Motivic Periods, Coleman Functions, and the Unit
Equation
An Ongoing Project
D. Corwin1
I. Dan-Cohen2
1 MIT/ENS
2 Ben
Paris
Gurion University of the Negev
Journées Algophantiennes, Bordeaux, June 2017
Corwin, Dan-Cohen (VFU)
Motivic Periods, Coleman Functions, and the Unit Equation
VLC 2014
1 / 21
Table of Contents
1
Motivation: The Unit Equation
2
Motivic Periods
3
Polylogarithmic Cocycles and Integral Points
4
Recent and Current Computations
Corwin, Dan-Cohen (VFU)
Motivic Periods, Coleman Functions, and the Unit Equation
VLC 2014
2 / 21
Table of Contents
1
Motivation: The Unit Equation
2
Motivic Periods
3
Polylogarithmic Cocycles and Integral Points
4
Recent and Current Computations
Corwin, Dan-Cohen (VFU)
Motivic Periods, Coleman Functions, and the Unit Equation
VLC 2014
3 / 21
Motivation: The Unit Equation
Let R be an integer ring with a finite set of primes inverted (= Ok [1/S])
and X = P1 \ {0, 1, ∞}.
Corwin, Dan-Cohen (VFU)
Motivic Periods, Coleman Functions, and the Unit Equation
VLC 2014
4 / 21
Motivation: The Unit Equation
Let R be an integer ring with a finite set of primes inverted (= Ok [1/S])
and X = P1 \ {0, 1, ∞}.
Theorem
There are finitely many x, y ∈ R × such that x + y = 1
Corwin, Dan-Cohen (VFU)
Motivic Periods, Coleman Functions, and the Unit Equation
VLC 2014
4 / 21
Motivation: The Unit Equation
Let R be an integer ring with a finite set of primes inverted (= Ok [1/S])
and X = P1 \ {0, 1, ∞}.
Theorem
There are finitely many x, y ∈ R × such that x + y = 1
Equivalently, |X (R)| < ∞.
Corwin, Dan-Cohen (VFU)
Motivic Periods, Coleman Functions, and the Unit Equation
VLC 2014
4 / 21
Motivation: The Unit Equation
Let R be an integer ring with a finite set of primes inverted (= Ok [1/S])
and X = P1 \ {0, 1, ∞}.
Theorem
There are finitely many x, y ∈ R × such that x + y = 1
Equivalently, |X (R)| < ∞.
Originally proven by Siegel using Diophantine approximation around 1929.
Corwin, Dan-Cohen (VFU)
Motivic Periods, Coleman Functions, and the Unit Equation
VLC 2014
4 / 21
Motivation: The Unit Equation
Let R be an integer ring with a finite set of primes inverted (= Ok [1/S])
and X = P1 \ {0, 1, ∞}.
Theorem
There are finitely many x, y ∈ R × such that x + y = 1
Equivalently, |X (R)| < ∞.
Originally proven by Siegel using Diophantine approximation around 1929.
Problem
Find X (R) for various R, or even find an algorithm.
Corwin, Dan-Cohen (VFU)
Motivic Periods, Coleman Functions, and the Unit Equation
VLC 2014
4 / 21
Motivation: The Unit Equation
Let R be an integer ring with a finite set of primes inverted (= Ok [1/S])
and X = P1 \ {0, 1, ∞}.
Theorem
There are finitely many x, y ∈ R × such that x + y = 1
Equivalently, |X (R)| < ∞.
Originally proven by Siegel using Diophantine approximation around 1929.
Problem
Find X (R) for various R, or even find an algorithm.
In 2004, Minhyong Kim gave a proof in the case k = Q using fundamental
groups and p-adic analytic Coleman functions.
Corwin, Dan-Cohen (VFU)
Motivic Periods, Coleman Functions, and the Unit Equation
VLC 2014
4 / 21
Motivation: The Unit Equation
Let R be an integer ring with a finite set of primes inverted (= Ok [1/S])
and X = P1 \ {0, 1, ∞}.
Theorem
There are finitely many x, y ∈ R × such that x + y = 1
Equivalently, |X (R)| < ∞.
Originally proven by Siegel using Diophantine approximation around 1929.
Problem
Find X (R) for various R, or even find an algorithm.
In 2004, Minhyong Kim gave a proof in the case k = Q using fundamental
groups and p-adic analytic Coleman functions.
Refined Problem (Chabauty-Kim Theory)
Find p-adic analytic (Coleman) functions on X (Qp ) that vanish on X (R).
Corwin, Dan-Cohen (VFU)
Motivic Periods, Coleman Functions, and the Unit Equation
VLC 2014
4 / 21
Table of Contents
1
Motivation: The Unit Equation
2
Motivic Periods
3
Polylogarithmic Cocycles and Integral Points
4
Recent and Current Computations
Corwin, Dan-Cohen (VFU)
Motivic Periods, Coleman Functions, and the Unit Equation
VLC 2014
5 / 21
Periods
Let X be an algebraic variety of dimension d over Q and D a normal
crossings divisor in X .
Corwin, Dan-Cohen (VFU)
Motivic Periods, Coleman Functions, and the Unit Equation
VLC 2014
6 / 21
Periods
Let X be an algebraic variety of dimension d over Q and D a normal
crossings divisor in X .
Definition
R
A period is a complex number equal to an integral γ ω, where ω is an
algebraic differential form of degree d on X , and ω is an element of the
relative homology Hd (X (C), D(C); Q).
Corwin, Dan-Cohen (VFU)
Motivic Periods, Coleman Functions, and the Unit Equation
VLC 2014
6 / 21
Periods
Let X be an algebraic variety of dimension d over Q and D a normal
crossings divisor in X .
Definition
R
A period is a complex number equal to an integral γ ω, where ω is an
algebraic differential form of degree d on X , and ω is an element of the
relative homology Hd (X (C), D(C); Q).
Examples
Algebraic numbers, π, ζ(n), log(n), Lik (n), · · ·
Corwin, Dan-Cohen (VFU)
Motivic Periods, Coleman Functions, and the Unit Equation
VLC 2014
6 / 21
Periods
Let X be an algebraic variety of dimension d over Q and D a normal
crossings divisor in X .
Definition
R
A period is a complex number equal to an integral γ ω, where ω is an
algebraic differential form of degree d on X , and ω is an element of the
relative homology Hd (X (C), D(C); Q).
Examples
Algebraic numbers, π, ζ(n), log(n), Lik (n), · · ·
One may deduce relations between periods using rules for linearity,
products, algebraic changes of variables, and Stokes’ Theorem.
Corwin, Dan-Cohen (VFU)
Motivic Periods, Coleman Functions, and the Unit Equation
VLC 2014
6 / 21
Periods
Let X be an algebraic variety of dimension d over Q and D a normal
crossings divisor in X .
Definition
R
A period is a complex number equal to an integral γ ω, where ω is an
algebraic differential form of degree d on X , and ω is an element of the
relative homology Hd (X (C), D(C); Q).
Examples
Algebraic numbers, π, ζ(n), log(n), Lik (n), · · ·
One may deduce relations between periods using rules for linearity,
products, algebraic changes of variables, and Stokes’ Theorem.
For example, one can theoretically deduce 6ζ(2) = π 2 in this way.
Corwin, Dan-Cohen (VFU)
Motivic Periods, Coleman Functions, and the Unit Equation
VLC 2014
6 / 21
Motivic Periods
Defintion
The ring P of effective motivic periods is the formal Q-algebra generated
by tuples (X , D, ω, γ) as in the previous slide, modulo relations coming
from linearity, algebraic change of variables, and Stokes’ Theorem.
Corwin, Dan-Cohen (VFU)
Motivic Periods, Coleman Functions, and the Unit Equation
VLC 2014
7 / 21
Motivic Periods
Defintion
The ring P of effective motivic periods is the formal Q-algebra generated
by tuples (X , D, ω, γ) as in the previous slide, modulo relations coming
from linearity, algebraic change of variables, and Stokes’ Theorem.
Conjecture (Kontsevich-Zagier)
The natural map I : P → C given by integration is injective.
Corwin, Dan-Cohen (VFU)
Motivic Periods, Coleman Functions, and the Unit Equation
VLC 2014
7 / 21
Motivic Periods
Defintion
The ring P of effective motivic periods is the formal Q-algebra generated
by tuples (X , D, ω, γ) as in the previous slide, modulo relations coming
from linearity, algebraic change of variables, and Stokes’ Theorem.
Conjecture (Kontsevich-Zagier)
The natural map I : P → C given by integration is injective.
Examples
We denote the corresponding “motivic special values” by ζ m (n), logm (n),
Lim
k (n), · · ·
Corwin, Dan-Cohen (VFU)
Motivic Periods, Coleman Functions, and the Unit Equation
VLC 2014
7 / 21
Motivic Periods
Defintion
The ring P of effective motivic periods is the formal Q-algebra generated
by tuples (X , D, ω, γ) as in the previous slide, modulo relations coming
from linearity, algebraic change of variables, and Stokes’ Theorem.
Conjecture (Kontsevich-Zagier)
The natural map I : P → C given by integration is injective.
Examples
We denote the corresponding “motivic special values” by ζ m (n), logm (n),
Lim
k (n), · · ·
Examples
Applying IBC to each, we obtain ζ p (n), logp (n), Lipk (n), · · ·
Corwin, Dan-Cohen (VFU)
Motivic Periods, Coleman Functions, and the Unit Equation
VLC 2014
7 / 21
De Rham Periods
Coleman integrals use de Rham cohomology (specifically, the Frobenius
and Hodge filtration) but not Betti cohomology. We therefore need:
Corwin, Dan-Cohen (VFU)
Motivic Periods, Coleman Functions, and the Unit Equation
VLC 2014
8 / 21
De Rham Periods
Coleman integrals use de Rham cohomology (specifically, the Frobenius
and Hodge filtration) but not Betti cohomology. We therefore need:
Definition
The ring Pdr of effective de Rham periods is a variant of P in which γ
represents a de Rham homology class. For each p, there is a map
IBC : Pdr → Qp given by Coleman integration.
Corwin, Dan-Cohen (VFU)
Motivic Periods, Coleman Functions, and the Unit Equation
VLC 2014
8 / 21
De Rham Periods
Coleman integrals use de Rham cohomology (specifically, the Frobenius
and Hodge filtration) but not Betti cohomology. We therefore need:
Definition
The ring Pdr of effective de Rham periods is a variant of P in which γ
represents a de Rham homology class. For each p, there is a map
IBC : Pdr → Qp given by Coleman integration.
Examples
We similarly write ζ dr (n), logdr (n), Lidr
k (n), · · ·
Corwin, Dan-Cohen (VFU)
Motivic Periods, Coleman Functions, and the Unit Equation
VLC 2014
8 / 21
Mixed Tate Periods
We will focus on a subring P dr,+ (R) ⊆ Pdr of effective mixed Tate de
Rham periods over R. These contain all periods coming from
unirational pairs (X , D) with good reduction over R.
Corwin, Dan-Cohen (VFU)
Motivic Periods, Coleman Functions, and the Unit Equation
VLC 2014
9 / 21
Mixed Tate Periods
We will focus on a subring P dr,+ (R) ⊆ Pdr of effective mixed Tate de
Rham periods over R. These contain all periods coming from
unirational pairs (X , D) with good reduction over R.
Furthermore, as Coleman integrals are path independent, the
Coleman version of ζ(2) is 0.
Corwin, Dan-Cohen (VFU)
Motivic Periods, Coleman Functions, and the Unit Equation
VLC 2014
9 / 21
Mixed Tate Periods
We will focus on a subring P dr,+ (R) ⊆ Pdr of effective mixed Tate de
Rham periods over R. These contain all periods coming from
unirational pairs (X , D) with good reduction over R.
Furthermore, as Coleman integrals are path independent, the
Coleman version of ζ(2) is 0.
Our Motivic Periods
We will therefore work with P u (R) : = P dr,+ (R)/ζ(2) and the integration
map IBC : P u (R) → Qp for p ∈ Spec(R).
Corwin, Dan-Cohen (VFU)
Motivic Periods, Coleman Functions, and the Unit Equation
VLC 2014
9 / 21
Mixed Tate Periods
We will focus on a subring P dr,+ (R) ⊆ Pdr of effective mixed Tate de
Rham periods over R. These contain all periods coming from
unirational pairs (X , D) with good reduction over R.
Furthermore, as Coleman integrals are path independent, the
Coleman version of ζ(2) is 0.
Our Motivic Periods
We will therefore work with P u (R) : = P dr,+ (R)/ζ(2) and the integration
map IBC : P u (R) → Qp for p ∈ Spec(R).
We note that an inclusion R ⊆ R 0 gives rise to an inclusion
P u (R) ⊆ P u (R 0 ) (e.g., P u (Q) contains P u (Z[1/S]) for all S).
Corwin, Dan-Cohen (VFU)
Motivic Periods, Coleman Functions, and the Unit Equation
VLC 2014
9 / 21
Why Motivic Periods
The reason working with motivic periods rather than ordinary periods is
useful is that they have a nice algebraic structure.
Corwin, Dan-Cohen (VFU)
Motivic Periods, Coleman Functions, and the Unit Equation
VLC 2014
10 / 21
Why Motivic Periods
The reason working with motivic periods rather than ordinary periods is
useful is that they have a nice algebraic structure.
Theorem (Deligne, Goncharov, Borel, ...)
P u (R) has the structure of a graded Hopf algebra, and as such is
abstractly isomorphic to an explicit free shuffle algebra. Assuming
Frac(R) = Q, it is the free shuffle algebra
Qh{{gp }p∈S , {f2n+1 }n≥1 }i,
where each gp has degree 1, and f2n+1 has degree 2n + 1.
Corwin, Dan-Cohen (VFU)
Motivic Periods, Coleman Functions, and the Unit Equation
VLC 2014
10 / 21
Why Motivic Periods
The reason working with motivic periods rather than ordinary periods is
useful is that they have a nice algebraic structure.
Theorem (Deligne, Goncharov, Borel, ...)
P u (R) has the structure of a graded Hopf algebra, and as such is
abstractly isomorphic to an explicit free shuffle algebra. Assuming
Frac(R) = Q, it is the free shuffle algebra
Qh{{gp }p∈S , {f2n+1 }n≥1 }i,
where each gp has degree 1, and f2n+1 has degree 2n + 1.
As a graded vector space, it’s the free non-commutative algebra in these
generators. However, it’s equipped with a commutative product denoted
by X.
Corwin, Dan-Cohen (VFU)
Motivic Periods, Coleman Functions, and the Unit Equation
VLC 2014
10 / 21
Table of Contents
1
Motivation: The Unit Equation
2
Motivic Periods
3
Polylogarithmic Cocycles and Integral Points
4
Recent and Current Computations
Corwin, Dan-Cohen (VFU)
Motivic Periods, Coleman Functions, and the Unit Equation
VLC 2014
11 / 21
Polylogarithms and Integral Points
Definition
Let O(ΠX )PL = Q[Liu0 = logu , Liu1 , Liu2 , · · · ] as a Q-algebra.
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Motivic Periods, Coleman Functions, and the Unit Equation
VLC 2014
12 / 21
Polylogarithms and Integral Points
Definition
Let O(ΠX )PL = Q[Liu0 = logu , Liu1 , Liu2 , · · · ] as a Q-algebra.
As before, let X = P1 \ {0, 1, ∞}. Let z ∈ X (Q). For each integer k,
one can define a motivic period Liuk (z) ∈ P u (Q).
Corwin, Dan-Cohen (VFU)
Motivic Periods, Coleman Functions, and the Unit Equation
VLC 2014
12 / 21
Polylogarithms and Integral Points
Definition
Let O(ΠX )PL = Q[Liu0 = logu , Liu1 , Liu2 , · · · ] as a Q-algebra.
As before, let X = P1 \ {0, 1, ∞}. Let z ∈ X (Q). For each integer k,
one can define a motivic period Liuk (z) ∈ P u (Q).
It follows that each z ∈ X (Q) defines a homomorphism
κ(z) : O(ΠX )PL → P u (Q) sending Liuk to Liuk (z).
Corwin, Dan-Cohen (VFU)
Motivic Periods, Coleman Functions, and the Unit Equation
VLC 2014
12 / 21
Polylogarithms and Integral Points
Definition
Let O(ΠX )PL = Q[Liu0 = logu , Liu1 , Liu2 , · · · ] as a Q-algebra.
As before, let X = P1 \ {0, 1, ∞}. Let z ∈ X (Q). For each integer k,
one can define a motivic period Liuk (z) ∈ P u (Q).
It follows that each z ∈ X (Q) defines a homomorphism
κ(z) : O(ΠX )PL → P u (Q) sending Liuk to Liuk (z).
Fact
z ∈ X (R) iff Image(κ(z)) ⊆ P u (R)
Corwin, Dan-Cohen (VFU)
Motivic Periods, Coleman Functions, and the Unit Equation
VLC 2014
12 / 21
Hopf Algebra Structure
There is furthermore a graded Hopf algebra structure on O(ΠX )PL , in
which Liuk has degree max(k, 1), and the reduced coproduct is given by:
Corwin, Dan-Cohen (VFU)
Motivic Periods, Coleman Functions, and the Unit Equation
VLC 2014
13 / 21
Hopf Algebra Structure
There is furthermore a graded Hopf algebra structure on O(ΠX )PL , in
which Liuk has degree max(k, 1), and the reduced coproduct is given by:
d
0
Liuk
=
k−1
X
(logu )Xi
i=1
Corwin, Dan-Cohen (VFU)
i!
⊗ Liuk−i .
Motivic Periods, Coleman Functions, and the Unit Equation
VLC 2014
13 / 21
Hopf Algebra Structure
There is furthermore a graded Hopf algebra structure on O(ΠX )PL , in
which Liuk has degree max(k, 1), and the reduced coproduct is given by:
d
0
Liuk
=
k−1
X
(logu )Xi
i=1
i!
⊗ Liuk−i .
Fact
For z ∈ X (Q), the homomorphism κ(z) is a homomorphism of graded
Hopf algebras.
Corwin, Dan-Cohen (VFU)
Motivic Periods, Coleman Functions, and the Unit Equation
VLC 2014
13 / 21
Hopf Algebra Structure
There is furthermore a graded Hopf algebra structure on O(ΠX )PL , in
which Liuk has degree max(k, 1), and the reduced coproduct is given by:
d
0
Liuk
=
k−1
X
(logu )Xi
i=1
i!
⊗ Liuk−i .
Fact
For z ∈ X (Q), the homomorphism κ(z) is a homomorphism of graded
Hopf algebras.
In particular, d 0 Liuk (z) =
Corwin, Dan-Cohen (VFU)
Pk−1
i=1
(logu (z))Xi
i!
⊗ Liuk−i (z).
Motivic Periods, Coleman Functions, and the Unit Equation
VLC 2014
13 / 21
Motivic Kim’s Cutter
For a prime p, this gives us a diagram:
−−−−→
X (R)


κy
X (Z)
HomGrHopf (O(ΠX )PL , P u (R))
Corwin, Dan-Cohen (VFU)
Motivic Periods, Coleman Functions, and the Unit Equation
VLC 2014
14 / 21
Motivic Kim’s Cutter
For a prime p, this gives us a diagram:
−−−−→
X (R)


κy
X (Z)
HomGrHopf (O(ΠX )PL , P u (R))
We recall the integration map IBC : P u (R) → Qp .
Corwin, Dan-Cohen (VFU)
Motivic Periods, Coleman Functions, and the Unit Equation
VLC 2014
14 / 21
Motivic Kim’s Cutter
For a prime p, this gives us a diagram:
−−−−→
X (R)


κy
X (Z)
HomGrHopf (O(ΠX )PL , P u (R))
We recall the integration map IBC : P u (R) → Qp .
This induces
IBC
HomGrHopf (O(ΠX )PL , P u (R)) −−
→ HomAlg (O(ΠX )PL , Qp ).
Corwin, Dan-Cohen (VFU)
Motivic Periods, Coleman Functions, and the Unit Equation
VLC 2014
14 / 21
Motivic Kim’s Cutter
For a prime p, this gives us a diagram:
X (R)


κy
−−−−→
X (Z)
I
BC
−→ HomAlg (O(ΠX )PL , Qp )
HomGrHopf (O(ΠX )PL , P u (R)) −−−
We recall the integration map IBC : P u (R) → Qp .
This induces
IBC
HomGrHopf (O(ΠX )PL , P u (R)) −−
→ HomAlg (O(ΠX )PL , Qp ).
Corwin, Dan-Cohen (VFU)
Motivic Periods, Coleman Functions, and the Unit Equation
VLC 2014
14 / 21
Motivic Kim’s Cutter
For a prime p, this gives us a diagram:
X (R)


κy
−−−−→
X (Z)
I
BC
−→ HomAlg (O(ΠX )PL , Qp )
HomGrHopf (O(ΠX )PL , P u (R)) −−−
We recall the integration map IBC : P u (R) → Qp .
This induces
IBC
HomGrHopf (O(ΠX )PL , P u (R)) −−
→ HomAlg (O(ΠX )PL , Qp ).
In addition, an arbitrary z ∈ X (Zp ) induces a homomorphism
O(ΠX )PL → Qp sending Liuk to Lipk (z).
Corwin, Dan-Cohen (VFU)
Motivic Periods, Coleman Functions, and the Unit Equation
VLC 2014
14 / 21
Motivic Kim’s Cutter
For a prime p, this gives us a diagram:
X (R)


κy
−−−−→
X (Z)


y
I
BC
−→ HomAlg (O(ΠX )PL , Qp )
HomGrHopf (O(ΠX )PL , P u (R)) −−−
We recall the integration map IBC : P u (R) → Qp .
This induces
IBC
HomGrHopf (O(ΠX )PL , P u (R)) −−
→ HomAlg (O(ΠX )PL , Qp ).
In addition, an arbitrary z ∈ X (Zp ) induces a homomorphism
O(ΠX )PL → Qp sending Liuk to Lipk (z).
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Motivic Periods, Coleman Functions, and the Unit Equation
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Motivic Kim’s Cutter, cont.
X (R)


κy
−−−−→
X (Zp )


y
I
BC
HomGrHopf (O(ΠX )PL , P u (R)) −−−
−→ HomAlg (O(ΠX )PL , Qp )
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Motivic Periods, Coleman Functions, and the Unit Equation
VLC 2014
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Motivic Kim’s Cutter, cont.
X (R)


κy
−−−−→
X (Zp )


y
I
BC
HomGrHopf (O(ΠX )PL , P u (R)) −−−
−→ HomAlg (O(ΠX )PL , Qp )
The above diagram is known as Kim’s Cutter.
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Motivic Periods, Coleman Functions, and the Unit Equation
VLC 2014
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Motivic Kim’s Cutter, cont.
X (R)


κy
−−−−→
X (Zp )


y
I
BC
HomGrHopf (O(ΠX )PL , P u (R)) −−−
−→ HomAlg (O(ΠX )PL , Qp )
The above diagram is known as Kim’s Cutter.
We may think of the two bottom objects as schemes (one over Q and
the other over Qp ).
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Motivic Periods, Coleman Functions, and the Unit Equation
VLC 2014
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Motivic Kim’s Cutter, cont.
X (R)


κy
−−−−→
X (Zp )


y
I
BC
HomGrHopf (O(ΠX )PL , P u (R)) −−−
−→ HomAlg (O(ΠX )PL , Qp )
The above diagram is known as Kim’s Cutter.
We may think of the two bottom objects as schemes (one over Q and
the other over Qp ).
After tensoring the first with Qp , the bottom arrow becomes a map of
schemes, and the right vertical arrow is Coleman-analytic.
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Motivic Periods, Coleman Functions, and the Unit Equation
VLC 2014
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Motivic Kim’s Cutter, cont.
X (R)


κy
−−−−→
X (Zp )


y
I
BC
HomGrHopf (O(ΠX )PL , P u (R)) −−−
−→ HomAlg (O(ΠX )PL , Qp )
The above diagram is known as Kim’s Cutter.
We may think of the two bottom objects as schemes (one over Q and
the other over Qp ).
After tensoring the first with Qp , the bottom arrow becomes a map of
schemes, and the right vertical arrow is Coleman-analytic.
Dimension counts show that this arrow is non-dominant, which is
what proves Siegel’s theorem.
Corwin, Dan-Cohen (VFU)
Motivic Periods, Coleman Functions, and the Unit Equation
VLC 2014
15 / 21
Motivic Kim’s Cutter, cont.
X (R)


κy
−−−−→
X (Zp )


y
I
BC
HomGrHopf (O(ΠX )PL , P u (R)) −−−
−→ HomAlg (O(ΠX )PL , Qp )
The above diagram is known as Kim’s Cutter.
We may think of the two bottom objects as schemes (one over Q and
the other over Qp ).
After tensoring the first with Qp , the bottom arrow becomes a map of
schemes, and the right vertical arrow is Coleman-analytic.
Dimension counts show that this arrow is non-dominant, which is
what proves Siegel’s theorem.
Therefore, there is a nonzero ideal ICK ⊆ O(ΠX )PL vanishing on the
image of the bottom arrow, known as the Chabauty-Kim ideal.
Corwin, Dan-Cohen (VFU)
Motivic Periods, Coleman Functions, and the Unit Equation
VLC 2014
15 / 21
Table of Contents
1
Motivation: The Unit Equation
2
Motivic Periods
3
Polylogarithmic Cocycles and Integral Points
4
Recent and Current Computations
Corwin, Dan-Cohen (VFU)
Motivic Periods, Coleman Functions, and the Unit Equation
VLC 2014
16 / 21
The Chabauty-Kim Ideal
Elements of ICK pull back to X (Zp ) to give Coleman functions that
vanish on X (R).
Corwin, Dan-Cohen (VFU)
Motivic Periods, Coleman Functions, and the Unit Equation
VLC 2014
17 / 21
The Chabauty-Kim Ideal
Elements of ICK pull back to X (Zp ) to give Coleman functions that
vanish on X (R).
General goal: Compute some of these functions and show that they
cut out the rational points.
Corwin, Dan-Cohen (VFU)
Motivic Periods, Coleman Functions, and the Unit Equation
VLC 2014
17 / 21
The Chabauty-Kim Ideal
Elements of ICK pull back to X (Zp ) to give Coleman functions that
vanish on X (R).
General goal: Compute some of these functions and show that they
cut out the rational points.
Theorem (Dan-Cohen, Wewers, 2013)
For R = Z[1/2], the following Coleman function is in ICK :
 p

Li4 (z) logp (z)Lip3 (z) (logp (z))3 Lip1 (z)
det  Lip4 ( 21 ) logp ( 12 )Lip3 ( 21 ) (logp ( 21 ))3 Lip1 ( 12 ) 
1
1
1
24
6
Corwin, Dan-Cohen (VFU)
Motivic Periods, Coleman Functions, and the Unit Equation
VLC 2014
17 / 21
The Chabauty-Kim Ideal
Elements of ICK pull back to X (Zp ) to give Coleman functions that
vanish on X (R).
General goal: Compute some of these functions and show that they
cut out the rational points.
Theorem (Dan-Cohen, Wewers, 2013)
For R = Z[1/2], the following Coleman function is in ICK :
 p

Li4 (z) logp (z)Lip3 (z) (logp (z))3 Lip1 (z)
det  Lip4 ( 21 ) logp ( 12 )Lip3 ( 21 ) (logp ( 21 ))3 Lip1 ( 12 ) 
1
1
1
24
6
In 2015, Dan-Cohen posted a preprint showing that, assuming certain
well-known conjectures, this could be made into an algorithm.
Corwin, Dan-Cohen (VFU)
Motivic Periods, Coleman Functions, and the Unit Equation
VLC 2014
17 / 21
Our Current Work
Our current work revolves around improving the algorithm, extending
to multiple polylogarithms, and verifying cases of Kim’s conjecture.
Corwin, Dan-Cohen (VFU)
Motivic Periods, Coleman Functions, and the Unit Equation
VLC 2014
18 / 21
Our Current Work
Our current work revolves around improving the algorithm, extending
to multiple polylogarithms, and verifying cases of Kim’s conjecture.
More specifically, we are working on Z[1/6].
Corwin, Dan-Cohen (VFU)
Motivic Periods, Coleman Functions, and the Unit Equation
VLC 2014
18 / 21
Our Current Work
Our current work revolves around improving the algorithm, extending
to multiple polylogarithms, and verifying cases of Kim’s conjecture.
More specifically, we are working on Z[1/6].
To do this, we need to compute a basis for P u (Z[1/6]) (up to a
certain degree) as linear combinations of explicit polylogarithms of
the form Liu (z) for z ∈ X (Q).
Corwin, Dan-Cohen (VFU)
Motivic Periods, Coleman Functions, and the Unit Equation
VLC 2014
18 / 21
Computations for Z[1/6]
To simplify notation, we let A denote P u (Z[1/6]). We let An denote
the nth graded piece.
Corwin, Dan-Cohen (VFU)
Motivic Periods, Coleman Functions, and the Unit Equation
VLC 2014
19 / 21
Computations for Z[1/6]
To simplify notation, we let A denote P u (Z[1/6]). We let An denote
the nth graded piece.
The abstract description shows that dim(A0 ) = 1, dim(A1 ) = 2,
dim(A2 ) = 4, dim(A3 ) = 9, and dim(A4 ) = 20.
Corwin, Dan-Cohen (VFU)
Motivic Periods, Coleman Functions, and the Unit Equation
VLC 2014
19 / 21
Computations for Z[1/6]
To simplify notation, we let A denote P u (Z[1/6]). We let An denote
the nth graded piece.
The abstract description shows that dim(A0 ) = 1, dim(A1 ) = 2,
dim(A2 ) = 4, dim(A3 ) = 9, and dim(A4 ) = 20.
In fact, A is a free polynomial algebra on infinitely many generators,
so we only need to find such generators. There are two in degree 1,
one in degree 2, three in degree 3, and five in degree 4.
Corwin, Dan-Cohen (VFU)
Motivic Periods, Coleman Functions, and the Unit Equation
VLC 2014
19 / 21
Computations for Z[1/6]
To simplify notation, we let A denote P u (Z[1/6]). We let An denote
the nth graded piece.
The abstract description shows that dim(A0 ) = 1, dim(A1 ) = 2,
dim(A2 ) = 4, dim(A3 ) = 9, and dim(A4 ) = 20.
In fact, A is a free polynomial algebra on infinitely many generators,
so we only need to find such generators. There are two in degree 1,
one in degree 2, three in degree 3, and five in degree 4.
Basic tool: use the reduced coproduct d 0 . It’s injective in degrees 2
and 4 and has a kernel of dimension one in degree 3, generated by
ζ u (3).
Corwin, Dan-Cohen (VFU)
Motivic Periods, Coleman Functions, and the Unit Equation
VLC 2014
19 / 21
Computations for Z[1/6]
To simplify notation, we let A denote P u (Z[1/6]). We let An denote
the nth graded piece.
The abstract description shows that dim(A0 ) = 1, dim(A1 ) = 2,
dim(A2 ) = 4, dim(A3 ) = 9, and dim(A4 ) = 20.
In fact, A is a free polynomial algebra on infinitely many generators,
so we only need to find such generators. There are two in degree 1,
one in degree 2, three in degree 3, and five in degree 4.
Basic tool: use the reduced coproduct d 0 . It’s injective in degrees 2
and 4 and has a kernel of dimension one in degree 3, generated by
ζ u (3).
Procedure: Inductively on k, Write down motivic periods of the form
Liuk (z) for z ∈ X (R), apply d 0 , check dependence lower degree.
Corwin, Dan-Cohen (VFU)
Motivic Periods, Coleman Functions, and the Unit Equation
VLC 2014
19 / 21
Computations for Z[1/6]
To simplify notation, we let A denote P u (Z[1/6]). We let An denote
the nth graded piece.
The abstract description shows that dim(A0 ) = 1, dim(A1 ) = 2,
dim(A2 ) = 4, dim(A3 ) = 9, and dim(A4 ) = 20.
In fact, A is a free polynomial algebra on infinitely many generators,
so we only need to find such generators. There are two in degree 1,
one in degree 2, three in degree 3, and five in degree 4.
Basic tool: use the reduced coproduct d 0 . It’s injective in degrees 2
and 4 and has a kernel of dimension one in degree 3, generated by
ζ u (3).
Procedure: Inductively on k, Write down motivic periods of the form
Liuk (z) for z ∈ X (R), apply d 0 , check dependence lower degree.
The non-injectivity of d 0 for k = 3 requires use of p-adic
approximation to determine rational multiples of ζ u (3).
Corwin, Dan-Cohen (VFU)
Motivic Periods, Coleman Functions, and the Unit Equation
VLC 2014
19 / 21
Useful References
The following are on arXiv:
Mixed Tate Motives and the Unit Equation, Ishai Dan-Cohen and
Stefan Wewers
Mixed Tate Motives and the Unit Equation II, Ishai Dan-Cohen
Single-Valued Motivic Periods, Francis Brown
Motivic Periods and P1 \ {0, 1, ∞}, Francis Brown
Notes on Motivic Periods, Francis Brown
Integral Points on Curves and Motivic Periods, Francis Brown
Our definition of motivic periods comes from Periods, Kontsevich and
Zagier (http://www.maths.ed.ac.uk/ aar/papers/kontzagi.pdf).
Corwin, Dan-Cohen (VFU)
Motivic Periods, Coleman Functions, and the Unit Equation
VLC 2014
20 / 21
Thank You!
Corwin, Dan-Cohen (VFU)
Motivic Periods, Coleman Functions, and the Unit Equation
VLC 2014
21 / 21