Mixed Tate motives and the unit equation Stefan Wewers, Ishai - - PowerPoint PPT Presentation

Mixed Tate motives and the unit equation

Stefan Wewers, Ishai Dan-Cohen

Ulm/Essen

December 4th, 2014

The unit equation

Notation:

◮ X := P1 − {0, 1, ∞} ◮ S finite set of primes ◮ R := Z[S−1]

Then X(R) = { (u, v) ∈ (R×)2 | u + v = 1 }.

The unit equation

Notation:

◮ X := P1 − {0, 1, ∞} ◮ S finite set of primes ◮ R := Z[S−1]

Then X(R) = { (u, v) ∈ (R×)2 | u + v = 1 }.

Theorem (Mahler,Siegel)

X(R) is finite.

The unit equation

Notation:

◮ X := P1 − {0, 1, ∞} ◮ S finite set of primes ◮ R := Z[S−1]

Then X(R) = { (u, v) ∈ (R×)2 | u + v = 1 }.

Theorem (Mahler,Siegel)

X(R) is finite.

Remark

X(R) is computable, using e.g. Baker’s method and LLL.

The Chabauty-Kim method

For p ∈ S define pro-unipotent algebraic groups over Qp: Πet

X = lim

← −

n

Πet,[n]

X

, ΠdR

X

= lim ← −

n

ΠdR,[n]

X

. E.g. Πet

X is the Malcev completion of πet 1 (X¯ Q,

01) over Qp (with GQ-action).

The Chabauty-Kim method

For p ∈ S define pro-unipotent algebraic groups over Qp: Πet

X = lim

← −

n

Πet,[n]

X

, ΠdR

X

= lim ← −

n

ΠdR,[n]

X

. E.g. Πet

X is the Malcev completion of πet 1 (X¯ Q,

01) over Qp (with GQ-action). For each z ∈ X(R) we get Πet,[n]

X

• torsor

Πet,[n]

X

( 01, z), classified by a family of elements κ(p)

n (z) ∈ Sel(p) S,n := H1 f (GS, Πet,[n] X

) in a Selmer variety.

The main diagram

Using p-adic Hodge theory and Coleman integration we obtain a commutative diagram X(R)

• κ(p)

n

• X(Zp)

αn

• Sel(p)

S,n h(p)

n

ΠdR,[n]

X

. Here h(p)

n

is an algebraic map, while αn is locally analytic.

The main diagram

Using p-adic Hodge theory and Coleman integration we obtain a commutative diagram X(R)

• κ(p)

n

• X(Zp)

αn

• Sel(p)

S,n h(p)

n

ΠdR,[n]

X

. Here h(p)

n

is an algebraic map, while αn is locally analytic. We set X(Zp)n := α−1

n

• h(p)

n (Sel(p) S,n)

• ⊂ X(Zp).

Then X(Zp)1 ⊃ X(Zp)2 ⊃ . . . ⊃ X(R).

Kim’s conjecture

X(R)

• κ(p)

n

• X(Zp)

αn

• Sel(p)

S,n h(p)

n

ΠdR,[n]

X

. .

Theorem (M. Kim, 2005)

For n ≫ 0 we have dim Sel(p)

S,n < dim ΠdR,[n] X

and hence |X(Zp)n| < ∞.

Corollary

X(R) is finite.

Kim’s conjecture

X(R)

• κ(p)

n

• X(Zp)

αn

• Sel(p)

S,n h(p)

n

ΠdR,[n]

X

. .

Theorem (M. Kim, 2005)

For n ≫ 0 we have dim Sel(p)

S,n < dim ΠdR,[n] X

and hence |X(Zp)n| < ∞.

Conjecture (M. Kim, 2012)

For n ≫ 0 we have X(R) = X(Zp)n.

Goals

Our goals:

◮ algorithm to compute the map

h(p)

n

: Sel(p)

S,n → ΠdR,[n] X ◮ test Kim’s conjecture numerically

Goals

Our goals:

◮ algorithm to compute the map

h(p)

n

: Sel(p)

S,n → ΠdR,[n] X ◮ test Kim’s conjecture numerically

Problems:

◮ Galois cohomology difficult to compute ◮ h(p) n

may involve transcendental numbers

Goals

Our goals:

◮ algorithm to compute the map

h(p)

n

: Sel(p)

S,n → ΠdR,[n] X ◮ test Kim’s conjecture numerically

Problems:

◮ Galois cohomology difficult to compute ◮ h(p) n

may involve transcendental numbers Solution: use mixed Tate motives

Mixed Tate motives

Let MTMS denote the category of mixed Tate motives over Q, unramified outside of S. This is a Tannakian category with a canonical fiber functor ω = gr∗

w : MTMS → Vec(Q).

Mixed Tate motives

Let MTMS denote the category of mixed Tate motives over Q, unramified outside of S. This is a Tannakian category with a canonical fiber functor ω = gr∗

w : MTMS → Vec(Q).

Set G mot

S

:= π1(MTMS, ω) = Umot

S

⋊ Gm. We will identify MTMS with the category of finite dimensional Z-graded representations of Umot

S

.

The ring of unipotent periods

We work with the graded Hopf algebra AS := O(Umot

S

) = ⊕n≥0AS,n = lim − →

n

A[n]

S .

The ring of unipotent periods

We work with the graded Hopf algebra AS := O(Umot

S

) = ⊕n≥0AS,n = lim − →

n

A[n]

S .

The known structure of Exti

MTMS(Q(n), Q(m))

shows that there is a noncanonical isomorphism AS ∼ = Qgℓ, ℓ ∈ S; f3, f5, . . ., with deg(gℓ) = 1, deg(fn) = n.

The motivic fundamental group

Deligne and Goncharov have constructed Πmot

X

:= πmot

1

(X, 01) = lim ← −

n

Πmot,[n]

X

. We view Πmot

X

as a pro-unipotent algebraic group with G mot

S

• action.

The motivic fundamental group

Deligne and Goncharov have constructed Πmot

X

:= πmot

1

(X, 01) = lim ← −

n

Πmot,[n]

X

. We view Πmot

X

as a pro-unipotent algebraic group with G mot

S

• action.

It has the following ‘explicit’ description: Πmot

X

= Spec Qe0, e1, where e0 = dz z , e1 = dz 1 − z . Action of Umot

S

determined by motivic multiple zeta values.

The motivic Selmer variety

Set SelS,n := H1(G mot

S

, Πmot,[n]

X

). This is an affine Q-variety, isomorphic to AN

Q.

For z ∈ X(R), the path torsor Πmot,[n]

X

( 01, z) defines a class κn(z) ∈ SelS,n. The resulting map κn : X(R) → SelS,n is the motivic Kummer map.

Motivic multiple polylogarithms

For z ∈ X(R) there exists a canonical path γdR

z

∈ Πmot

X

( 01, z) (not fixed by GS!),

Motivic multiple polylogarithms

For z ∈ X(R) there exists a canonical path γdR

z

∈ Πmot

X

( 01, z) (not fixed by GS!), defining a cocycle cn(z) : G mot

S

→ Πmot,[n]

X

∈ Z 1(G mot

S

, Πmot,[n]

X

) representing κn(z).

Motivic multiple polylogarithms

For z ∈ X(R) there exists a canonical path γdR

z

∈ Πmot

X

( 01, z) (not fixed by GS!), defining a cocycle cn(z) : G mot

S

→ Πmot,[n]

X

∈ Z 1(G mot

S

, Πmot,[n]

X

) representing κn(z). The coefficients of the dual map Qe0, e1 → AS, w → Limot

w

(z), are the motivic multi polylogarithms. Formally, Limot

w

(z) =

• γdR

z

w.

The p-adic period map

For p ∈ S there is a p-adic period map perp : AS → Qp (Chatzistamatiou-¨ Unver). One shows that Li(p)

w (z) := perp(Limot w

(z)) ∈ Qp, where Li(p)

w

: X(Zp) → Qp are Furusho’s p-adic multiple polylogarithms (which appear in Kim’s diagram).

The map hn

Lemma

Every class in SelS,n is represented by a unique cocycle c whose dual map c∗ : Qe0, e1 → AS respects the grading.

The map hn

Lemma

Every class in SelS,n is represented by a unique cocycle c whose dual map c∗ : Qe0, e1 → AS respects the grading. We define hn : SelS,n × Umot

S

→ Πmot,[n]

X

by hn([c], σ) := c(σ).

The map hn

Lemma

Every class in SelS,n is represented by a unique cocycle c whose dual map c∗ : Qe0, e1 → AS respects the grading. We define hn : SelS,n × Umot

S

→ Πmot,[n]

X

by hn([c], σ) := c(σ). Then h(p)

n

= hn( · , σp) : Sel(p)

S,n ∼

= SelS,n ⊗ Qp → ΠdR,[n]

X

, where σp ∈ Umot

S

(Qp) corresponds to perp.

Exhaustion of AS by iterated integrals

Assumption (Exhaustion)

There exists ¯ S ⊃ S and a set I (resp. In) of motivic iterated integrals of the form I mot(a0; a1, . . . , ar; ar+1), with ai ∈ X(¯ R) ∪ {0, 1, ∞} and such that A¯

S (resp. A[n] ¯ S ) is

generated by I (resp. In).

Exhaustion of AS by iterated integrals

Assumption (Exhaustion)

There exists ¯ S ⊃ S and a set I (resp. In) of motivic iterated integrals of the form I mot(a0; a1, . . . , ar; ar+1), with ai ∈ X(¯ R) ∪ {0, 1, ∞} and such that A¯

S (resp. A[n] ¯ S ) is

generated by I (resp. In).

Remark

◮ For ¯

S = ∅ (resp. ¯ S = {2}), this is a theorem of Brown (resp. Deligne).

Exhaustion of AS by iterated integrals

Assumption (Exhaustion)

There exists ¯ S ⊃ S and a set I (resp. In) of motivic iterated integrals of the form I mot(a0; a1, . . . , ar; ar+1), with ai ∈ X(¯ R) ∪ {0, 1, ∞} and such that A¯

S (resp. A[n] ¯ S ) is

generated by I (resp. In).

Remark

◮ For ¯

S = ∅ (resp. ¯ S = {2}), this is a theorem of Brown (resp. Deligne).

◮ True for A[2] ¯ S if ¯

S contains all primes ≤ max S.

Exhaustion of AS by iterated integrals

Assumption (Exhaustion)

There exists ¯ S ⊃ S and a set I (resp. In) of motivic iterated integrals of the form I mot(a0; a1, . . . , ar; ar+1), with ai ∈ X(¯ R) ∪ {0, 1, ∞} and such that A¯

S (resp. A[n] ¯ S ) is

generated by I (resp. In).

Remark

◮ For ¯

S = ∅ (resp. ¯ S = {2}), this is a theorem of Brown (resp. Deligne).

◮ True for A[2] ¯ S if ¯

S contains all primes ≤ max S.

◮ Goncharov conjectures this if ¯

S contains all primes.

Decomposition

Theorem (D.-C.,W.,Brown)

Assuming exhaustion, there is an ‘algorithm’ which determines

• 1. a subset B ⊂ I such that AS = Q[B],
• 2. expression of all elements of I as polynomials in B.

Decomposition

Theorem (D.-C.,W.,Brown)

Assuming exhaustion, there is an ‘algorithm’ which determines

• 1. a subset B ⊂ I such that AS = Q[B],
• 2. expression of all elements of I as polynomials in B.

Remark

◮ This ‘algorithm’ is inexact in the sense that coefficients in (2)

can only be approximated p-adically to any desired precision.

Decomposition

Theorem (D.-C.,W.,Brown)

Assuming exhaustion, there is an ‘algorithm’ which determines

• 1. a subset B ⊂ I such that AS = Q[B],
• 2. expression of all elements of I as polynomials in B.

Remark

◮ This ‘algorithm’ is inexact in the sense that coefficients in (2)

can only be approximated p-adically to any desired precision.

◮ Uses the p-adic period map perp.

Decomposition

Theorem (D.-C.,W.,Brown)

Assuming exhaustion, there is an ‘algorithm’ which determines

• 1. a subset B ⊂ I such that AS = Q[B],
• 2. expression of all elements of I as polynomials in B.

Remark

◮ This ‘algorithm’ is inexact in the sense that coefficients in (2)

can only be approximated p-adically to any desired precision.

◮ Uses the p-adic period map perp. ◮ If successful, one obtains a proof that B has property (1).

Decomposition

Theorem (D.-C.,W.,Brown)

Assuming exhaustion, there is an ‘algorithm’ which determines

• 1. a subset B ⊂ I such that AS = Q[B],
• 2. expression of all elements of I as polynomials in B.

Remark

◮ This ‘algorithm’ is inexact in the sense that coefficients in (2)

can only be approximated p-adically to any desired precision.

◮ Uses the p-adic period map perp. ◮ If successful, one obtains a proof that B has property (1). ◮ We also obtain an explicit isomorphism AS ∼

= Qgℓ, f2m+1.

Explicit description of SelS,n and hn

The map c∗ : Qe0, e1 → AS ∼ = Qgℓ, f3, f5, . . . corresponding to a cocycle is uniquely determined by xℓ = coeff. of gℓ in φ(e0), yℓ = coeff. of gℓ in φ(e1), zw = coeff. of fn in φ(w), where w runs over Lyndon words of degree n odd (e1 < e0).

Explicit description of SelS,n and hn

The map c∗ : Qe0, e1 → AS ∼ = Qgℓ, f3, f5, . . . corresponding to a cocycle is uniquely determined by xℓ = coeff. of gℓ in φ(e0), yℓ = coeff. of gℓ in φ(e1), zw = coeff. of fn in φ(w), where w runs over Lyndon words of degree n odd (e1 < e0). Hence SelS,n = Spec Q[xℓ, yℓ, zw | ℓ ∈ S, deg(w) ≤ n, odd].

Explicit description of SelS,n and hn

The map c∗ : Qe0, e1 → AS ∼ = Qgℓ, f3, f5, . . . corresponding to a cocycle is uniquely determined by xℓ = coeff. of gℓ in φ(e0), yℓ = coeff. of gℓ in φ(e1), zw = coeff. of fn in φ(w), where w runs over Lyndon words of degree n odd (e1 < e0). Hence SelS,n = Spec Q[xℓ, yℓ, zw | ℓ ∈ S, deg(w) ≤ n, odd]. Assuming decomposition, we obtain an explicit description of the maps hn.

Example: n = 2

Recall that SelS,2 = Spec Q[xℓ, yℓ | ℓ ∈ S], Πmot,[2]

X

=   1 a c 1 b 1  

Example: n = 2

Recall that SelS,2 = Spec Q[xℓ, yℓ | ℓ ∈ S], Πmot,[2]

X

=   1 a c 1 b 1  

Theorem (D.-C., W.)

Assume that S = {ℓ | ℓ ≤ ℓmax}. Then h2(xℓ, yℓ) = (a, b, c), with a =

• ℓ

logmot(ℓ)xℓ, b =

• ℓ

Limot

1

(ℓ)yℓ, and c =

• ℓ1,ℓ2

cℓ1,ℓ2xℓ1yℓ2, where cℓ1,ℓ2 are explicit polynomials in logmot(ℓ), Limot

2

(ℓ), ℓ ∈ S.

Example: S = {2}

Assume S = {2}, n = 4. Then A[4]

S = Q[logmot(2), ζmot(3), Limot 4

(1/2)] ∼ = Qg2, f3.

Example: S = {2}

Assume S = {2}, n = 4. Then A[4]

S = Q[logmot(2), ζmot(3), Limot 4

(1/2)] ∼ = Qg2, f3. The subset X(Zp)4 is defined by the two equations F2(z) = Li(p)

2 (z) + 1

2 log(p)(z) log(p)(1 − z) and

Example: S = {2}

Assume S = {2}, n = 4. Then A[4]

S = Q[logmot(2), ζmot(3), Limot 4

(1/2)] ∼ = Qg2, f3. The subset X(Zp)4 is defined by the two equations F2(z) = Li(p)

2 (z) + 1

2 log(p)(z) log(p)(1 − z) and F4(z) = Li(p)

4 (z) + 7

8c log(p)(z)Li(p)

3 (z)

+ ( 4 21c + 1 24) log(p)(z)3 log(p)(1 − z), where c = log(p)(2)3 24ζ(p)(3) + Li(p)

4 (1/2)

log(p)(2)ζ(p)(3) .

Verification of Kim’s conjecture

We have checked numerically that X(Zp)4 = X(Z[1/2]) = {−1, 2, 1/2} for p = 3, 5, . . . , 29. Both functions F2(z) and F4(z) tend to have more than 3 zeroes.

References

• I. Dan-Cohen, S. Wewers

Explicit Chabauty-Kim theory for the thrice punctured line in depth two, Proceedings of the London Math Society, to appear. .

• I. Dan-Cohen, S. Wewers

Mixed Tate motives and the unit equation, arXiv:1311.7008v2

• F. Brown

Notes on motivic periods and the unit equation, letter to I. Dan-Cohen

References

• I. Dan-Cohen, S. Wewers

Explicit Chabauty-Kim theory for the thrice punctured line in depth two, Proceedings of the London Math Society, to appear. .

• I. Dan-Cohen, S. Wewers

Mixed Tate motives and the unit equation, arXiv:1311.7008v2

• F. Brown

Notes on motivic periods and the unit equation, letter to I. Dan-Cohen Thank you for your attention!