w∈T H 1(Gw , H 1(X, But either Bloch-Kato or Fontaine-Mazur implies that Sh1 n = 0 for n ≥ 2.

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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.

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Remarks on future direction. Need a non-abelian method of KolyvaginKato. X/Q genus 1. ¯ Qp)) such Kato produces c ∈ H 1(Γ, H1(X, that the map ¯ Qp)(1))→ H 1(Γ, H 1(X, exp∗ 1 1 ¯ →H (Γp, H (X, 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.

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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?

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