#  $M_{0,5}$: Towards the Chabauty-Kim method in higher dimensions

Ishai Dan-Cohen and David Jarossay

###### Abstract.

If $Z$ is an open subscheme of $\operatorname{Spec}\mathbb{Z}$, $X$ is a
sufficiently nice $Z$-model of a smooth curve over $\mathbb{Q}$, and $p$ is a
closed point of $Z$, the Chabauty-Kim method leads to the construction of
locally analytic functions on $X({\mathbb{Z}_{p}})$ which vanish on $X(Z)$; we
call such functions “Kim functions”. At least in broad outline, the method
generalizes readily to higher dimensions. In fact, in some sense, the surface
$M_{0,5}$ should be easier than the previously studied curve
$M_{0,4}=\mathbb{P}^{1}\setminus\\{0,1,\infty\\}$ since its points are closely
related to those of $M_{0,4}$, yet they face a further condition to
integrality. This is mirrored by a certain _weight advantage_ we encounter,
because of which, $M_{0,5}$ possesses _new Kim functions_ not coming from
$M_{0,4}$. Here we focus on the case “$\mathbb{Z}[1/6]$ in half-weight 4”,
where we provide a first nontrivial example of a Kim function on a surface.

Central to our approach to Chabauty-Kim theory (as developed in works by S.
Wewers, D. Corwin, and the first author) is the possibility of separating the
geometric part of the computation from its arithmetic context. However, we
find that in this case the geometric step grows beyond the bounds of standard
algorithms running on current computers. Therefore, some ingenuity is needed
to solve this seemingly straightforward problem, and our new Kim function is
huge.

Both authors were supported by ISF grant 726/17.

###### Contents

  1. 1 Introduction
  2. 2 The polylogarithmic Lie algebra in abstraction
  3. 3 The polylogarithmic Hopf algebra in abstraction
  4. 4 The geometric step
  5. 5 The arithmetic step
  6. 6 The polylogarithmic quotient
  7. 7 The $p$-adic unipotent Albanese map
  8. 8 Summary and construction of $F^{\mathrm{BC}}$

##  1\. Introduction

The Chabauty-Kim method, introduced in [Kim1, Kim2], extends the classical
Chabauty method in two (related) directions. By going to higher quotients of
the fundamental group (where the Chabauty method stops at the abelianization)
it produces $p$-adic analytic functions (“Kim functions”) which vanish on
integral points, beyond the Chabauty bound. Thus, it can be applied in cases
where the Chabauty method does not apply. However, even in cases where the
Chabauty method does apply to produce a $p$-adic analytic function which can
be used to bound the set of integral points, it rarely produces a sharp bound.
As one climbs up the tower of unipotent quotients, however, the Chabauty-Kim
method produces more functions. Together, these may be used to give a sharp
bound. Indeed, according to Kim’s conjecture [BDCKW], this should be the case
for (suitable integral models) of all hyperbolic curves.

Going exactly one step beyond the abelian quotient leads to the so-called
_quadratic Chabauty method_. In a growing number of cases [BD2, BDCKW, BD1,
BBM2, BBM1, BB, BKK, Kim3] this has been worked out to produce numerical
results, and those results have been used to provide numerical evidence for
the conjecture. Of particular note is the work [BDM+] whose final point-count
(apart from verifying another case of the conjecture) solved an old and
sought-after problem in arithmetic.

The methods of Dan-Cohen–Wewers [DCW1, DCW2, DC] and Corwin–Dan-Cohen [CDC1,
CDC2], while so far limited to the simplest of all cases
($X=\mathbb{P}^{1}\setminus\\{0,1,\infty\\}$), have been particularly
successful in going beyond the quadratic level. These articles incorporate the
methods of mixed Tate motives and motivic iterated integrals (see, for
instance, [DG, Gon, Bro1, Bro3]). A key point is the possibility of extracting
the geometric aspects of the computation from their natural arithmetic
surroundings. The result is an algorithm which includes among its
subalgorithms a _geometric step_ and an _arithmetic step_. The arithmetic step
involves a search for enough motivic iterated integrals to generate suitable
portions of the mixed Tate Hopf algebra, and its halting is conditional on
conjectures of Goncharov, refined somewhat with respect to ramification.
Before embarking on the present work, we regarded the geometric step as being
comparatively simple, as it may, in principle, be solved by standard
algorithms.

Kim’s method generalizes naturally to higher dimensions. The connection with
the section conjecture suggests that a suitable generalization of Kim’s
conjecture may hold for anabelian varieties. In this article we take a
conservative step in this direction. Kim’s conjecture for $X=M_{0,4}$ over
$Z\subset\operatorname{Spec}\mathbb{Z}$ implies Kim’s conjecture for
$Y=M_{0,5}$ (the moduli space of genus $0$ curves with $5$ marked points) over
$Z$. Nevertheless, as we go up from $M_{0,4}$ to $M_{0,5}$, we encounter a
_weight advantage_ , which allows us to construct Kim functions on
$M_{0,5}({\mathbb{Z}_{p}})$ not coming from $M_{0,4}$ (see §8.7). 111temp:
We’ve added a paragraph explaining this. Our first opportunity to take
advantage of this weight advantage occurs for $\mathbb{Z}[1/6]$ in half-weight
$4$, and it is this one case that we focus on in the present work.

Our conclusions (so far) are somewhat mixed. Much of the work for $M_{0,4}$
generalizes readily. However, the geometric step via standard computational
methods has turned out to be computationally infeasible. With a careful (but
elementary) analysis of the geometric step (and a certain method involving
resultants) we are nevertheless able to produce a new Kim function, which
turns out to be huge.

To state our result, let us recall Kim’s method in outline. Our purpose here
is only to fix notation and terminology, and we refer the reader for instance
to [BDCKW] for a general introduction, and to [DCW2] for our mixed Tate
version. The Chabauty-Kim method applied to a smooth mixed-Tate variety $Y$
over $Z\subset\operatorname{Spec}\mathbb{Z}$, a prime of good reduction $p\in
Z$, and a finite type ${\mathbb{G}_{m}}$-equivariant quotient

$\pi_{1}^{\mathrm{un}}(Y,b)\twoheadrightarrow\pi^{\prime}$

of the unipotent fundamental group of $Y$ at the $Z$-integral base-point $b$,
revolves around a commuting diagram

$\textstyle{Y(Z)\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\kappa}$$\textstyle{Y({\mathbb{Z}_{p}})\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\alpha}$$\textstyle{H^{1}(\pi_{1}^{\operatorname{\mathbf{MT}}}(Z,{\rm{dR}}),\pi^{\prime}_{\rm{dR}})_{\mathbb{Q}_{p}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{LR}$$\textstyle{\pi^{\prime}_{{\rm{dR}},{\mathbb{Q}_{p}}}.}$

which we refer to as _Kim’s cutter_. Here
$\pi_{1}^{\operatorname{\mathbf{MT}}}(Z,{\rm{dR}})$ is the mixed Tate Galois
group of $Z$ at the de Rham fiber functor, the decoration “dR” denotes de Rham
realization, $\kappa$ is the unipotent Kummer map, $\alpha$ is the unipotent
Albanese map — a morphism of $p$-adic analytic spaces, and $\mathrm{LR}$,
which is a map of finite-type ${\mathbb{Q}_{p}}$-varieties (in our case,
affine spaces) is obtained by a combination of localization and realization.
The coordinate ring $A^{\prime}$ of $\pi^{\prime}$ is graded by the
${\mathbb{G}_{m}}$-action hiding inside the action of

$\pi_{1}^{\operatorname{\mathbf{MT}}}(Z,{\rm{dR}})={\mathbb{G}_{m}}\ltimes\pi_{1}^{\mathrm{un}}(Z,{\rm{dR}})$

on $\pi^{\prime}_{\rm{dR}}$ and we refer to the graded degree of a function as
its _half-weight_. If $f\in A^{\prime}_{{\mathbb{Q}_{p}},n}$ is a function of
half-weight $n$ such that $\mathrm{LR}^{\sharp}(f)=0$ then
$f^{\mathrm{BC}}:=\alpha^{\sharp}f$ is a Besser-Coleman function on
$Y({\mathbb{Z}_{p}})$ which vanishes on $Y(Z)$. We refer to such a function as
a _$p$ -adic Kim function for $Y$ over $Z$ in half-weight $n$_. Let $p$ be any
prime not equal to $2$ or $3$.

######  Theorem 1.1.

The function $F^{BC}$ on $M_{0,5}({\mathbb{Z}_{p}})$ constructed in §8.1 below
is a $p$-adic Kim function for $M_{0,5}$ over $\mathbb{Z}[1/6]$ in half-weight
four222up to a small $p$-adic error.

Unlike the unipotent fundamental group of
$\mathbb{P}^{1}\setminus\\{0,1,\infty\\}$, which has been studied to death,
working with the unipotent fundamental group of $M_{0,5}$ requires that we
address some basic issues ourselves. This mostly concerns a certain analog of
the polylogarithmic quotient. For instance, in §6, we revisit (and generalize
in a straightforward way) the beautiful proof due to Deligne [Del] and
Deligne–Goncharov [DG] that the polylogarithmic quotient is semisimple. We
also give an explicit algebra-basis for the algebra of functions on our
polylogarithmic quotient in §3.

Since the polylogarithmic quotient is not fixed under automorphisms of
$M_{0,5}$, translation by automorphisms gives rise to inequivalent Kim
functions. We may then ask if the vanishing locus of $F^{\mathrm{BC}}$ and its
conjugates is finite, or even equal to the set of integral points. Precedent
for computations of this sort may be found in [BBBM, Dog, Has]. Unfortunately,
the large size of $F^{\mathrm{BC}}$ presents a hurdle to computation. We hope
to face this challenge in a separate future work.

The tower of moduli spaces of curves with marked points, and especially its
first two steps $M_{0,4}$ and $M_{0,5}$, plays a central role in
Grothendieck’s vision for anabelian geometry, and, relatedly, in motivating
relations between multiple zeta values (complex and $p$-adic) [Fur2, FJ, BF,
OU] (see also [And, Chapter 25] and [GF] and the references there). We hope
that further investigation may shed some light, in one direction, on the
interaction between Kim’s cutter and the tower. In another direction, we hope
to better understand how the geometry of the tower controls relations between
motivic iterated integrals. In turn, this may lead to a better understanding
of the ramification of motivic iterated integrals and hence to more precise
$S$-integral refinements of Goncharov’s conjectures. As explained in [DCW2,
DC, CDC1, CDC2], our algorithms for $\mathbb{P}^{1}\setminus\\{0,1,\infty\\}$
rely on such statements for halting, and a better understanding will lead to
faster and more elegant algorithms. As we explain in [DCJ1], our methods with
resultants also help to clarify and simplify the geometric step for
$\mathbb{P}^{1}\setminus\\{0,1,\infty\\}$ and for punctured lines in general.

This article does not include introductory material on the moduli space
$M_{0,5}$; the facts we use, which we learned for instance from [GM, Bro2] as
well as the references given above, are summarized in §6. This article is also
written in correct logical order, which, at least in this case, runs counter
to the natural flow of exposition. Indeed, sections 2–5 make no mention of
$M_{0,5}$. Most readers will want to start with §6 and to refer back as
needed.

Speaking of _order_ , a word is in order concerning the order of
multiplication in fundamental groups. For many reasons, it seems to us far
more natural to let $\alpha\beta$ denote first $\beta$, then $\alpha$. For
instance, this is the notation used in _category theory_ , which is why it is
sometimes referred to as the “functorial order”. The reverse “lexicographic”
order, it seems to us, leads to the systematic reversal of a vast swath of
mathematics. However, for reasons we do not understand, there appears to be
quite a tradition of using lexicographic order, and the authors of [CDC1] in
particular, chose to follow this tradition. Thus, we’re forced to resolve this
conflict within the body of the article by using both orderings and spelling
out where and how we transition between them. For this purpose, we prefer not
to think of the question of ordering as being merely a matter of _notation_.
Rather, given a Tannakian category, we have a _functorial fundamental
groupoid_ and a _lexicographic fundamental groupoid_ , and the one is the
opposite of the other. Short of adding decorations everywhere, we attempt to
make clear which sort of fundamental groupoid is being used where, and some
sections begin with a declaration that holds throughout the section.

Finally, the reader may have noticed a footnote, according to which, for
Theorem 1.1 to be precise we would need to bound the error incurred by our
$p$-adic approximations. This task, while somewhat tedious, presents no
particular difficulty. Since our purpose here is to demonstrate a method (its
promise, and its challenges), we have chosen not to carry this out. If the
reader is disturbed by this logical wrinkle, she may view the main result of
this work as a fully fleshed-out algorithm which associates to every
$\epsilon>0$ a Besser-Coleman function $F^{\mathrm{BC}}_{\epsilon}$ on
$M_{0,5}({\mathbb{Z}_{p}})$ which is within $\epsilon$ of a Kim function. The
particular function we construct is then an _example_ with $\epsilon$ fixed.
As explained for instance in [DC], such an algorithm suffices for the
application to integral points and the verification of Kim’s conjecture.

In fact, we do not expect our formula for $F^{\mathrm{BC}}_{\epsilon}$ in
terms of polylogarithms to change as $\epsilon$ shrinks further. Proving this
would require proving that the formulas for decomposition of certain motivic
polylogarithms in terms of shuffle coordinates on the mixed Tate Galois group
obtained via computations of certain $p$-adic periods in §5 hold precisely
(and not only to within $\epsilon$). Some methods for doing so are
demonstrated for instance in [DCW2, CDC1] (along with attribution to those who
taught us these methods). But these are not needed for the application to
integral points.

### Acknowlegements

We would like to thank Jennifer Balakrishnan, Amnon Besser and Hidekazu
Furusho for their interest and encouragement as well as for helpful
conversations. In particular, we thank Balakrishnan for her participation in
our attempts to approach the “geometric step” via computer computation.
Finally, we are grateful to the referee for many helpful comments and
suggestions.

##  2\. The polylogarithmic Lie algebra in abstraction

### 2.1.

Let $L$ be the free Lie algebra on the set $\Gamma$ of generators

$e_{1},e_{11},e_{2},e_{22},e_{12}$

modulo the relations

$[e_{1},e_{2}]=[e_{11},e_{2}]=[e_{1},e_{22}]=0$
$[e_{11},e_{22}]=-[e_{11},e_{12}]=[e_{22},e_{12}]=[e_{2}-e_{1},e_{12}].$

Let $N$ be the Lie ideal generated by $e_{11},e_{22},e_{12}$. We define the
_polylogarithmic Lie algebra (for $M_{0,5}$)_ by

$L^{\mathrm{PL}}=L/[N,N].$ 333temp: L’ changed to $L^{\mathrm{PL}}$ to match
the notation $\pi^{\mathrm{PL}}$ for the polylogarthmic fundamental group,
here and in all the article

Note that in $L^{\mathrm{PL}}$ we have

$[e_{1},e_{12}]=[e_{2},e_{12}].$

######  Lemma 2.2.

In $L^{\mathrm{PL}}$, we have

$(\operatorname{ad}e_{1})^{n}(e_{12})=(\operatorname{ad}e_{2})^{n}(e_{12}).$

###### Proof.

Since $e_{1},e_{2}$ commute, we have, by the Jacobi identity, for all $Y$,

$[e_{1},[e_{2},Y]]=[e_{2},[e_{1},Y]].$

Hence by induction

$(\operatorname{ad}e_{1})^{n}([e_{2},Y])=[e_{2},(\operatorname{ad}e_{1})^{n}(Y)].$

We use (**) to establish (*) by induction on $n$. As noted above, the case
$n=1$ follows directly from the defining relations. Suppose the lemma holds up
to $n\geq 1$. Then

$\displaystyle(\operatorname{ad}e_{1})^{n+1}(e_{12})$
$\displaystyle=(\operatorname{ad}e_{1})^{n}([e_{1},e_{12}])$
$\displaystyle=(\operatorname{ad}e_{1})^{n}([e_{2},e_{12}])$
$\displaystyle=[e_{2},(\operatorname{ad}e_{1})^{n}(e_{12})]$
$\displaystyle=[e_{2},(\operatorname{ad}e_{2})^{n}(e_{12})]$
$\displaystyle=(\operatorname{ad}e_{2})^{n+1}(e_{12})$

which establishes the induction step and hence the lemma. ∎

######  Lemma 2.3.

The Lie algebra $L^{\mathrm{PL}}$ has vector space basis $S=$

$e_{1},e_{2},(\operatorname{ad}e_{1})^{n}(e_{11}),(\operatorname{ad}e_{2})^{n}(e_{22}),(\operatorname{ad}e_{1})^{n}(e_{12})=(\operatorname{ad}e_{2})^{n}(e_{12})\quad(n\geq
0).$

Among these basis elements we have the evident brackets

$[e_{i},(\operatorname{ad}e_{i})^{n}(e_{jk})]=(\operatorname{ad}e_{i})^{n+1}(e_{jk})$

and all other (ordered) brackets vanish.

The proof spans segments 2.4-2.6.

### 2.4.

We first show that $S$ spans $L^{\mathrm{PL}}$. For this, by induction, it’s
enough to show that for any generator $Y$ and any basis element $Z$, the
bracket $[Y,Z]$ is again a basis element or zero444temp:added “or zero”. We
check case by case by direct elementary calculation. If $Z=e_{i}$ ($i=1$ or
$2$) this is clear. Next we consider the case

$Z=(\operatorname{ad}e_{1})^{n}(e_{11}).$

If $Y=e_{1}$ ok; if $Y=e_{jk}\in N$ then the bracket is zero. If $Y=e_{2}$ we
show

$[e_{2},(\operatorname{ad}e_{1})^{n}(e_{11})]=0$

by induction on $n$. The base case is a defining relation. For $n\geq 1$ we
have

$\displaystyle[e_{2},$
$\displaystyle(\operatorname{ad}e_{1})^{n}(e_{11})]=[e_{2},[e_{1},(\operatorname{ad}e_{1})^{n-1}(e_{11})]]$
$\displaystyle=-[(\operatorname{ad}e_{1})^{n-1}(e_{11}),[e_{2},e_{1}]]-[e_{1},[(\operatorname{ad}e_{1})^{n-1}(e_{11}),e_{2}]]$
$\displaystyle=[e_{1},[e_{2},(\operatorname{ad}e_{1})^{n-1}(e_{11})]]$
$\displaystyle=[e_{1},0]$

by the inductive hypothesis as desired. The case

$Z=(\operatorname{ad}e_{2})^{n}(e_{22})$

follows by symmetry. Finally, the case

$Z=(\operatorname{ad}e_{1})^{n}(e_{12})=(\operatorname{ad}e_{2})^{n}(e_{22})$

is immediate. This completes the verification that $S$ spans
$L^{\mathrm{PL}}$.

### 2.5.

We turn to the linear independence. There’s an obvious homomorphism from
$L^{\mathrm{PL}}$ to the free abelian Lie algebra on the 5 generators, hence
the set of generators is linearly independent. Since $L^{\mathrm{PL}}$ is
defined by a homogeneous ideal, it’s enough to show that for each $n$, the
three purported basis elements in degree $n$ are linearly independent. Let
$L_{1}$ denote the polylogarithmic Lie algebra for
$\mathbb{P}^{1}\setminus\\{0,1,\infty\\}$ with vector space basis

$Y_{1},Y_{11},(\operatorname{ad}Y_{1})(Y_{11}),(\operatorname{ad}Y_{1})^{2}(Y_{11}),\dots,$

let $L_{2}$ denote the same Lie algebra with basis denoted

$Y_{2},Y_{22},(\operatorname{ad}Y_{2})(Y_{22}),(\operatorname{ad}Y_{2})^{2}(Y_{22}),\dots,$

and let $L^{\prime\prime}=L_{1}\oplus L_{2}$. Let $L(\Gamma)$ denote the free
Lie algebra on the set of generators $\Gamma$. The homomorphism

$L(\Gamma)\to L^{\prime\prime}$ $X_{i}\mapsto Y_{i},\quad X_{ii}\mapsto
Y_{ii},\quad X_{12}\mapsto 0$

factors through $L^{\mathrm{PL}}\to L^{\prime\prime}$. In $L^{\prime\prime}$
the set

$\\{(\operatorname{ad}Y_{1})^{n}(Y_{11}),(\operatorname{ad}Y_{2})^{n}(Y_{22})\\}$

is linearly independent for every $n\geq 0$. Consequently, the set

$\\{(\operatorname{ad}e_{1})^{n}(e_{11}),(\operatorname{ad}e_{2})^{n}(e_{22})\\}$

is linearly independent for every $n\geq 0$.

Define a Lie algebra $L^{\prime\prime\prime}$ to be the vector space with
basis the symbols

$Y_{1},Y_{2},Z_{0},Z_{1},Z_{2},\dots$

and only nonzero ordered brackets

$[Y_{1},Z_{i}]=[Y_{2},Z_{i}]=Z_{i+1}.$

To verify the Jacobi identity, note that it’s enough to show vanishing of

$J\\{A,B,C\\}:=[A,[B,C]]+[C,[A,B]]+[B,[C,A]]$ 555temp: commas added in the
Jacobi identity

on subsets of the set of basis elements of size 3. For all but two types of
subsets all terms vanish immediately; those are $\\{Y_{i},Y_{j},Z_{k}\\}$ and
$\\{Y_{i},Z_{j},Z_{k}\\}$ where a short calculation completes the
verification.

The homomorphism

$\phi:L(\Gamma)\to L^{\prime\prime\prime}$ $e_{i}\mapsto Y_{i},\quad
e_{ii}\mapsto 0,\quad e_{12}\mapsto Z_{0}$

factors modulo the relations 2.1(R). The ideal $N\subset L$ maps to the ideal
$M\subset L^{\prime\prime\prime}$ generated by $Z_{0}$. Since every
nonvanishing bracket of basis elements of $L^{\prime\prime\prime}$ is up to
sign a $Z_{i}$ and those commute, $[M,M]=0$, hence $\phi$ factors through
$L^{\mathrm{PL}}\to L^{\prime\prime\prime}$. This shows that for each $n$

$(\operatorname{ad}e_{i})^{n}(e_{12})\neq 0.$

Now consider an arbitrary vanishing linear combination

$0=a(\operatorname{ad}e_{1})^{n}(e_{11})+b(\operatorname{ad}e_{2})^{n}(e_{22})+c(\operatorname{ad}e_{i})^{n}(e_{12}).$

Acting by $\operatorname{ad}e_{1}$ and by $\operatorname{ad}e_{2}$, we obtain

$\displaystyle 0$
$\displaystyle=a(\operatorname{ad}e_{1})^{n+1}(e_{11})+c(\operatorname{ad}e_{i})^{n+1}(e_{12})$
$\displaystyle=b(\operatorname{ad}e_{2})^{n+1}(e_{22})+c(\operatorname{ad}e_{i})^{n+1}(e_{12}).$

By the linear independence of the sets (*) and the nonvanishing (**) it
follows that

$a=b=c=0.$

### 2.6.

Finally, the statement regarding the brackets among the basis elements is
immediate from the defining relations, from the vanishing 2.5(*) and from its
symmetrical twin

$[e_{1},(\operatorname{ad}e_{2})^{n}(e_{22})]=0.$

This completes the proof of Lemma 2.3.

##  3\. The polylogarithmic Hopf algebra in abstraction

### 3.1.

666Temp. We’ve added this paragraph (generalities on free prounipotent
groups).

We fix a base field $k$ of characteristic $0$. We refer the reader to §2 of
[DCW2] for an efficient review of the basics of free prounipotent groups in
notation similar to ours. We recall from loc. cit. that if $S$ is a set, then
the _free prounipotent $k$-group on $S$_, which we denote here by $\pi(S)$, is
the prounipotent group associated to the free pronilpotent Lie algebra
$\mathfrak{n}(S)$ on $S$. In particular, the elements of $S$ do _not_ belong
to $\pi(S)$. If $w$ is a word in $S$, then $f_{w}$ denotes the function on
$\pi(S)$ associated to $w$. Let $\mathcal{U}(S)$ denote the completed
universal enveloping algebra of $\mathfrak{n}(S)$ and let $A(S)$ denote the
coordinate ring of $\pi(S)$. We recall that there’s a nondegenerate pairing

$\langle\cdot,\cdot\rangle:\mathcal{U}(S)\times A(S)\to k.$

### 3.2.

Let $L(\Gamma)$ denote the free Lie algebra on the set of generators

$\Gamma=\\{e_{1},e_{11},e_{2},e_{22},e_{12}\\}.$

Let $\hat{L}(\Gamma)$ be the pronilpotent completion of $L(\Gamma)$, let
$\pi(\Gamma)$ be the associated prounipotent group, let
$\hat{\mathcal{U}}(\Gamma)$ be the completed universal enveloping algebra and
let $A(\Gamma)$ be the associated Hopf algebra. Recall that $A(\Gamma)$ is a
shuffle algebra and that the natural linear transformation from the
topological dual

$\hat{\mathcal{U}}(\Gamma)^{\lor}\to A(\Gamma)$

is an isomorphism of Hopf algebras. If $W$ is a word in the set $\Gamma$ of
generators, we denote the associated element of $A(\Gamma)$ by $f_{W}$.

### 3.3.

Recall that $L^{\mathrm{PL}}$ denotes the polylogarithmic Lie algebra (2.1).
777temp: changed everywhere $\pi^{\prime}$ to $\pi^{\mathrm{PL}}$,
$L^{\prime}$ to $L^{\mathrm{PL}}$ etc and added a comment about it. We repeat
the notation of §3.2 with the decoration ‘PL’ everywhere; thus,
$\hat{L}^{\mathrm{PL}}$ denotes the pronilpotent completion,
$\hat{\mathcal{U}}^{\mathrm{PL}}$ the completed universal enveloping algebra,
$\pi^{\mathrm{PL}}$ the prounipotent group, and
$A^{\mathrm{PL}}=\mathcal{O}(\pi^{\mathrm{PL}})$ the associated Hopf algebra.
There’s a natural surjection

$\hat{L}(\Gamma)\twoheadrightarrow\hat{L}^{\mathrm{PL}},$

hence an associated inclusion

$A(\Gamma)\supset A^{\mathrm{PL}}$

of Hopf algebras.

### 3.4.

By a _Lie word_ $W\in L(\Gamma)$ in the set of generators $\Gamma$ we mean an
element of the form

$[Y_{1}[Y_{2}[\cdots[Y_{n-1},Y_{n}]\cdots]]]$

with $Y_{i}\in\Gamma$. We set

$\mathrm{depth}(e_{i})=0,$ $\mathrm{depth}(e_{jk})=1,$ 888temp. $X_{i}$ and
$X_{jk}$ replaced by $e_{i}$ and $e_{jk}$

and we define the depth of a Lie word to be the sum of the depths of its
letters $Y_{i}$. 999Temp. A trivial yet confusingly-worded remark to which the
referee objected has been removed. We describe elements of the free monoid
generated by $\Gamma$ as _associative words_ in order to distinguish them from
Lie words. If $W$, $W^{\prime}$ are associative words, we say that
$W^{\prime}$ is a _subword_ of $W$ if there exist words $W_{1}$, $W_{2}$ such
that $W=W_{1}W^{\prime}W_{2}$. If $\sum a_{i}w_{i}$ is a finite linear
combination of associative words in $\Gamma$, we let $f_{\sum a_{i}w_{i}}=\sum
a_{i}f_{w_{i}}$ denote the dual vector in $A(\Gamma)$.

### 3.5.

Let $\mathfrak{a}$ denote the kernel of

$L(\Gamma)\twoheadrightarrow L^{\mathrm{PL}}.$

The Lie ideal $\mathfrak{a}<L(\Gamma)$ is generated by all Lie words of depth
$2$ together with the elements

$[e_{1},e_{2}],[e_{11},e_{2}],[e_{1},e_{22}],[e_{1},e_{12}]-[e_{2},e_{12}].$

######  Lemma 3.6.

The elements

$f_{e_{1}},f_{e_{2}},f_{e_{11}e_{1}^{n}},f_{e_{22}e_{2}^{n}},f_{e_{12}(e_{1}+e_{2})^{n}}\quad
n\geq 0$

of $A(\Gamma)$ form an algebra basis of $A^{\mathrm{PL}}$, by which we mean
that $A^{\mathrm{PL}}$ is a polynomial algebra freely generated by these
elements. 101010temp: added explanation on “algebra basis”

The proof of Lemma 3.6 spans segments 3.7-3.11.

### 3.7.

We first show that the elements 3.6(*) are contained in $A^{\mathrm{PL}}$. Let
$f$ be one of them. It’s enough to show that $f$ vanishes on the two-sided
ideal $\mathcal{U}\mathfrak{a}\mathcal{U}$. Thus, referring to §3.5, it’s
enough to show that for any of the Lie-word generators
$\lambda\in\mathfrak{a}$ described there and any two associative words
$W,W^{\prime}\in\mathcal{U}$,

$f(W\lambda W^{\prime})=0.$

This is clear for all but the two cases

$f_{e_{12}(e_{1}+e_{2})^{n}}(W[e_{1},e_{2}]W^{\prime})\quad\mbox{and}\quad
f_{e_{12}(e_{1}+e_{2})^{n}}(W([e_{1},e_{12}]-[e_{2},e_{12}])W^{\prime}).$

In turn, some simple but tedious combinatorics show that these vanish as well.

### 3.8.

The bijections

$(\mathcal{U}(\Gamma)/I^{n})^{\lor}=\mathcal{U}(\Gamma)/I^{n}$

induced by the basis of associative words in $\Gamma$ endow

$A(\Gamma)=\lim_{\to}(\mathcal{U}(\Gamma)/I^{n})^{\lor}$

with a second product $\ast$ which satisfies

$f_{v}\ast f_{w}:=f_{vw}.$

The operation $\ast$ also induces left and right actions of $A(\Gamma)$ on
$A(\Gamma)\otimes A(\Gamma)$ in an obvious way.

######  Lemma 3.9.

For any letter $e$ and any linear combination of associative words $\sum
a_{i}w_{i}$, the deconcatenation coproduct in the shuffle algebra $A(\Gamma)$
satisfies

$\Delta(f_{(\sum a_{i}w_{i})e})=(\Delta f_{\sum a_{i}w_{i}})\ast
f_{e}+f_{(\sum a_{i}w_{i})e}\varotimes 1,$

and, symmetrically

$\Delta(f_{e(\sum a_{i}w_{i})})=f_{e}\ast(\Delta f_{\sum
a_{i}w_{i}})+1\varotimes f_{e(\sum a_{i}w_{i})}.$

######  Lemma 3.10.

The deconcatenation coproduct in the shuffle algebra $A(\Gamma)$ satisfies

$\Delta f_{(e_{1}+e_{2})^{n}}=\sum_{i+j=n}f_{(e_{1}+e_{2})^{i}}\varotimes
f_{(e_{1}+e_{2})^{j}}.$

The proofs of Lemmas 3.9 and 3.10 are straightforward and we omit them.

### 3.11.

Let $A^{\prime\prime}$ be the subalgebra of $A^{\mathrm{PL}}$ generated by the
elements 3.6(*). Then $A^{\prime\prime}$ is closed under coproduct and is a
Hopf subalgebra of $A^{\mathrm{PL}}$. 111111temp: removed the expression
“closed under counit”, and added that it’s a Hopf subalgebra Combining lemmas
3.9 and 3.10, we have

$\displaystyle\Delta f_{e_{12}(e_{1}+e_{2})^{n}}$
$\displaystyle=f_{e_{12}}\ast\Delta f_{(e_{1}+e_{2})^{n}}+1\otimes
f_{e_{12}(e_{1}+e_{2})^{n}}$
$\displaystyle=f_{e_{12}}\ast\sum_{i+j=n}f_{(e_{1}+e_{2})^{i}}\varotimes
f_{(e_{1}+e_{2})^{j}}+1\otimes f_{e_{12}(e_{1}+e_{2})^{n}}$
$\displaystyle=\sum_{i+j=n}f_{e_{12}(e_{1}+e_{2})^{i}}\varotimes
f_{(e_{1}+e_{2})^{j}}+1\otimes f_{e_{12}(e_{1}+e_{2})^{n}}\in
A^{\prime\prime}\otimes A^{\prime\prime}.$

It follows that

$\phi:\operatorname{Spec}A^{\mathrm{PL}}\to\operatorname{Spec}A^{\prime\prime}$

is a surjection of graded prounipotent groups. Let $J^{\prime}$,
$J^{\prime\prime}$ denote the respective augmentation ideals, and consider the
induced map of Lie coalgebras

$(L^{\mathrm{PL}})^{\lor}=J^{\prime}/J^{\prime
2}\xleftarrow{\varphi}J^{\prime\prime}/J^{\prime\prime 2}.$

Under $\varphi$, the images of the generators 3.6(*) map to the dual basis of
the Lie algebra basis 2.3. Hence it must be an isomorphism. Hence
$A^{\prime\prime}=A^{\mathrm{PL}}$. This completes the proof of Lemma 3.6.

### 3.12.

121212temp: added §3.11, which contains the alternative proof of Lemma 3.5
suggested by the referee

We record nother way to finish the proof of Lemma 3.6 after §3.6, indicated by
the referee. The functor corresponding to the group scheme $\pi^{\mathrm{PL}}$
is given by

$R\mapsto\hat{L}^{\mathrm{PL}}_{R}:=\varprojlim(L^{\mathrm{PL}}\otimes
R)/(\Gamma^{i}L^{\mathrm{PL}}\otimes R)$

where $\Gamma^{i}L^{\mathrm{PL}}$ denotes the descending central series of
$L^{\mathrm{PL}}$. By Lemma 2.3, any element $x$ of
$\hat{L}^{\mathrm{PL}}_{R}$ can be written in a unique way as

$x=a_{1}e_{1}+a_{2}e_{2}+\sum_{n\geq
0}b_{n}\operatorname{ad}_{e_{1}}^{n}(e_{11})+\sum_{n\geq
0}c_{n}\operatorname{ad}_{e_{2}}^{n}(e_{22})+\sum_{n\geq
0}d_{n}\operatorname{ad}_{e_{1}}^{n}(e_{12})$

where $a_{i},b_{i},c_{i},d_{i}\in R$. By §3.6, $f_{e_{1}}$, $f_{e_{2}}$,
$f_{e_{11}e_{1}^{n}}$ etc. factor through $\pi^{\mathrm{PL}}$ so they can be
applied to this equality. This yields

$a_{i}=f_{e_{i}}(x),\text{ }b_{n}=(-1)^{n}f_{e_{11}e^{n}_{1}}(x),$
$c_{n}=(-1)^{n}f_{e_{22}e_{2}^{n}}(x),\text{
}d_{n}=(-1)^{n}f_{e_{12}(e_{1}+e_{2})^{n}}(x).$

Thus the elements $f_{\lambda}\in\mathcal{O}(\pi^{\mathrm{PL}})$ form a
complete set of coordinates on $\pi^{\mathrm{PL}}$.

##  4\. The geometric step

### 4.1.

Let $K(\tau,\upsilon,\sigma)$ denote the fraction field of the free
prounipotent group on three generators $\tau,\upsilon,\sigma$ over
$\mathbb{Q}$. The elements

$f_{\tau},f_{\upsilon},f_{\tau\upsilon},f_{\sigma},f_{\tau\tau\upsilon},f_{\tau\upsilon\upsilon},f_{\tau\sigma},f_{\upsilon\sigma},f_{\tau\tau\tau\upsilon},f_{\tau\tau\upsilon\upsilon},f_{\tau\upsilon\upsilon\upsilon}$

(3.1) 131313temp: added reference for the notation $f_{W}$ are algebraically
independent, and in what follows we may equally and equivalently work over the
sub-field of transcendence degree 11 generated by them.

We consider the alphabet 141414temp. $S$ changed to $\Gamma$

$\Gamma=\\{e_{1},e_{2},e_{11},e_{22},e_{12}\\}$

and the set of elements of the noncommutative polynomial ring
$\mathbb{Q}\langle\Gamma\rangle$

$\Lambda_{\geq-\infty}=\bigcup_{i=1}^{\infty}\Lambda_{-i},\mbox{ where
}\Lambda_{1}=\Gamma,$ $\mbox{ and
}\Lambda_{-i}=\\{e_{11}e_{1}^{i-1},e_{22}e_{2}^{i-1},e_{12}(e_{1}+e_{2})^{i-1}\\}\mbox{
for }i\geq 2.$

Since we’ll be working primarily with $\Lambda_{\geq-4}$, we abbreviate

$\Lambda:=\Lambda_{\geq-4}.$

We declare $\tau,\upsilon$ to have weight $-1$ and $\sigma$ to have weight
$-3$; we declare the elements of $\Gamma$ to have weight $-1$. We consider the
polynomial rings $\mathbb{Q}[\\{f_{\lambda}\\}_{\lambda}]$ and
$\mathbb{Q}[\\{\Phi^{\rho}_{\lambda}\\}_{\lambda,\rho}]$ where $\lambda$
ranges over $\Lambda$ and $\rho$ ranges over the three generators
$\\{\tau,\upsilon,\sigma\\}$ and is required to have weight equal to that of
$\lambda$. Explicitly, the complete list of algebra generators of
$\mathbb{Q}[\\{\Phi^{\rho}_{\lambda}\\}_{\lambda,\rho}]$ is

$\displaystyle\Phi^{\tau}_{e_{1}},\Phi^{\tau}_{e_{2}},\Phi^{\tau}_{e_{11}},\Phi^{\tau}_{e_{22}},\Phi^{\tau}_{e_{12}}$
$\displaystyle\Phi^{\upsilon}_{e_{1}},\Phi^{\upsilon}_{e_{2}},\Phi^{\upsilon}_{e_{11}},\Phi^{\upsilon}_{e_{22}},\Phi^{\upsilon}_{e_{12}},$
$\displaystyle\Phi^{\sigma}_{e_{11}e_{1}e_{1}},\Phi^{\sigma}_{e_{22}e_{2}e_{2}},\Phi^{\sigma}_{e_{12}(e_{1}+e_{2})^{2}},.$

Thus, $\mathbb{Q}[\\{f_{\lambda}\\}_{\lambda}]$ has Krull dimension $14$
whereas $\mathbb{Q}[\\{\Phi^{\rho}_{\lambda}\\}_{\lambda,\rho}]$ has Krull
dimension $13$. Define

$\theta:\mathcal{A}:=K(\tau,\upsilon,\sigma)[\\{f_{\lambda}\\}_{\lambda}]\to\mathcal{S}:=K(\tau,\upsilon,\sigma)[\\{\Phi^{\rho}_{\lambda}\\}_{\rho,\lambda}]$

by

$\displaystyle\theta(f_{e_{1}})$
$\displaystyle=f_{\tau}\Phi^{\tau}_{e_{1}}+f_{\upsilon}\Phi^{\upsilon}_{e_{1}}$
$\displaystyle\theta(f_{e_{2}})$
$\displaystyle=f_{\tau}\Phi^{\tau}_{e_{2}}+f_{\upsilon}\Phi^{\upsilon}_{e_{2}}$
$\displaystyle\theta(f_{e_{11}})$
$\displaystyle=f_{\tau}\Phi^{\tau}_{e_{11}}+f_{\upsilon}\Phi^{\upsilon}_{e_{11}}$
$\displaystyle\theta(f_{e_{22}})$
$\displaystyle=f_{\tau}\Phi^{\tau}_{e_{22}}+f_{\upsilon}\Phi^{\upsilon}_{e_{22}}$
$\displaystyle\theta(f_{e_{12}})$
$\displaystyle=f_{\tau}\Phi^{\tau}_{e_{12}}+f_{\upsilon}\Phi^{\upsilon}_{e_{12}}$
$\displaystyle\theta(f_{e_{11}e_{1}})$
$\displaystyle=f_{\tau\tau}\Phi^{\tau}_{e_{11}}\Phi^{\tau}_{e_{1}}+f_{\tau\upsilon}\Phi^{\tau}_{e_{11}}\Phi^{\upsilon}_{e_{1}}+f_{\upsilon\tau}\Phi^{\upsilon}_{e_{11}}\Phi^{\tau}_{e_{1}}+f_{\upsilon\upsilon}\Phi^{\upsilon}_{e_{11}}\Phi^{\upsilon}_{e_{1}}$
$\displaystyle\theta(f_{e_{22}e_{2}})$ $\displaystyle=\mbox{similar}$
$\displaystyle\theta(f_{e_{12}(e_{1}+e_{2})})$
$\displaystyle=f_{\tau\tau}\Phi^{\tau}_{e_{12}}\Phi^{\tau}_{e_{1}+e_{2}}+\cdots$
$\displaystyle\theta(f_{e_{11}e_{1}e_{1}})$
$\displaystyle=f_{\tau\tau\tau}\Phi^{\tau}_{e_{11}}\Phi^{\tau}_{e_{1}}\Phi^{\tau}_{e_{1}}+\cdots+f_{\upsilon\upsilon\upsilon}\Phi^{\upsilon}_{e_{11}}\Phi^{\upsilon}_{e_{1}}\Phi^{\upsilon}_{e_{1}}+f_{\sigma}\Phi^{\sigma}_{e_{11}e_{1}e_{1}}$
$\displaystyle\theta(f_{e_{22}e_{2}e_{2}})$ $\displaystyle=\mbox{similar}$
$\displaystyle\theta(f_{e_{12}(e_{1}+e_{2})^{2}})$
$\displaystyle=\mbox{similar}$ $\displaystyle\theta(f_{e_{11}e_{1}^{3}})$
$\displaystyle=f_{\tau^{4}}\Phi^{\tau}_{e_{11}}(\Phi^{\tau}_{e_{1}})^{3}+\cdots+f_{\upsilon^{4}}\Phi^{\upsilon}_{e_{11}}(\Phi^{\upsilon}_{e_{1}})^{3}+f_{\sigma\tau}\Phi^{\sigma}_{e_{11}e_{1}e_{1}}\Phi^{\tau}_{e_{1}}+f_{\sigma\upsilon}\Phi^{\sigma}_{e_{11}e_{1}e_{1}}\Phi^{\upsilon}_{e_{1}}$
$\displaystyle\theta(f_{e_{22}e_{2}^{3}})$ $\displaystyle=\mbox{similar}$
$\displaystyle\theta(f_{e_{12}(e_{1}+e_{2})^{3}})$
$\displaystyle=\mbox{similar}$

where

$\Phi^{\tau}_{e_{1}+e_{2}}=\Phi^{\tau}_{e_{1}}+\Phi^{\tau}_{e_{2}}\quad\mbox{and}\quad\Phi^{\upsilon}_{e_{1}+e_{2}}=\Phi^{\upsilon}_{e_{1}}+\Phi^{\upsilon}_{e_{2}}.$

### 4.2.

Let $\pi(\tau,\upsilon,\sigma)$ be the graded free prounipotent group on three
generators in weights $-1,-1,-3$. Let $\pi^{\mathrm{PL}}$ denote the
prounipotent $\mathbb{Q}$-group associated to the Lie algebra
$L^{\mathrm{PL}}$ of §2.1. Let $A^{\mathrm{PL}}$ denote the coordinate ring of
$\pi^{\mathrm{PL}}$. Let $A^{\mathrm{PL}}_{[\leq n]}$ denote the subalgebra
generated by elements in graded degree $\leq n$. This is a Hopf-subalgebra and
we let

$\pi^{\mathrm{PL}}_{\geq-n}=\operatorname{Spec}A^{\mathrm{PL}}_{[\leq n]}$

be the associated quotient of $\pi^{\mathrm{PL}}$. 151515Temp. Added
definition of $\pi^{\mathrm{PL}}_{\geq-n}$. Let

${\bf
Z}^{1}(\pi(\tau,\upsilon,\sigma),\pi^{\mathrm{PL}}_{\geq-4})^{\mathbb{G}_{m}}$

denote the functor from $\mathbb{Q}$-algebras to sets sending

$R\mapsto
Z^{1}(\pi(\tau,\upsilon,\sigma)_{R},(\pi^{\mathrm{PL}}_{\geq-4})_{R})^{\mathbb{G}_{m}},$

the pointed set of ${\mathbb{G}_{m}}$-equivariant cocycles for the trivial
group-action. We refer to an $R$-valued point of ${\bf
Z}^{1}(\pi(\tau,\upsilon,\sigma),\pi^{\mathrm{PL}}_{\geq-4})^{\mathbb{G}_{m}}$
as an _$R$ -family of cocycles_ or an _$R$ -cocycle_ for short; we omit the
repeating phrase “for $R$ an arbitrary $\mathbb{Q}$-algebra”. Let
$\mathfrak{ev}$ denote the map

$\pi(\tau,\upsilon,\sigma)\times{\bf
Z}^{1}(\pi(\tau,\upsilon,\sigma),\pi^{\mathrm{PL}}_{\geq-4})^{\mathbb{G}_{m}}\to\pi(\tau,\upsilon,\sigma)\times\pi^{\mathrm{PL}}_{\geq-4}$

given on $R$-points by

$(\gamma,c)\mapsto(\gamma,c(\gamma)).$

Let $\mathfrak{ev}_{K}$ denote the map

${\bf
Z}^{1}(\pi(\tau,\upsilon,\sigma),\pi^{\mathrm{PL}}_{\geq-4})^{\mathbb{G}_{m}}_{K(\tau,\upsilon,\sigma)}\to(\pi^{\mathrm{PL}}_{\geq-4})_{K(\tau,\upsilon,\sigma)}$

obtained from $\mathfrak{ev}$ by base-change.

######  Proposition 4.3.

In the situation and the notation of segments 4.1-4.2, there’s a commuting
square of functors

$\textstyle{{\bf
Z}^{1}(\pi(\tau,\upsilon,\sigma),\pi^{\mathrm{PL}}_{\geq-4})^{\mathbb{G}_{m}}_{K(\tau,\upsilon,\sigma)}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\mathfrak{ev}_{K}}$$\textstyle{(\pi^{\mathrm{PL}}_{\geq-4})_{K(\tau,\upsilon,\sigma)}}$$\textstyle{\operatorname{Spec}K(\tau,\upsilon,\sigma)[\\{\Phi^{\rho}_{\lambda}\\}_{\rho,\lambda}]\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\operatorname{Spec}\theta}$$\scriptstyle{\sim}$$\textstyle{\operatorname{Spec}K(\tau,\upsilon,\sigma)[\\{f_{\lambda}\\}_{\lambda}]\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\sim}$

in which the vertical maps are isomorphisms.

The proof of proposition 4.3 spans segments 4.4-4.9.

### 4.4.

Let $U$ be a prounipotent $\mathbb{Q}$-group with coordinate ring $A$ and let
$\mathcal{U}$ be the completed universal enveloping algebra of
$\operatorname{Lie}U$. Then there’s an isomorphism between $A$ and the
topological dual $\mathcal{U}^{\lor}$. Given $f\in A$ and $w\in\mathcal{U}$ we
denote the action of the linear functional associated to $f$ on $w$ by

$\langle f,w\rangle.$ 161616Temp. Added definition of bracket.

If $\lambda$ is a linear combination of elements of $\Lambda$ and $w$ is an
associative word in the alphabet $\\{\tau,\upsilon,\sigma\\}$, we define

$\phi^{w}_{\lambda}:{\bf
Z}^{1}(\pi(\tau,\upsilon,\sigma),\pi^{\mathrm{PL}}_{\geq-4})^{\mathbb{G}_{m}}\to\mathbb{A}^{1}_{\mathbb{Q}}$

by the formula

$\phi^{w}_{\lambda}(c)=\langle c^{\sharp}f_{\lambda},w\rangle$

on $R$-valued points. Note that for any ${\mathbb{G}_{m}}$-equivariant
$R$-cocycle $c$ and $\lambda\in\Lambda$,

$c^{\sharp}f_{\lambda}=\sum_{w}\phi^{w}_{\lambda}(c)f_{w}$

is homogeneous of weight equal to the weight of $\lambda$. Thus,
$\phi^{w}_{\lambda}=0$ unless
$\operatorname{wt}(w)=\operatorname{wt}(\lambda)$. Note also that
$\phi^{w}_{\lambda}$ is linear in the subscript, that is

$\phi^{w}_{\sum_{i}a_{i}\lambda_{i}}=\sum_{i}a_{i}\phi^{w}_{\lambda_{i}}.$

For future use, we formulate and prove the following proposition in the
slightly more general setting of a free graded prounipotent group
$\pi(\Sigma)$ on the graded 171717By a “graded set” we mean a family of sets
indexed by integers. set

$\Sigma=\Sigma_{-1}\cup\Sigma_{-2}\cup\Sigma_{-3}\cup\cdots\cup\Sigma_{-n}$

with $\Sigma_{-1}$ an arbitrary finite set, $\Sigma_{i}=0$ for $i$ even
$\leq-2$, $\Sigma_{i}=\\{\sigma_{-i}\\}$ of size one for $i$ odd $\leq-3$, and
with $\pi^{\mathrm{PL}}$ in place of $\pi^{\mathrm{PL}}_{\geq-4}$. 181818Temp.
Clarified setup.

######  Proposition 4.5.

We continue with the situation and the notation of segments 4.2, 4.4. Let

$c:\pi(\Sigma)_{R}\to\pi^{\mathrm{PL}}_{R}$

be a ${\mathbb{G}_{m}}$-equivariant $R$-cocycle. 191919Temp. We’ve clarified
that this proposition pertains to ${\mathbb{G}_{m}}$-equivariant cocycles.
Then for $0\leq r\leq n$, $\tau_{1},\dots,\tau_{r}\in\Sigma_{-1}$ and
$\sigma\in\Sigma_{r-n}$, we have

(1) $\displaystyle\phi^{\sigma\tau_{1}\cdots\tau_{r}}_{e_{11}e_{1}^{n-1}}(c)$
$\displaystyle=\phi^{\sigma}_{e_{11}e_{1}^{n-r-1}}(c)\phi^{\tau_{1}}_{e_{1}}(c)\cdots\phi^{\tau_{r}}_{e_{1}}(c)$
(2) $\displaystyle\phi^{\sigma\tau_{1}\cdots\tau_{r}}_{e_{22}e_{2}^{n-1}}(c)$
$\displaystyle=\phi^{\sigma}_{e_{22}e_{2}^{n-r-1}}(c)\phi^{\tau_{1}}_{e_{2}}(c)\cdots\phi^{\tau_{r}}_{e_{2}}(c)$
(3)
$\displaystyle\phi^{\sigma\tau_{1}\cdots\tau_{r}}_{e_{12}(e_{1}+e_{2})^{n-1}}(c)$
$\displaystyle=\phi^{\sigma}_{e_{12}(e_{1}+e_{2})^{n-r-1}}(c)\phi^{\tau_{1}}_{e_{1}+e_{2}}(c)\cdots\phi^{\tau_{r}}_{e_{1}+e_{2}}(c),$

and if $w$ is any word not occurring in the above equations then

$\phi^{w}_{\lambda}(c)=0.$

Conversely, given arbitrary elements $a^{\rho}_{\lambda}\in R$ for
$\rho\in\Sigma$ and $\lambda\in\Lambda_{\geq-\infty}$ of equal half-weight,
202020Temp. Added “of equal weight”. there exists one and only one
${\mathbb{G}_{m}}$-equivariant $R$-cocycle $c$ satisfying

$\phi^{\rho}_{\lambda}(c)=a^{\rho}_{\lambda}.$ 212121Temp. Corrected
quantifiers.

Proposition 4.5 and its proof are similar to Proposition 3.10 of [CDC1]. The
proof spans segments 4.6-4.8.

### 4.6.

We begin with a formal calculation, in which $\Sigma_{-1}$ may be an arbitrary
finite set, and $\\{a^{\tau}\\}_{\tau\in\Sigma_{-1}}$ a family of commuting
coefficients. In this abstract setting, we claim that

$\left(\sum_{\tau\in\Sigma_{-1}}a^{\tau}f_{\tau}\right)^{\Sha
n}=n!\sum_{\tau_{1},\dots,\tau_{n}\in\Sigma_{-1}}a^{\tau_{1}}\cdots
a^{\tau_{n}}f_{\tau_{1}\cdots\tau_{n}}.$

Indeed, the left side of the equation

$\displaystyle=\sum_{\tau_{1},\dots,\tau_{n}}(a^{\tau_{1}}f_{\tau_{1}})\mbox{\tiny{$\tiny{\Sha}$}}\cdots\mbox{\tiny{$\tiny{\Sha}$}}(a^{\tau_{n}}f_{\tau_{n}})$
$\displaystyle=\sum_{\tau_{1},\dots,\tau_{n}}a^{\tau_{1}}\cdots
a^{\tau_{n}}\left(\underset{\mbox{\tiny{of}
}\tau_{1},\dots,\tau_{n}}{\sum_{\mbox{\tiny{permutations}
}p}}f_{\tau_{1}^{p}\cdots\tau_{n}^{p}}\right)$
$\displaystyle=\sum_{p}\underbrace{\sum_{\tau_{1},\dots,\tau_{n}}a^{\tau_{1}}\cdots
a^{\tau_{n}}f_{\tau_{1}^{p}\cdots\tau_{n}^{p}}}_{\mbox{\tiny{independent of}
}p},$

which equals the right side of the equation.

### 4.7.

Returning to our concrete situation, we focus on equation 4.5(3) and,
simultaneously, on the case $\lambda=e_{12}(e_{1}+e_{2})^{n-1}$ of equation
4.5(4). We have, tautologically

$c^{\sharp}f_{e_{12}(e_{1}+e_{2})^{n}}=\sum_{\big{\\{}w\;\big{|}\;\operatorname{wt}w=n+1\big{\\}}}\phi^{w}_{e_{12}(e_{1}+e_{2})^{n}}(c)f_{w}.$

We wish to compute the coproduct $\Delta$ of both sides, remembering that
$c^{\sharp}$, since it corresponds to 222222Temp. Clarified role of cocycle
condition. a cocycle for the trivial group action, preserves the coproduct. On
the right side, we have

$\Delta({\rm
RHS})=\sum_{\big{\\{}w^{\prime},w^{\prime\prime}\;\big{|}\;\operatorname{wt}(w^{\prime})+\operatorname{wt}(w^{\prime\prime})=n+1\big{\\}}}\phi^{w^{\prime}w^{\prime\prime}}_{e_{12}(e_{1}+e_{2})^{n}}(c)f_{w^{\prime}}\otimes
f_{w^{\prime\prime}}.$

On the left we have

$\displaystyle\Delta({\rm LHS})=(c^{\sharp}\otimes
c^{\sharp})\left(\sum_{i+j=n}f_{e_{12}(e_{1}+e_{2})^{i}}\otimes\frac{f_{e_{1}+e_{2}}^{\Sha
j}}{j!}+1\otimes f_{e_{12}(e_{1}+e_{2})^{n}}\right)$ by Lemmas 3.9 and 3.10,
$\displaystyle=\sum_{i+j=n}\left(\sum_{u}\phi^{u}_{e_{12}(e_{1}+e_{2})^{i}}(c)f_{u}\right)\otimes\frac{\left(\sum_{\tau\in\Sigma_{-1}}\phi^{\tau}_{e_{1}+e_{2}}(c)f_{\tau}\right)^{\Sha
j}}{j!}$
$\displaystyle+1\otimes\sum_{v}\phi^{v}_{e_{12}(e_{1}+e_{2})^{n}}(c)f_{v}$
$\displaystyle=\underset{\tau_{1},\dots,\tau_{j}}{\underset{u}{\underset{i+j=n}{\sum}}}\phi^{u}_{e_{12}(e_{1}+e_{2})^{i}}(c)\phi_{e_{1}+e_{2}}^{\tau_{1}}(c)\cdots\phi_{e_{1}+e_{2}}^{\tau_{j}}(c)f_{u}\otimes
f_{\tau_{1}\cdots\tau_{j}}$
$\displaystyle+\sum_{v}\phi^{v}_{e_{12}(e_{1}+e_{2})^{n}}(c)1\otimes f_{v}$
232323Temp. Corrected $c^{\sharp}$ to $c^{\sharp}\otimes c^{\sharp}$.
Corrected missing argument ‘$c$’.

by §4.6. Taking the coefficient of $f_{v}\otimes f_{\tau}$ with
$\tau\in\Sigma_{-1}$ and $v$ an arbitrary word of length $n\geq 1$, we obtain

$\phi^{v\tau}_{e_{12}(e_{1}+e_{2})^{n}}(c)=\phi^{v}_{e_{12}(e_{1}+e_{2})^{n-1}}(c)\phi^{\tau}_{e_{1}+e_{2}}(c)$

while taking the coefficient of $f_{v}\otimes f_{\sigma}$ with
$\sigma\in\Sigma_{i}$ for $i<-1$ and $v$ an arbitrary word of length
$n+1-i\geq 1$, we obtain

$\phi^{v\sigma}_{e_{12}(e_{1}+e_{2})^{n}}(c)=0.$

Equations 4.5(3) and (4)($\lambda=e_{12}(e_{1}+e_{2})^{n-1}$) follow.242424
Temp. We removed a footnote here concerning the possibility of using the
infinitesimal coaction.

### 4.8.

We turn to the second clause of the proposition. Equations 4.7(B) and (C) show
explicitly how all coefficients of $c^{\sharp}f_{\lambda}$ in equation 4.7(A)
are determined by the values $\phi^{\rho}_{\lambda}$ for $\rho\in\Sigma$ an
individual letter. Alternatively, 252525temp: We’ve added the referee’s
argument here, which is much simpler and clearer than ours. this is just an
immediate consequence of the fact that the elements $f_{\lambda}$ for
$\lambda\in\Lambda_{\geq-\infty}$ form an algebra basis of $\pi^{\mathrm{PL}}$
on the one hand, and that the elements $\rho\in\Sigma$ form a free set of
generators for the prounipotent group $\pi(\Sigma)$ on the other. The fact
that the generators are not elements of the group may cause some confusion, so
we take the time to spell out the existence: given elements
$a^{\rho}_{\lambda}\in R$ as in the proposition, we define

$c:\pi(\Sigma)_{R}\to\pi^{\mathrm{PL}}_{R}$

to be the unique homomorphism such that

$\langle(\operatorname{Lie}c)(\rho),f_{\lambda}\rangle=a^{\rho}_{\lambda}$

whenever the half-weights of $\rho$ and $\lambda$ are equal, and $=0$
otherwise. Here, as usual, we regard $(\operatorname{Lie}c)(\rho)$ as
belonging to the universal enveloping algebra of $\pi^{\mathrm{PL}}$, and the
bracket refers to the natural pairing between the coordinate ring and the
universal enveloping algebra. This completes the proof of Proposition 4.5.

######  4.9.

We now prove Proposition 4.3. 262626temp. We’ve added a proof of Proposition
4.3. The second clause of Proposition 4.5 says that the map

$\operatorname{Spec}\mathbb{Q}[\\{\Phi^{\rho}_{\lambda}\\}_{\rho,\lambda}]\leftarrow{\bf
Z}^{1}(\pi(\tau,\upsilon,\sigma),\pi^{\mathrm{PL}}_{\geq-4})^{\mathbb{G}_{m}}$

induced by the functions

$\mathbb{A}^{1}_{\mathbb{Q}}\xleftarrow{\Phi^{\rho}_{\lambda}}{\bf
Z}^{1}(\pi(\tau,\upsilon,\sigma),\pi^{\mathrm{PL}}_{\geq-4})^{\mathbb{G}_{m}}$

is an isomorphism of $\mathbb{Q}$-schemes. Lemma 3.6 implies that the map

$\operatorname{Spec}\mathbb{Q}[\\{f_{\lambda}\\}_{\operatorname{wt}\lambda\geq-4}]\leftarrow\pi^{\mathrm{PL}}_{\geq-4}$

induced by the functions $f_{\lambda}$ is also an isomorphism of
$\mathbb{Q}$-schemes.

Let $A(\tau,\upsilon,\sigma)$ denote the coordinate ring of
$\pi(\tau,\upsilon,\sigma)$. In terms of the functions $\Phi^{\rho}_{\lambda}$
on ${\bf
Z}^{1}(\pi(\tau,\upsilon,\sigma),\pi^{\mathrm{PL}}_{\geq-4})^{\mathbb{G}_{m}}$,
the functions $f_{\lambda}$ on $\pi^{\mathrm{PL}}_{\geq-4}$ and the functions
$f_{w}$ on $\pi(\tau,\upsilon,\sigma)$, the universal cocycle evaluation map
$\mathfrak{ev}$ of §4.2(*) is computed as follows. Since the map
$\mathfrak{ev}$ commutes with the first projection, we have

$\mathfrak{ev}^{\sharp}(f_{w})=f_{w}$

for all words $w$ in the generators $\\{\tau,\upsilon,\sigma\\}$. Let $Q$ be
the set of words which occur in equations 1-3 of Proposition 4.5 and let
$Q_{n}\subset Q$ denote the words of half-weight $n$. We use the same
equations to define new functions $\Phi^{w}_{\lambda}$ on the space of
equivariant cocycles (replace each lower-case $\phi$ by an upper-case $\Phi$).
We then have for $(\gamma,c)$ an $R$-valued point of

$\pi(\tau,\upsilon,\sigma)\times{\bf
Z}^{1}(\pi(\tau,\upsilon,\sigma),\pi_{\geq-4}^{\mathrm{PL}})^{\mathbb{G}_{m}}$

and $\lambda\in\Lambda_{n}$,

$\displaystyle(\mathfrak{ev}^{\sharp}f_{\lambda})(\gamma,c)$
$\displaystyle=(c^{\sharp}f_{\lambda})(\gamma)$
$\displaystyle=\Big{(}\sum_{\operatorname{wt}w=n}\phi^{w}_{\lambda}(c)f_{w}\Big{)}(\gamma)$
$\displaystyle=\Big{(}\sum_{w\in
Q_{n}}f_{w}\Phi^{w}_{\lambda}\Big{)}(\gamma,c)$

where the last equation holds by Proposition 4.5. Hence we have

$\mathfrak{ev}^{\sharp}f_{\lambda}=\sum_{w\in Q_{n}}f_{w}\Phi^{w}_{\lambda}.$

These equations are written out explicitly in §4.1. This completes the proof
of Proposition 4.3.

We now turn to the problem of constructing a nonzero element in the kernel of
$\theta$.

### 4.10.

Consider the following element of $\mathcal{A}[X,Y]$:

$Q_{1}(X,Y)=-\Big{[}(f_{\tau\upsilon}f_{\upsilon}-f_{\tau}f_{\upsilon\upsilon})X+(f_{\tau\tau}f_{\upsilon}-f_{\tau}f_{\upsilon\upsilon})Y\Big{]}f_{e_{11}e_{1}^{3}}+\\\
\Big{[}f_{\upsilon}f_{e_{11}e_{1}}-(f_{\upsilon\tau}Y+f_{\upsilon\upsilon}X)f_{e_{11}}\Big{]}(f_{\tau^{4}}Y^{3}+f_{\tau\upsilon^{3}}X^{3}+(f_{\tau(\tau^{2}\mathcyr{sh}\upsilon)})XY^{2}+(f_{\tau(\upsilon^{2}\mathcyr{sh}\tau)})YX^{2})\\\
+\Big{[}-f_{\tau}f_{e_{11}e_{1}}+(f_{\tau\upsilon}X+f_{\tau\tau}Y)f_{e_{11}}\Big{]}(f_{\upsilon\tau^{3}}Y^{3}+f_{\upsilon^{4}}X^{3}+(f_{\upsilon(\tau^{2}\mathcyr{sh}\upsilon)})XY^{2}+(f_{\upsilon(\upsilon^{2}\mathcyr{sh}\tau)})YX^{2})+\\\
(f_{\sigma\tau}Y+f_{\sigma\upsilon}X)\frac{1}{f_{\sigma}}\Big{[}f_{e_{11}e_{1}^{2}}-\Big{[}f_{\upsilon}f_{e_{11}e_{1}}-(f_{\upsilon\tau}Y+f_{\upsilon\upsilon}X)f_{e_{11}}\Big{]}\
(f_{\tau\tau\tau}Y^{2}+f_{\tau\upsilon\upsilon}X^{2}+f_{\tau(\tau\mathcyr{sh}\upsilon)}XY)\\\
+\Big{[}-f_{\tau}f_{e_{11}e_{1}}+(f_{\tau\upsilon}X+f_{\tau\tau}Y)f_{e_{11}}\Big{]}(f_{\upsilon\tau\tau}Y^{2}+f_{\upsilon\upsilon\upsilon}X^{2}+f_{\upsilon(\tau\mathcyr{sh}\upsilon)}XY)\Big{]}$

and let $a_{i,j}$ denote the coefficient of the monomial $X^{i}Y^{j}$. We use
the elements $a_{i,j}\in\mathcal{A}$ to construct a collection of elements of
$\mathcal{A}[Y]$: we define

$A_{1}(Y)=-(f_{\tau\upsilon}-\frac{1}{2}f_{\tau}f_{\upsilon})\Big{[}f_{\upsilon\upsilon}f_{e_{11}}\Big{]}$
$B_{1}(Y)=(f_{\tau\upsilon}-\frac{1}{2}f_{\tau}f_{\upsilon})\Big{[}f_{\upsilon}f_{e_{11}e_{1}}-f_{\upsilon\tau}Yf_{e_{11}}\Big{]}\\\
+(f_{\upsilon\tau}-\frac{1}{2}f_{\upsilon}f_{\tau})\Big{[}f_{\tau\upsilon}f_{e_{11}}\Big{]}Y-\Big{[}(f_{\tau\upsilon}f_{\upsilon}-f_{\tau}f_{\upsilon\upsilon})\Big{]}\Big{[}f_{e_{11}e_{1}}-\frac{1}{2}f_{e_{11}}f_{e_{1}}\Big{]}$
$C_{1}(Y)=(f_{\upsilon\tau}-\frac{1}{2}f_{\upsilon}f_{\tau})\Big{[}-f_{\tau}f_{e_{11}e_{1}}+f_{\tau\tau}Yf_{e_{11}}\Big{]}Y-(f_{\tau\tau}f_{\upsilon}-f_{\tau}f_{\upsilon\upsilon})Y\Big{[}f_{e_{11}e_{1}}-\frac{1}{2}f_{e_{11}}f_{e_{1}}\Big{]}$
${Q}_{1,a}(Y)=\sum_{i=0}^{4}\sum_{j=0}^{4}a_{ij}(2A_{1})^{4-i}Y^{j}\sum_{k^{\prime}=0}^{\lfloor\frac{i}{2}\rfloor}{i\choose
2k^{\prime}}(B_{1}^{2}-4A_{1}C_{1})^{k^{\prime}}(-B_{1})^{i-2k^{\prime}}$
${Q}_{1,b}(Y)=\sum_{i=0}^{4}\sum_{j=0}^{4}a_{ij}(2A)^{4-i}Y^{j}\sum_{k^{\prime}=0}^{\lfloor\frac{i-1}{2}\rfloor}{i\choose
2k^{\prime}+1}(B_{1}^{2}-4A_{1}C_{1})^{k^{\prime}}(-B_{1})^{i-2k^{\prime}-1}$
(4) $P_{1}(Y)={Q}_{1,a}^{2}-(B_{1}^{2}-4A_{1}C_{1}){Q}_{1,b}^{2}.$

This last polynomial $P_{1}(Y)$ is divisible by a power of $Y$ and we let
$p_{1}(Y)$ be the result of dividing by this factor. Polynomials
$p_{2}(Y),p_{3}(Y)\in\mathcal{A}[Y]$ are defined similarly with
$(e_{11},e_{1})$ replaced respectively by $(e_{22},e_{2})$ and
$(e_{12},e_{1}+e_{2})$. Finally, we define $F\in\mathcal{A}$ to be the double-
resultant

$F=Res_{Y}\Big{(}Res_{X}\big{(}p_{1}(X),p_{2}(Y-X)\big{)},p_{3}(Y)\Big{)}.$

######  Proposition 4.11.

The element $F$ of $\mathcal{A}$ defined above is a nonzero element of the
kernel of $\theta:\mathcal{A}\to\mathcal{S}$.

###### Proof.

Needless to say, given a sufficiently powerful computer this could be easily
checked via direct computation. Indeed, we do rely on computer verification
for the claim that $F\neq 0$ [DCJ2]. We nevertheless give a somewhat more
nuanced account of the second claim, highlighting those steps in the
construction which require something more than direct manipulation.

We denote by $Q_{1}^{\theta}(X,Y)$ the image of $Q_{1}(X,Y)$ in
$\mathcal{S}[X,Y]$, and similarly for
$A_{1}^{\theta}(Y),B_{1}^{\theta}(Y),C_{1}^{\theta}(Y)$. Direct manipulation
shows on the one hand that

$Q_{1}^{\theta}(\Phi^{\upsilon}_{e_{1}},\Phi^{\tau}_{e_{1}})=0,$

and on the other hand that

$A^{\theta}_{1}(\Phi^{\tau}_{e_{1}})\cdot(\Phi^{\upsilon}_{e_{1}})^{2}+B^{\theta}_{1}(\Phi^{\tau}_{e_{1}})\cdot\Phi^{v}_{e_{1}}+C^{\theta}_{1}(\Phi^{\tau}_{e_{1}})=0.$

Let

$\Delta_{1}(Y)=B_{1}(Y)^{2}-4A_{1}(Y)C_{1}(Y),$

denote by $\mathcal{A}[Y,\delta_{1}]$ the $\mathcal{A}[Y]$-algebra

$\mathcal{A}[Y,\delta_{1}]:=\mathcal{A}[Y,t]/(t^{2}-\Delta_{1}(Y))$

and denote by $\delta_{1}$ the equivalence class of $t$; denote by
$\mathcal{S}[Y,\delta_{1}^{\theta}]$ the $\mathcal{S}[Y]$-algebra

$\mathcal{S}[Y,\delta^{\theta}_{1}]:=\mathcal{S}[Y,u]/(u^{2}-\Delta^{\theta}_{1}(Y)),$

and let $\delta_{1}^{\theta}$ denote the equivalence class of $u$. We denote
the induced homomorphism

$\mathcal{S}[Y,\delta_{1}^{\theta}]\leftarrow\mathcal{A}[Y,\delta_{1}]$

as well as the homomorphism of localizations

$\mathcal{S}[Y,\delta_{1}^{\theta},{A_{1}^{\theta}}^{-1}]\leftarrow\mathcal{A}[Y,\delta_{1},A_{1}^{-1}]$

simply by $\theta$. We sometimes write ‘$\delta_{1}(Y)$’ in place of
‘$\delta_{1}$’ in order to emphasize that it’s contained in an algebra over a
polynomial algebra in $Y$ and can be specialized to particular values of $Y$
in any $\mathcal{A}$-algebra. We then denote by $\delta_{1}^{\theta}(?)$ the
specialization of $\delta_{1}^{\theta}$ at $Y=?$. Similarly, we sometimes
write ‘$\delta_{1}^{\theta}(Y)$’ and ‘$\delta_{1}^{\theta}(?)$’. In this
notation, we have the equation

$\left(\Phi^{\upsilon}_{e_{1}}-\frac{-B^{\theta}_{1}(\Phi^{\tau}_{e_{1}})+\delta_{1}^{\theta}(\Phi^{\tau}_{e_{1}})}{2A_{1}^{\theta}(\Phi^{\tau}_{e_{1}})}\right)\left(\Phi^{\upsilon}_{e_{1}}-\frac{-B_{1}^{\theta}(\Phi^{\tau}_{e_{1}})-\delta_{1}^{\theta}(\Phi^{\tau}_{e_{1}})}{2A_{1}^{\theta}(\Phi^{\tau}_{e_{1}})}\right)=0$

in the ring

$\mathcal{S}[\delta_{1}^{\theta}(\Phi^{\tau}_{e_{1}}),{A_{1}^{\theta}}(\Phi^{\tau}_{e_{1}})^{-1}]=\mathcal{S}_{A_{1}^{\theta}(\Phi^{\tau}_{e_{1}})}[s]/(s^{2}-\Delta^{\theta}_{1}(\Phi^{\tau}_{e_{1}}))$

where the subscript denotes localization and
$\delta_{1}^{\theta}(\Phi^{\tau}_{e_{1}})$ corresponds to $s$.

This ring is integral. This follows from the following general fact. If $R$ is
an integral domain with function field $K$ and $f\in R[x]$ is monic and
irreducible over $K$, then $K[t]/(f)$ is again integral. Indeed,

$R[t]/(f)\leftarrow R$

is flat, so $K[t]/(f)\leftarrow R[t]/(f)$ is obtained from an injective map
via flat base-change. Since $K[t]/(f)$ is a field, it follows that $R[t]/(f)$
is integral.

Consequently, we have

$\Phi^{\upsilon}_{e_{1}}=\frac{-B^{\theta}_{1}(\Phi^{\tau}_{e_{1}})+\epsilon\delta_{1}^{\theta}(\Phi^{\tau}_{e_{1}})}{2A_{1}^{\theta}(\Phi^{\tau}_{e_{1}})}$

in the ring

$\mathcal{S}[\delta_{1}^{\theta}(\Phi^{\tau}_{e_{1}}),{A_{1}^{\theta}}(\Phi^{\tau}_{e_{1}})^{-1}]$

for some $\epsilon\in\\{1,-1\\}$.

Direct calculation in the ring $\mathcal{A}[Y,\delta_{1}(Y),A_{1}(Y)^{-1}]$
shows that we have

$Q_{1}\left(\frac{-B_{1}(Y)+\epsilon\delta_{1}(Y)}{2A_{1}(Y)},Y\right)=\frac{Q_{1,a}(Y)+\epsilon\delta_{1}(Y)Q_{1,b}(Y)}{(2A_{1}(Y))^{4}}.$

It follows that in $\mathcal{S}[A^{\theta}_{1}(\Phi^{\tau}_{e_{1}})^{-1}]$ we
have

$\displaystyle
P_{1}^{\theta}(\Phi^{\tau}_{e_{1}})=\big{(}Q_{1,a}^{\theta}(\Phi^{\tau}_{e_{1}})-\epsilon\delta_{1}^{\theta}(\Phi^{\tau}_{e_{1}})Q_{1,b}^{\theta}(\Phi^{\tau}_{e_{1}})\big{)}$
$\displaystyle\cdot\big{(}Q_{1,a}^{\theta}(\Phi^{\tau}_{e_{1}})+\epsilon\delta_{1}^{\theta}(\Phi^{\tau}_{e_{1}})Q_{1,b}^{\theta}(\Phi^{\tau}_{e_{1}})\big{)}$
$\displaystyle=(2A_{1}^{\theta}(\Phi^{\tau}_{e_{1}}))^{4}\big{(}Q_{1,a}^{\theta}(\Phi^{\tau}_{e_{1}})-\epsilon\delta_{1}^{\theta}(\Phi^{\tau}_{e_{1}})Q_{1,b}^{\theta}(\Phi^{\tau}_{e_{1}})\big{)}$
$\displaystyle\cdot
Q_{1}^{\theta}\left(\frac{-B_{1}(\Phi^{\tau}_{e_{1}})+\epsilon\delta_{1}(\Phi^{\tau}_{e_{1}})}{2A_{1}(\Phi^{\tau}_{e_{1}})},\Phi^{\tau}_{e_{1}}\right)$
$\displaystyle=(2A_{1}^{\theta}(\Phi^{\tau}_{e_{1}}))^{4}\big{(}Q_{1,a}^{\theta}(\Phi^{\tau}_{e_{1}})-\epsilon\delta_{1}^{\theta}(\Phi^{\tau}_{e_{1}})Q_{1,b}^{\theta}(\Phi^{\tau}_{e_{1}})\big{)}$
$\displaystyle\cdot
Q^{\theta}_{1}(\Phi^{\upsilon}_{e_{1}},\Phi^{\tau}_{e_{1}})$
$\displaystyle=0.$

Since the localization map
$\mathcal{S}\to\mathcal{S}[A^{\theta}_{1}(\Phi^{\tau}_{e_{1}})^{-1}]$ is
injective, it follows that

$P_{1}^{\theta}(\Phi^{\tau}_{e_{1}})=0$

already in $\mathcal{S}$. Since $\mathcal{S}$ is integral, it follows also
that

$p_{1}^{\theta}(\Phi^{\tau}_{e_{1}})=0$

Similarly, we have $p_{2}^{\theta}(\Phi^{\tau}_{e_{2}})=0,$ and

$p_{3}^{\theta}(\Phi^{\tau}_{e_{1}}+\Phi^{\tau}_{e_{2}})=0.$

Consequently, $\theta(F)=0,$ as claimed. ∎

##  5\. The arithmetic step

Throughout this section, for $Z$ an open subscheme of
$\operatorname{Spec}(\mathbb{Z})$, we let
$\pi_{1}^{\mathrm{MT}}(Z)={\mathbb{G}_{m}}\ltimes\pi_{1}^{\mathrm{un}}(Z)$
denote the lexicographic mixed Tate Galois group of $Z$. We will focus
primarily on our special case $Z=\operatorname{Spec}(\mathbb{Z}[1/6])$, yet
along the way will have occasion to make statements that hold equally for
arbitrary $Z$. 272727temp: We’ve clarified that we do indeed go back and forth
between the general case and the particular case, and we’ve endeavored to
signal these changes more clearly below. We also let
$X=\mathbb{P}^{1}\setminus\\{0,1,\infty\\}$ and we let
$\pi_{1}^{\mathrm{un}}(X,{1_{0}})$ denote the lexicographic unipotent
fundamental group of $X$. We let $\pi^{\mathrm{PL}}(X)$ denote its
polylogarithmic quotient. We let $\mathcal{U}(Z)$ denote the completed
universal enveloping algebra of $\pi_{1}^{\mathrm{un}}(Z)$. In this section we
recall from [CDC1] the construction of generators
$\tau_{2},\tau_{3},\sigma\in\mathcal{U}(\mathbb{Z}[1/6])$ and write the
ensuing shuffle coordinates as polynomials in unipotent motivic
$n$-logarithms. We find it helpful to have several different notations
available: we denote the generators of half-weight $-1$ by $\tau_{2},\tau_{3}$
when we wish to emphasize the associated primes, by $\tau,\upsilon$ when we
wish instead to lighten the notation, and simply by $2,3$ when we wish to
lighten notation while nevertheless emphasizing the associated primes
(especially when words in the generators occur as subscripts).

### 5.1.

Recall that the de Rham realization $\pi_{1}^{\mathrm{un}}(X,1_{0})^{\rm{dR}}$
of the unipotent fundamental group of $X$ is free prounipotent on two
generators, determined by the choice of 1-forms

$\frac{dt}{t},\quad\frac{dt}{1-t}$

which define a basis of $H_{\rm{dR}}^{1}(X_{\mathbb{Q}})$282828temp: changed
to indicate $H_{\rm{dR}}^{1}(X_{\mathbb{Q}})$ instead of
$H^{\rm{dR}}_{1}(X_{\mathbb{Q}})$ and the fact that the generators of the
fundamental group depend on the 1-forms and not just their classes. We here
denote the corresponding generators by

$d_{0},\quad d_{1}.$

Moreover, the torsor $\pi_{1}^{\mathrm{un}}(X,1_{0},a)^{\rm{dR}}$ of unipotent
de Rham paths from $1_{0}$ to $a$ is canonically trivialized by a special “de
Rham” path which we denote by $p^{\rm{dR}}$. In our lexicographic ordering,
the motivic polylogarithm $\operatorname{Li}^{\mathfrak{u}}_{n}(a)$ for $a\in
X(Z)$ and $n\geq 1$ is defined to be the composite 292929temp: Added
definition of $o(p^{\rm{dR}})$.

$\pi_{1}^{\mathrm{un}}(Z)\xrightarrow{o(p^{\rm{dR}})}\pi_{1}^{\mathrm{un}}(X,1_{0},a)\xrightarrow{f_{d_{0}^{n-1}d_{1}}}\mathbb{A}^{1}_{\mathbb{Q}}$

where $o(p^{\rm{dR}})$ denotes the orbit map

$\gamma\mapsto\gamma p^{\rm{dR}}.$

The motivic logarithm $\log^{\mathfrak{u}}(a)$ is defined similarly with
$f_{d_{0}}$ in place of $f_{d_{0}^{n-1}d_{1}}$.303030temp: added definition of
$\log^{\mathfrak{u}}(a)$.

### 5.2.

We let $A(Z)=\mathcal{O}(\pi_{1}^{\mathrm{un}}(Z))$. Recall that there’s a
canonical isomorphism of $\mathbb{Q}$-vector spaces

$A(Z)_{n}=\mathcal{U}(Z)^{\lor}_{-n}.$

Recall that $A(Z)_{1}$ has basis $\log^{\mathfrak{u}}(q)$ for $q\notin Z$.
313131temp: clarified the difference between Z a general open in
Spec($\mathbb{Z}$) and $Z=\operatorname{Spec}(\mathbb{Z}[1/6]$, by saying
“from now on we take…” From now on we take
$Z=\operatorname{Spec}\mathbb{Z}[1/6]$. In this case a basis of $A(Z)_{1}$ is
given by $\log^{\mathfrak{u}}(2),\log^{\mathfrak{u}}(3)$. We define
$\tau_{q}\in\mathcal{U}(Z)_{-1}$ to be the dual of $\log^{\mathfrak{u}}(q)$
with respect to this basis.

### 5.3.

We let $E(Z)_{n}\subset A(Z)_{n}$ denote the space of extensions and we let
$D(Z)_{n}\subset A(Z)_{n}$ denote the space of decomposables. According to
Proposition 4.7 of [CDC1], the elements
$\operatorname{Li}^{\mathfrak{u}}_{3}(-2)$,
$\operatorname{Li}^{\mathfrak{u}}_{3}(3)$ span a subspace $P(Z)_{3}$ of
$A(\mathbb{Z}[1/6])_{3}$ complementary to $E(Z)_{3}+D(Z)_{3}$ 323232temp:
added (Z) for homogeneity of notations. Based on this arbitrary choice, we let
$\sigma\in\mathcal{U}(\mathbb{Z}[1/6])_{-3}$ be the unique element which pairs
with $P(Z)_{3}+D(Z)_{3}$ to zero and pairs with $\zeta^{\mathfrak{u}}(3)$ to
$1$.

### 5.4.

The $\mathbb{Q}$-vector space $\mathcal{U}(\mathbb{Z}[1/6])_{-i}$ for
$i=1,2,3,4$ has a vector space basis consisting of associative words of half-
weight $-i$ in the generators $\\{\tau_{2},\tau_{3},\sigma\\}$. If $w$ is such
a word, we define $f_{w}\in A(\mathbb{Z}[1/6])_{i}$ to be the function dual to
$w$ with respect to this basis. The choice of ordering
$\tau_{2}<\tau_{3}<\sigma$ gives rise to a set of Lyndon words whose
duals333333temp: changed $\tau,\upsilon$ to $\tau_{2},\tau_{3}$

$\displaystyle f_{\tau_{2}},f_{\tau_{3}}$ $\displaystyle f_{\tau_{2}\tau_{3}}$
$\displaystyle
f_{\sigma},f_{\tau_{2}\tau_{2}\tau_{3}},f_{\tau_{2}\tau_{3}\tau_{3}}$
$\displaystyle
f_{\tau_{2}\sigma},f_{\upsilon\sigma},f_{\tau_{2}\tau_{2}\tau_{2}\tau_{3}},f_{\tau_{2}\tau_{2}\tau_{3}\tau_{3}},f_{\tau_{2}\tau_{3}\tau_{3}\tau_{3}}$

form an algebra basis of the subalgebra $A(Z)_{[\leq 4]}$ of $A(Z)$ generated
by elements of degree $\leq 4$. 343434temp: added definition of $A(Z)_{[\leq
4]}$. We refer to these as _shuffle coordinates_ on
$\pi_{1}^{\mathrm{un}}(Z)_{\geq-4}$.

### 5.5.

Set

$\displaystyle\mathcal{E}_{1}=\\{\log^{\mathfrak{u}}(2),\log^{\mathfrak{u}}(3)\\}$
$\displaystyle\mathcal{P}_{1}=\emptyset$
$\displaystyle\mathcal{E}_{2}=\emptyset$
$\displaystyle\mathcal{P}_{2}=\\{\operatorname{Li}^{\mathfrak{u}}_{2}(-2)\\}$
$\displaystyle\mathcal{E}_{3}=\\{\zeta^{\mathfrak{u}}(3)\\}$
$\displaystyle\mathcal{P}_{3}=\\{\operatorname{Li}^{\mathfrak{u}}_{3}(-2),\operatorname{Li}^{\mathfrak{u}}_{3}(3)\\}$
$\displaystyle\mathcal{E}=\bigcup_{i=1}^{3}\mathcal{E}_{i}$
$\displaystyle\mathcal{P}=\bigcup_{i=1}^{3}\mathcal{P}_{i}.$

Then $\mathcal{E}\cup\mathcal{P}$ forms a second algebra basis of $A(Z)_{[\leq
3]}$. We refer to its elements as _polylogarithmic coordinates_.

###  5.6. Remarks concerning functoriality

We wish to import computations carried out for
$A(\operatorname{Spec}\mathbb{Z}[1/2])$ to
$A(\operatorname{Spec}\mathbb{Z}[1/6])$. For this purpose, we temporarily
allow $Z$ to vary among the open subschemes of
$\operatorname{Spec}\mathbb{Z}$. The structures discussed above
($\pi_{1}^{\mathrm{un}}(Z)$, $\mathcal{U}(Z)_{-n}$, $A(Z)_{n}$, $E(Z)_{n}$,
$D(Z)_{n}$) are functorial with respect to $Z$. An inclusion
$\iota:Z^{\prime}\subset Z$ of open subschemes of
$\operatorname{Spec}\mathbb{Z}$ (corresponding to an inclusion of finite sets
of primes $S^{\prime}\supset S$) gives rise to a surjection

$\iota_{*}:\pi_{1}^{\mathrm{un}}(Z^{\prime})\twoheadrightarrow\pi_{1}^{\mathrm{un}}(Z)$

and an injection

$\iota^{\sharp}:A(Z^{\prime})\supset A(Z).$

In terms of any choice of homogeneous free generators of
$\pi_{1}^{\mathrm{un}}(Z^{\prime})$, (*) corresponds to the quotient by the
normal subgroup generated by $\tau_{q}$ for $q\in Z\setminus Z^{\prime}$, and
so, $A(Z)$ is the corresponding shuffle subalgebra. In particular, a set of
generators $\Sigma^{\prime}$ for $\pi_{1}^{\mathrm{un}}(Z^{\prime})$ gives
rise to a set of generators $\Sigma$ for $\pi^{\mathrm{un}}_{1}(Z)$. If $\rho$
is a generator such that $\iota_{*}\rho\neq 0$, we denote $\iota_{*}\rho$
again by $\rho$. With this notational convention, $\Sigma$ is obtained from
$\Sigma^{\prime}$ simply by removing the generators $\tau_{q}$, $q\in
Z\setminus Z^{\prime}$, and $\iota^{\sharp}(f_{w})$ (for $w$ any word in the
generators $\Sigma$) is equal to $f_{w}$.

### 5.7.

We return to the case $Z=\operatorname{Spec}\mathbb{Z}[1/6]$. In view of the
remarks concerning functoriality (5.6), the generators $\tau_{2},\sigma$ of
$\pi_{1}^{\mathrm{un}}(\mathbb{Z}[1/6])$ may be viewed as generators also of
$\pi_{1}^{\mathrm{un}}(\mathbb{Z}[1/2])$. The $\mathbb{Q}$-vector space
$A(\mathbb{Z}[1/2])_{3}$ is spanned by the two subspaces

$E(\mathbb{Z}[1/2])_{3}\quad\mbox{and}\quad D(\mathbb{Z}[1/2])_{3}.$

Thus, as an element of $\pi_{1}^{\mathrm{un}}(\mathbb{Z}[1/2])$, $\sigma$ may
be characterized as the unique element of $\mathcal{U}(\mathbb{Z}[1/2])_{-3}$
which pairs trivially with $D(\mathbb{Z}[1/2])_{3}$ and pairs to $1$ with
$\zeta^{\mathfrak{u}}(3)$. In particular, it does not depend on any arbitrary
choices.

######  Proposition 5.8.

In the situation and the notation of segments 5.2-5.7, we have

$\displaystyle f_{\tau_{q}}$
$\displaystyle=\log^{\mathfrak{u}}(q)\quad(q=2,3),$ $\displaystyle
f_{\sigma\tau_{2}}$
$\displaystyle=-\frac{7}{8}\left(\frac{\log^{\mathfrak{u}}(2)^{4}}{24}+\operatorname{Li}^{\mathfrak{u}}_{4}(1/2)\right),$
$\displaystyle f_{\sigma\tau_{3}}$
$\displaystyle=\frac{3}{13}\left(6\operatorname{Li}^{\mathfrak{u}}_{4}(3)-\frac{1}{4}\operatorname{Li}^{\mathfrak{u}}_{4}(9)\right).$

###### Proof.

See section 4.3 of [CDC1]. The discussion of functoriality in segments 5.6,
5.7 above, shows that the second equation, which, as interpreted in loc. cit.,
takes place in $A(\mathbb{Z}[1/2])$, holds equally in $A(\mathbb{Z}[1/6])$,
with no conflict of notation. ∎

######  5.9.

Since

$\log^{\mathfrak{u}}(2)\zeta^{\mathfrak{u}}(3)=f_{\tau}\tiny{\Sha}f_{\sigma}=f_{\tau\sigma}+f_{\sigma\tau}$

and similarly for $f_{\upsilon}$ we obtain

$\displaystyle f_{\tau\sigma}$
$\displaystyle=\log^{\mathfrak{u}}(2)\zeta^{\mathfrak{u}}(3)+\frac{7}{8}\left(\frac{\log^{\mathfrak{u}}(2)^{4}}{24}+\operatorname{Li}^{\mathfrak{u}}_{4}(1/2)\right)$
$\displaystyle=(7/8)\operatorname{Li}^{\mathfrak{u}}_{4}(1/2)+(7/192)\log^{\mathfrak{u}}(2)^{4}+\log^{\mathfrak{u}}(2)\zeta^{\mathfrak{u}}(3)$

and

$\displaystyle f_{\upsilon\sigma}$
$\displaystyle=\log^{\mathfrak{u}}(3)\zeta^{\mathfrak{u}}(3)-\frac{3}{13}\left(6\operatorname{Li}^{\mathfrak{u}}_{4}(3)-\frac{1}{4}\operatorname{Li}^{\mathfrak{u}}_{4}(9)\right)$
$\displaystyle=\log^{\mathfrak{u}}(3)\zeta^{\mathfrak{u}}(3)-(18/13)\operatorname{Li}^{\mathfrak{u}}_{4}(3)+(3/52)\operatorname{Li}^{\mathfrak{u}}_{4}(9).$

######  Proposition 5.10.

Let $Z\subset\operatorname{Spec}\mathbb{Z}$ be an open subscheme with
complement

$S=(\operatorname{Spec}\mathbb{Z})\setminus Z.$

Let $\mathcal{U}(Z)$ denote the completed universal enveloping algebra of
$\pi_{1}^{\mathrm{un}}(Z)$. For each $q\in S$, let
$\tau_{q}\in\mathcal{U}(Z)_{-1}$ be the unique element such that
$\langle\log^{\mathfrak{u}}(q^{\prime}),\tau_{q}\rangle=1$ if $q=q^{\prime}$
and $0$ otherwise. For $n$ odd $\geq 3$, pick arbitrarily free generators
$\sigma_{n}\in\mathcal{U}(Z)_{-n}$ such that
$\langle\zeta^{\mathfrak{u}}(n),\sigma_{n}\rangle=1$. For a word $w$ in the
free generators $\tau_{q},\sigma_{n}$, let $f_{w}$ denote the element of
$A(Z)$ dual to $w$ with respect to the basis formed by such words. We denote
the $q$-adic valuation on $\mathbb{Q}$ associated to a prime $q$ by $v_{q}$.
Then, independently of the choice of generators $\sigma_{n}$,

$\log^{\mathfrak{u}}(q)=f_{\tau_{q}},$
$\zeta^{\mathfrak{u}}(n)=f_{\sigma_{n}},$

and for any $a\in(\mathbb{P}^{1}\setminus\\{0,1,\infty\\})(Z)$ and any $n\geq
1$,

$\displaystyle\operatorname{Li}^{\mathfrak{u}}_{n}(a)=\underset{q_{1},\dots,q_{n-r}\in
S}{\sum_{3\leq r\mbox{ odd}\leq
n}}\langle\operatorname{Li}^{\mathfrak{u}}_{r}(a),\sigma_{r}\rangle$
$\displaystyle v_{q_{1}}(a)\cdots
v_{q_{n-r}}(a)f_{\sigma_{r}\tau_{q_{1}}\cdots\tau_{q_{n-r}}}$
$\displaystyle-\sum_{q_{1},\dots,q_{n}\in S}v_{q_{1}}(1-a)v_{q_{2}}(a)\cdots
v_{q_{n}}(a)f_{\tau_{q_{1}}\cdots\tau_{q_{n}}}.$

###### Proof.

This is (a corrected version of) Remark 5.3 of [CDC2]. ∎

######  5.11.

We allow ourselves to replace words in $\tau_{2},\tau_{3}$ with words in
$2,3$. Applying Proposition 5.10, in half-weight 2 we obtain

$\displaystyle(\log^{\mathfrak{u}}2)^{2}$ $\displaystyle=2f_{22}$
$\displaystyle(\log^{\mathfrak{u}}2)(\log^{\mathfrak{u}}3)$
$\displaystyle=f_{23}+f_{32}$ $\displaystyle(\log^{\mathfrak{u}}3)^{2}$
$\displaystyle=2f_{33}$
$\displaystyle\operatorname{Li}^{\mathfrak{u}}_{2}(-2)$
$\displaystyle=-f_{32},$

hence,

$\displaystyle f_{22}$ $\displaystyle=\frac{1}{2}(\log^{\mathfrak{u}}2)^{2}$
$\displaystyle f_{23}$
$\displaystyle=(\log^{\mathfrak{u}}2)(\log^{\mathfrak{u}}3)+\operatorname{Li}^{\mathfrak{u}}_{2}(-2)$
$\displaystyle f_{32}$
$\displaystyle=-\operatorname{Li}^{\mathfrak{u}}_{2}(-2)$ $\displaystyle
f_{33}$ $\displaystyle=\frac{1}{2}(\log^{\mathfrak{u}}3)^{2}$

and in half-weight 3 we obtain,

$\displaystyle(\log^{\mathfrak{u}}2)^{3}$ $\displaystyle=6f_{222}$
$\displaystyle(\log^{\mathfrak{u}}2)^{2}(\log^{\mathfrak{u}}3)$
$\displaystyle=2(f_{223}+f_{232}+f_{322})$
$\displaystyle(\log^{\mathfrak{u}}2)(\log^{\mathfrak{u}}3)^{2}$
$\displaystyle=2(f_{233}+f_{323}+f_{332})$
$\displaystyle(\log^{\mathfrak{u}}3)^{3}$ $\displaystyle=6f_{333}$
$\displaystyle(\log^{\mathfrak{u}}2)\operatorname{Li}^{\mathfrak{u}}_{2}(-2)$
$\displaystyle=-(f_{232}+2f_{322})$
$\displaystyle(\log^{\mathfrak{u}}3)\operatorname{Li}^{\mathfrak{u}}_{2}(-2)$
$\displaystyle=-(2f_{332}+f_{323})$
$\displaystyle\operatorname{Li}^{\mathfrak{u}}_{3}(-2)$
$\displaystyle=-f_{322}$
$\displaystyle\operatorname{Li}^{\mathfrak{u}}_{3}(3)$
$\displaystyle=-f_{233}$ $\displaystyle\zeta^{\mathfrak{u}}(3)$
$\displaystyle=f_{\sigma},$

hence

$f_{223}=-\operatorname{Li}^{\mathfrak{u}}_{3}(-2)+(\log^{\mathfrak{u}}2)\operatorname{Li}^{\mathfrak{u}}_{2}(-2)+\frac{1}{2}(\log^{\mathfrak{u}}2)^{2}\log^{\mathfrak{u}}3$
$f_{233}=-\operatorname{Li}^{\mathfrak{u}}_{3}(3).$

######  5.12.

In half weight 4, we expand only those polynomials in the polylogarithmic
coordinates needed to convert the remaining shuffle coordinates
$f_{2223},f_{2233},f_{2333}$. We find,

$\displaystyle(\log^{\mathfrak{u}}2)^{3}(\log^{\mathfrak{u}}3)$
$\displaystyle=6(f_{2223}+f_{2232}+f_{2322}+f_{3222})$
$\displaystyle(\log^{\mathfrak{u}}2)^{2}\operatorname{Li}^{\mathfrak{u}}_{2}(-2)$
$\displaystyle=-2(f_{2232}+2f_{2322}+3f_{3222})$
$\displaystyle(\log^{\mathfrak{u}}2)\operatorname{Li}^{\mathfrak{u}}_{3}(-2)$
$\displaystyle=-(f_{2322}+3f_{3222}).$

We are able to eliminate $f_{3222}$ with the help of

$\displaystyle\operatorname{Li}^{\mathfrak{u}}_{4}(-2)$
$\displaystyle=\langle\operatorname{Li}^{\mathfrak{u}}_{3}(-2),\sigma\rangle
v_{2}(-2)f_{\sigma 2}-v_{3}(3)v_{2}(-2)^{3}f_{3222}$
$\displaystyle=-f_{3222},$

to obtain

$\displaystyle f_{2223}=\operatorname{Li}^{\mathfrak{u}}_{4}(-2)$
$\displaystyle-(\log^{\mathfrak{u}}2)\operatorname{Li}^{\mathfrak{u}}_{3}(-2)$
$\displaystyle+\frac{1}{2}(\log^{\mathfrak{u}}2)^{2}\operatorname{Li}^{\mathfrak{u}}_{2}(-2)+\frac{1}{6}(\log^{\mathfrak{u}}2)^{3}(\log^{\mathfrak{u}}3).$

######  5.13.

We have

$\displaystyle X(\mathbb{Z}[1/6])=\\{2,\frac{1}{2},-1\\}$
$\displaystyle\cup\\{3,\frac{1}{3},\frac{2}{3},\frac{3}{2},-\frac{1}{2},-2\\}$
$\displaystyle\cup\\{4,\frac{1}{4},\frac{4}{3},\frac{3}{4},-\frac{1}{3},-3\\}$
$\displaystyle\cup\\{-\frac{1}{8},\frac{1}{9},\frac{9}{8},\frac{8}{9},9,-8\\}$

divided into $S_{3}$-orbits. We have for any $a\in X(Z)$ (or more generally,
any ${\mathbb{G}_{m}}$-equivariant cocycle)

$\Delta^{\prime}\operatorname{Li}^{\mathfrak{u}}_{n}(a)=\sum_{i=1}^{n-1}\operatorname{Li}^{\mathfrak{u}}_{n-i}(a)\otimes\frac{(\log^{\mathfrak{u}}(a))^{i}}{i!}.$

Let $\Delta^{\prime}_{3}$ denote the reduced coproduct

$A_{3}(Z)\to A_{1}(Z)\otimes A_{2}(Z)\oplus A_{2}(Z)\otimes A_{1}(Z)$

and let $\Delta^{\prime}_{1,2}$ denote its composite with the projection onto
the factor $A_{1}(Z)\otimes A_{2}(Z)$. We recall that $E_{i}(Z)\subset
A_{i}(Z)$ denotes the space of extensions
$\operatorname{Ext}^{1}_{Z}\big{(}\mathbb{Q}(0),\mathbb{Q}(i)\big{)}$, equal
to the kernel of the reduced coproduct. We recall from Corollary 4.4 of [CDC1]
that

$\ker(\Delta^{\prime}_{1,2})=\ker(\Delta^{\prime}_{3})=E_{3}=\mathbb{Q}\zeta^{\mathfrak{u}}(3).$

######  5.14.

We record shuffle decompositions of decomposables.

$\displaystyle(\log^{\mathfrak{u}}2)^{2}(\log^{\mathfrak{u}}3)^{2}$
$\displaystyle=4f_{2233}+4f_{2323}+4f_{3223}+4f_{2332}+4f_{3232}+4f_{3322}$
$\displaystyle(\log^{\mathfrak{u}}2)(\log^{\mathfrak{u}}3)\operatorname{Li}^{\mathfrak{u}}_{2}(-2)$
$\displaystyle=-2f_{2332}-3f_{3232}-4f_{3322}-f_{2323}-2f_{3223}$
$\displaystyle(\log^{\mathfrak{u}}3)\operatorname{Li}^{\mathfrak{u}}_{3}(-2)$
$\displaystyle=-2f_{3322}-f_{3232}-f_{3223}$
$\displaystyle(\log^{\mathfrak{u}}2)\operatorname{Li}^{\mathfrak{u}}_{3}(3)$
$\displaystyle=-2f_{2233}-f_{2323}-f_{2332}$

We note the following relation between $f_{2233}$ and $f_{3322}$:

$(1/4)(\log^{\mathfrak{u}}2)^{2}(\log^{\mathfrak{u}}3)^{2}+(\log^{\mathfrak{u}}3)\operatorname{Li}^{\mathfrak{u}}_{3}(-2)+(\log^{\mathfrak{u}}2)\operatorname{Li}_{3}(3)=-f_{2233}-f_{3322}$

######  5.15.

We record expansions of $\Delta^{\prime}(L)$ for $L$ in the polylogarithmic
basis
$\\{(\log^{\mathfrak{u}}2)^{2},(\log^{\mathfrak{u}}2)(\log^{\mathfrak{u}}3),(\log^{\mathfrak{u}}3)^{2},\operatorname{Li}^{\mathfrak{u}}_{2}(-2)\\}$
of $A_{2}$ in the basis for $A_{1}$ in a table. We also include
$\Delta^{\prime}(\operatorname{Li}^{\mathfrak{u}}_{2}(3))$ and
$\Delta^{\prime}(\operatorname{Li}^{\mathfrak{u}}_{2}(\frac{2}{3}))$. We use
the abbreviations $l$ for $\log^{\mathfrak{u}}$ and $L$ for
$\operatorname{Li}^{\mathfrak{u}}$.

$\begin{matrix}&l(2)^{2}&l(2)l(3)&l(3)^{2}&L_{2}(-2)&&L_{2}(\frac{2}{3})&\operatorname{Li}^{\mathfrak{u}}_{2}(3)\\\
l(2)\otimes l(2)&2&0&0&0&|&0&0\\\ l(2)\otimes l(3)&0&1&0&0&|&0&-1\\\
l(3)\otimes l(2)&0&1&0&-1&|&1&0\\\ l(3)\otimes
l(3)&0&0&2&0&|&-1&0\end{matrix}$

Using $\Delta^{\prime}:A_{2}\xrightarrow{\sim}A_{1}\otimes A_{1}$, and the
above table, we find

$\displaystyle\operatorname{Li}^{\mathfrak{u}}_{2}(\frac{2}{3})$
$\displaystyle=-\frac{1}{2}(\log^{\mathfrak{u}}3)^{2}-\operatorname{Li}^{\mathfrak{u}}_{2}(-2)$
$\displaystyle\operatorname{Li}^{\mathfrak{u}}_{2}(3)$
$\displaystyle=-\log^{\mathfrak{u}}(2)\log^{\mathfrak{u}}(3)-\operatorname{Li}^{\mathfrak{u}}_{2}(-2).$

######  5.16.

We write $\operatorname{Li}^{\mathfrak{u}}_{3}(\frac{2}{3})$ as a polynomial
in $\mathcal{E}\cup\mathcal{P}$. For each polylogarithmic basis element in
$A_{3}(Z)$, we expand $\Delta_{2,1}(L)\in A_{2}(Z)\otimes A_{1}(Z)$ in the
basis induced by our polylogarithmic basis for $A_{1}(Z)$ and $A_{2}(Z)$. We
record the result in a matrix along with the expansion of
$\Delta_{2,1}(\operatorname{Li}^{\mathfrak{u}}_{3}(\frac{2}{3}))$ in the
rightmost column.

$\tiny\begin{matrix}&l(2)^{3}&l(2)^{2}l(3)&l(2)l(3)^{2}&l(3)^{3}&l(2)L_{2}(-2)&l(3)L_{2}(-2)&L_{3}(-2)&L_{3}(3)&L_{3}(\frac{2}{3})\\\
l(2)^{2}\otimes l(2)&3&0&0&0&0&0&0&0&0\\\ l(2)^{2}\otimes
l(3)&0&1&0&0&0&0&0&0&0\\\ l(2)l(3)\otimes l(2)&0&2&0&0&-1&0&0&0&0\\\
l(2)l(3)\otimes l(3)&0&0&2&0&0&0&0&-1&0\\\ l(3)^{2}\otimes
l(2)&0&0&1&0&0&-1&0&0&-\frac{1}{2}\\\ l(3)^{2}\otimes
l(3)&0&0&0&3&0&0&0&0&\frac{1}{2}\\\ L_{2}(-2)\otimes
l(2)&0&0&0&0&0&0&1&0&-1\\\ L_{2}(-2)\otimes
l(3)&0&0&0&0&0&1&0&-1&1\end{matrix}$

From this and the exact sequence

$0\to E_{3}(Z)\to A_{3}(Z)\xrightarrow{\Delta_{2,1}}A_{2}(Z)\otimes
A_{1}(Z)\to 0$

we find that

$\operatorname{Li}^{\mathfrak{u}}_{3}(\frac{2}{3})\equiv-\frac{1}{2}\log^{\mathfrak{u}}(2)\log^{\mathfrak{u}}(3)^{2}+\frac{1}{6}(\log^{\mathfrak{u}}3)^{3}-\operatorname{Li}^{\mathfrak{u}}_{3}(-2)-\operatorname{Li}^{\mathfrak{u}}_{3}(3)\mod\zeta^{\mathfrak{u}}(3).$

The Sage code

    Q = Qp(13)

    def l(z):
         return Q(z).log()

    def Li(n,z):
         return Q(z).polylog(n)

    def zeta(n):
         return 2^(n-1)*Li(n,-1)/(1-2^(n-1))

    q = (Li(3,2/3) + (1/2)*l(2)*l(3)^2 \
    - (1/6)*l(3)^3 + Li(3,-2) + Li(3,3))/zeta(3)

    r = q.rational_reconstruction()
    print(r)

outputs the number $1$. Hence, at least up to the chosen precision, we have

$\operatorname{Li}^{\mathfrak{u}}_{3}(\frac{2}{3})=-\frac{1}{2}\log^{\mathfrak{u}}(2)\log^{\mathfrak{u}}(3)^{2}+\frac{1}{6}(\log^{\mathfrak{u}}3)^{3}-\operatorname{Li}^{\mathfrak{u}}_{3}(-2)-\operatorname{Li}^{\mathfrak{u}}_{3}(3)+\zeta^{\mathfrak{u}}(3).$

######  5.17.

We apply Proposition 5.8 to $\operatorname{Li}_{4}(2/3)$ and
$\operatorname{Li}_{4}(4/3)$ :

(4)
$\operatorname{Li}_{4}(2/3)=\langle\operatorname{Li}_{3}(2/3),\sigma_{3}\rangle(f_{\sigma_{3}\tau_{2}}-f_{\sigma_{3}\tau_{3}})\\\
+f_{3222}-(f_{3322}+f_{3232}+f_{3223})+(f_{3233}+f_{3323}+f_{3332})-f_{3333}$
(5)
$\operatorname{Li}_{4}(4/3)=\langle\operatorname{Li}_{3}(4/3),\sigma_{3}\rangle(2f_{\sigma_{3}\tau_{2}}-f_{\sigma_{3}\tau_{3}})\\\
+8f_{3222}-4(f_{3322}+f_{3232}+f_{3223})+2(f_{3233}+f_{3323}+f_{3332})-f_{3333}$

We have

(6) $f_{3233}+f_{3323}+f_{3332}=f_{2}f_{333}-f_{2333}.$

We have

(7) $\log(3)\operatorname{Li}_{3}(-2)=-f_{3322}-(f_{3322}+f_{3232}+f_{3223}).$

### 5.18.

By (6) and (7), we can regard (4) and (5) as a linear system of equations in
$(f_{3322},f_{2333})$:

$\begin{array}[]{l}\operatorname{Li}_{4}(2/3)-\langle\operatorname{Li}_{3}(2/3),\sigma_{3}\rangle(f_{\sigma_{3}\tau_{2}}-f_{\sigma_{3}\tau_{3}})-f_{3222}-\log(3)\operatorname{Li}_{3}(-2)-f_{2}f_{333}+f_{3333}\\\
=f_{3322}-f_{2333}\\\
\operatorname{Li}_{4}(4/3)-\langle\operatorname{Li}_{3}(4/3),\sigma_{3}\rangle(2f_{\sigma_{3}\tau_{2}}-f_{\sigma_{3}\tau_{3}})-8f_{3222}-4\log(3)\operatorname{Li}_{3}(-2)-2f_{2}f_{333}+f_{3333}\\\
=4f_{3322}-2f_{2333}\end{array}$

If we denote

$\operatorname{Li}_{4}(2/3)-\langle\operatorname{Li}_{3}(2/3),\sigma_{3}\rangle(f_{\sigma_{3}\tau_{2}}-f_{\sigma_{3}\tau_{3}})-f_{3222}-\log(3)\operatorname{Li}_{3}(-2)-f_{2}f_{333}+f_{3333}$

by $E$ and

$\displaystyle\operatorname{Li}_{4}(4/3)-\langle\operatorname{Li}_{3}(4/3),\sigma_{3}\rangle(2f_{\sigma_{3}\tau_{2}}$
$\displaystyle-f_{\sigma_{3}\tau_{3}})-8f_{3222}$
$\displaystyle-4\log(3)\operatorname{Li}_{3}(-2)-2f_{2}f_{333}+f_{3333}$

by $F$, then we find that

$\left\\{\begin{array}[]{l}f_{3322}=-E+\frac{F}{2}\\\
f_{2333}=-\frac{E}{2}+\frac{F}{2}.\end{array}\right.$

The shuffle coordinates appearing in the above expressions for $E$ and $F$
have all been expanded in motivic polylogarithms above. It remains to compute
the coefficient
$\langle\operatorname{Li}^{\mathfrak{u}}_{3}(4/3),\sigma_{3}\rangle$.

### 5.19.

We have

$\Delta\operatorname{Li}_{3}(4/3)=\operatorname{Li}_{2}(4/3)\otimes\log(4/3)+\operatorname{Li}_{1}(4/3)\otimes\frac{1}{2}(\log(4/3))^{2}.$

By Proposition 5.10,

$\operatorname{Li}_{2}(4/3)=2f_{32}-f_{33}$

and

$f_{32}=-\operatorname{Li}_{2}(-2).$

Thus,

$\operatorname{Li}_{2}(4/3)=-2\operatorname{Li}_{2}(-2)-\frac{1}{2}(\log(3))^{2}.$

Additionally,

$\operatorname{Li}_{1}(4/3)=-\log(1-4/3)=-\log(-1/3)=-\log(-3)=-\log(3),$
$\log(4/3)=2\log(2)-\log(3),$
$\log(4/3)^{2}=4\log(2)^{2}-4\log(2)\log(3)+\log(3)^{2}.$

So

$\Delta\operatorname{Li}_{3}(4/3)=(-2\operatorname{Li}_{2}(-2)-\frac{1}{2}(\log(3))^{2})\otimes(2\log(2)-\log(3))\\\
+(-\log(3))\otimes(2\log(2)^{2}-2\log(2)\log(3)+\frac{1}{2}\log(3)^{2}).$

### 5.20.

We deduce

$\Delta_{2,1}\operatorname{Li}_{3}(4/3)=(-2\operatorname{Li}_{2}(-2)\otimes(2\log(2)+-\frac{1}{2}(\log(3))^{2})\otimes(2\log(2)\\\
+(-2\operatorname{Li}_{2}(-2)\otimes-\log(3))+-\frac{1}{2}(\log(3))^{2}\otimes-\log(3)\\\
=-4\operatorname{Li}_{2}(-2)\otimes\log(2)-\log(3)^{2}\otimes\log(2)\\\
+2\operatorname{Li}_{2}(-2)\otimes\log(3)+\frac{1}{2}(\log(3))^{2}\otimes\log(3).$

We use the table of §5.16 and the fact that $\ker\Delta_{2,1}$ is generated by
$\zeta(3)$. Let us denote by $A,B,C,D$ the four last lines of the table :

$A=\log(3)^{2}\otimes\log(2)$

$B=\log(3)^{2}\otimes\log(3)$

$C=\operatorname{Li}_{2}(-2)\otimes\log(2)$

$D=\operatorname{Li}_{2}(-2)\otimes\log(3)$

We have found

(8) $\Delta_{2,1}(\operatorname{Li}_{3}(4/3))=-A+\frac{1}{2}B-4C+2D.$

According to the table,

$\Delta_{2,1}(\log(2)\log(3)^{2}+2\operatorname{Li}_{3}(3))=A-2D$

$\Delta_{2,1}(\log(3)^{3})=3B$

$\Delta_{2,1}(\operatorname{Li}_{3}(-2))=C$

$\Delta_{2,1}(\log(3)\operatorname{Li}_{2}(-2))=-A+D.$

Thus

(9)
$\left\\{\begin{array}[]{l}\Delta_{2,1}(-\log(2)\log(3)^{2}-2\operatorname{Li}_{3}(3)+2\log(3)\operatorname{Li}_{2}(-2))=A\\\
\Delta_{2,1}(\frac{1}{3}\log(3)^{3})=B\\\
\Delta_{2,1}(\operatorname{Li}_{3}(-2))=C\\\
\Delta_{2,1}(-\log(2)\log(3)^{2}-2\operatorname{Li}_{3}(3)-\log(3)\operatorname{Li}_{2}(-2))=D\end{array}\right.$

### 5.21.

Define

$a=-\log(2)\log(3)^{2}-2\operatorname{Li}_{3}(3)-2\log(3)\operatorname{Li}_{2}(-2)$
$b=\log(2)\log(3)^{2}+2\operatorname{Li}_{3}(3)+\log(3)\operatorname{Li}_{2}(-2).$

Comparing (8) and (9) we deduce

$\operatorname{Li}_{3}(4/3)-(-(-\log(2)\log(3)^{2}-2\operatorname{Li}_{3}(3)-2\log(3)\operatorname{Li}_{2}(-2))\\\
+\frac{1}{6}\log(3)^{3}-4\operatorname{Li}_{3}(-2)+2(-\log(2)\log(3)^{2}-2\operatorname{Li}_{3}(3)-\log(3)\operatorname{Li}_{2}(-2)))\\\
=\operatorname{Li}_{3}(4/3)+\left(a-\frac{1}{6}\log(3)^{3}+4\operatorname{Li}_{3}(-2)+2b\right)\in\ker\Delta_{2,1}$

Thus the above element is a multiple of $\zeta(3)$. The coefficient is
precisely $\langle\operatorname{Li}_{3}(4/3),\sigma_{3}\rangle$. Computation
using a computer algebra system shows that the $p$-adic period of

$\langle\operatorname{Li}_{3}(4/3),\sigma_{3}\rangle=\frac{\operatorname{Li}_{3}(4/3)+\left(a-\frac{1}{6}\log(3)^{3}+4\operatorname{Li}_{3}(-2)+2b\right)}{\zeta(3)}$

is equal to $-\frac{1}{3}$ for several primes (in particular, up to high
$p$-adic precision).

### 5.22.

Assembling the previous computations, we have

$E=\operatorname{Li}_{4}(2/3)+\bigg{(}\frac{7}{8}\bigg{(}\frac{\log(2)^{4}}{24}+\operatorname{Li}_{4}(1/2)\bigg{)}+\frac{3}{13}\bigg{(}6\operatorname{Li}_{4}(3)-\frac{1}{4}\operatorname{Li}_{4}(9)\bigg{)}\bigg{)}\\\
+\operatorname{Li}_{4}(-2)-\log(3)\operatorname{Li}_{3}(-2)-\log(2)\frac{\log(3)^{3}}{3!}+\frac{\log(3)^{4}}{4!}$
$F=\operatorname{Li}_{4}(4/3)-(1/3)\bigg{(}\frac{7}{4}\bigg{(}\frac{\log(2)^{4}}{24}+\operatorname{Li}_{4}(1/2)\bigg{)}+\frac{3}{13}\bigg{(}6\operatorname{Li}_{4}(3)-\frac{1}{4}\operatorname{Li}_{4}(9)\bigg{)}\bigg{)}\\\
+8\operatorname{Li}_{4}(-2)-4\log(3)\operatorname{Li}_{3}(-2)-2\log(2)\frac{\log(3)^{3}}{3!}+\frac{\log(3)^{4}}{4!}$

### 5.23.

Summarizing the results of our computations, we have the following equalities
up to high precision. In longer equations we abbreviate
$l=\log^{\mathfrak{u}},L=\operatorname{Li}^{\mathfrak{u}}$.

$\displaystyle f_{\tau}=\log^{\mathfrak{u}}(2)$ $\displaystyle
f_{\upsilon}=\log^{\mathfrak{u}}(3)$ $\displaystyle
f_{\tau\upsilon}=(\log^{\mathfrak{u}}2)(\log^{\mathfrak{u}}3)+\operatorname{Li}^{\mathfrak{u}}_{2}(-2)$
$\displaystyle f_{\sigma}=\zeta^{\mathfrak{u}}(3)$ $\displaystyle
f_{\tau\tau\upsilon}=-\operatorname{Li}^{\mathfrak{u}}_{3}(-2)+(\log^{\mathfrak{u}}2)\operatorname{Li}^{\mathfrak{u}}_{2}(-2)+\frac{1}{2}(\log^{\mathfrak{u}}2)^{2}\log^{\mathfrak{u}}3$
$\displaystyle
f_{\tau\upsilon\upsilon}=-\operatorname{Li}^{\mathfrak{u}}_{3}(3)$
$\displaystyle
f_{\tau\sigma}=(7/8)\operatorname{Li}^{\mathfrak{u}}_{4}(1/2)+(7/192)\log^{\mathfrak{u}}(2)^{4}+\log^{\mathfrak{u}}(2)\zeta^{\mathfrak{u}}(3)$
$\displaystyle
f_{\upsilon\sigma}=\log^{\mathfrak{u}}(3)\zeta^{\mathfrak{u}}(3)-(18/13)\operatorname{Li}^{\mathfrak{u}}_{4}(3)+(3/52)\operatorname{Li}^{\mathfrak{u}}_{4}(9)$
$\displaystyle
f_{\tau\tau\tau\upsilon}=\operatorname{Li}^{\mathfrak{u}}_{4}(-2)-(\log^{\mathfrak{u}}2)\operatorname{Li}^{\mathfrak{u}}_{3}(-2)+(1/2)(\log^{\mathfrak{u}}2)^{2}\operatorname{Li}^{\mathfrak{u}}_{2}(-2)$
$\displaystyle+(1/6)(\log^{\mathfrak{u}}2)^{3}(\log^{\mathfrak{u}}3)$
$\displaystyle
f_{\tau\tau\upsilon\upsilon}=(7/144)l(2)^{4}-(1/4)l(2)^{2}l(3)^{2}+(1/48)l(3)^{4}$
$\displaystyle+2l(3)L_{3}(-2)+l2L_{3}(3)+L_{4}(2/3)$
$\displaystyle+(7/6)L_{4}(1/2)+(3/2)L_{4}(3)-(1/16)L_{4}(9)$
$\displaystyle-3L_{4}(-2)-(1/2)L_{4}(4/3)$ $\displaystyle
f_{\tau\upsilon\upsilon\upsilon}=-(35/1152)l(2)^{4}-(1/12)l(2)l(3)^{3}$
$\displaystyle-(3/2)l(3)L_{3}(-2)-(1/2)L_{4}(2/3)$
$\displaystyle-(35/48)L_{4}(1/2)-(12/13)L_{4}(3)$
$\displaystyle+(1/26)L_{4}(9)+(7/2)L_{4}(-2)+(1/2)L_{4}(4/3)$

##  6\. The polylogarithmic quotient

Throughout this section, Tannakian fundamental groups are endowed with the
usual _functorial_ product.

### 6.1.

The literature on motivic tangential base-points for unirational varieties of
dimension greater than 1 is not fully fleshed out. The theory is nevertheless
regarded as known, as it amounts to a fairly straightforward generalization of
the 1-dimensional case, complemented by techniques for bootstrapping to higher
dimensions in §4 of Deligne-Goncharov [DG]. $p$-Adic aspects are discussed in
Ünver [Ü1]. Here we provide an outline of the construction and verify that our
integrality conditions on tangential base-points ensure that the associated
fundamental groups are unramified. We begin with the $l$-adic realization.

### 6.2.

Let $Z$ be an open subscheme of $\operatorname{Spec}\mathbb{Z}$,
$\overline{X}\to Z$ a smooth proper morphism whose generic fiber is
unirational, $D\subset\overline{X}$ a relative simple normal crossings divisor
whose irreducible components are smooth and absolutely irreducible, and let
$X$ denote its complement in $\overline{X}$. By a $Z$-integral base-point we
mean either a section of $X\to Z$ or a nonvanishing $Z$-family of tangent
vectors along a stratum of $\overline{X}$ which are not tangent to any
boundary divisor.

353535Temp. Response to referee comment titled “Motivic fundamental groups at
tangential basepoints”: In segments 6.3–6.7 we’ve added an outline of the
construction.

### 6.3.

Let us be more explicit about our assumptions on a $Z$-integral tangent vector
$v$. We are provided with a $Z$-point

$x:Z\to\overline{X}$

of the compactification. Let

$T_{x}=\operatorname{Spec}\operatorname{Sym}^{*}\mathcal{T}_{x}^{\lor}$

denote the normal bundle to $x$ in $\overline{X}$. We use $\mathcal{O}_{Z}$
both for the structure sheaf and for the coordinate ring of $Z$. We may
equivalently think of $\mathcal{T}_{x}$ as a quasi-coherent sheaf on $Z$ or as
a module over the coordinate ring $\mathcal{O}_{Z}$, and we do not distinguish
between these notationally. In this notation,

$v\in\operatorname{Hom}_{Z}(Z,T_{x})=\mathcal{T}_{x}$

is a section which is nowhere tangent to the boundary divisors.

This last phrase may be interpreted in several equivalent ways; what we need
is the following. Let
$\hat{D}_{Z}=\operatorname{Spec}\mathcal{O}_{Z}\llbracket t\rrbracket$, let

$\hat{D}_{Z}^{\circ}=\operatorname{Spec}\mathcal{O}_{Z}((t)),$

and let $T_{0}$ denote the normal bundle to $\hat{D}$ along the zero section
$\\{t=0\\}$. Then there’s a map

$h:\hat{D}\to\bar{X}$

which maps the zero section $\\{t=0\\}$ to $x$, such that
$\hat{D}^{\circ}_{Z}$ maps to $X$ and such that the induced map of normal
bundles

$T_{0}\to T_{x}\quad\mbox{maps}\quad 1\mapsto v.$

Fix a prime $l$ of $\mathbb{Z}$. Let $\iota$ denote the natural map

$\hat{D}^{\circ}_{Z}\to\mathbb{A}^{1}_{Z}\setminus\\{0\\}.$

Pullback along $\iota$ induces an equivalence of categories of finite étale
coverings. Consequently, $\iota_{*}$ induces a monoidal equivalence of
categories of lisse $\mathbb{Q}_{l}$-sheaves.

We let $\operatorname{Lisse}_{\mathbb{Q}_{l}}(\dots)$ denote the category of
lisse sheaves. Let $g$ denote the map

$\operatorname{Spec}\overline{\mathbb{Q}}\to Z$

induced by the choice of an algebraic closure $\overline{\mathbb{Q}}$ of
$\mathbb{Q}$. The composite (diagonal solid arrow below)

$\textstyle{\operatorname{Lisse}_{\mathbb{Q}_{l}}(X)\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{h^{*}_{v}}$$\scriptstyle{v^{*}}$$\scriptstyle{\omega_{v}}$$\textstyle{\operatorname{Lisse}_{\mathbb{Q}_{l}}(\hat{D}^{\circ}_{Z})\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\iota_{*}}$$\textstyle{\operatorname{Lisse}_{\mathbb{Q}_{l}}(\mathbb{A}^{1}_{Z}\setminus\\{0\\})\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{1^{*}}$$\textstyle{\operatorname{Lisse}_{\mathbb{Q}_{l}}(Z)\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{g^{*}}$$\textstyle{\operatorname{\bf{Vect}}\mathbb{Q}_{l}}$

defines a “tangential” fiber functor on the category of Lisse
$\mathbb{Q}_{l}$-sheaves associated to the $Z$-integral tangent vector $v$. We
note the intermediate composite, denoted $v^{*}$, for future use. We also note
that the same construction defines tangential fiber functors over various
base-extensions of $X$ ($X_{{\mathbb{Q}_{p}}}$, $X_{\overline{\mathbb{Q}}}$,
…) and we continue to use the same notation $\omega_{v}$. A similar
construction at the level of Galois categories of finite étale coverings
provides us with a notion of tangential fiber functors for profinite étale
fundamental groups, and there’s an obvious compatibility between the two
constructions.

We claim that

$\pi_{1}^{l}(X_{\bar{\mathbb{Q}}},v)=\mathrm{Aut}\,^{\otimes}(\omega_{v})$

is unramified at $p$. By this we mean the following.

######  Proposition 6.4.

Fix arbitrarily an embedding
$\overline{\mathbb{Q}}\subset\overline{\mathbb{Q}}_{p}$. There’s an associated
decomposition group

$G_{\mathbb{Q}}\supset G_{p}=G_{\mathbb{Q}_{p}}.$

Then the induced action of $G_{p}$ on $\pi_{1}^{l}(X_{\bar{\mathbb{Q}}},v)$
factors through

$G_{p}\twoheadrightarrow G_{\mathbb{F}_{p}}=\hat{\mathbb{Z}}.$

###### Proof.

Our construction of the $l$-adic tangential fiber functor $\omega_{v}$ applies
equally over $Z_{p}=\operatorname{Spec}{\mathbb{Z}_{p}}$ and the verification
of the above statement may take place over $Z_{p}$. For this purpose we
temporarily replace $Z$ by $Z_{p}$. We let $f$ denote the structure morphism

$X\to Z_{p}.$

We define a lisse $\mathbb{Q}_{l}$-sheaf $\mathcal{F}$ on $X$ to be
_relatively unipotent_ if $\mathcal{F}$ admits a filtration by lisse
subsheaves such that

$\operatorname{gr}\mathcal{F}\simeq f^{*}\mathcal{G}$

for some lisse sheaf $\mathcal{G}$ on $Z_{p}$. We define _relatively lisse
sheaves on $X_{\mathbb{Q}_{p}}$_ similarly. We decorate “un” to indicate full
subcategories of unipotent objects and “run” to indicate full subcategories of
relatively unipotent objects. We have a diagram of $\mathbb{Q}_{l}$-Tannakian
categories

$\textstyle{\operatorname{Lisse}^{\mathrm{un}}_{\mathbb{Q}_{l}}X_{\overline{\mathbb{Q}}_{p}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{v^{*}}$$\textstyle{\operatorname{Lisse}^{\mathrm{run}}_{\mathbb{Q}_{l}}X_{\mathbb{Q}_{p}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{v^{*}}$$\textstyle{\operatorname{Lisse}^{\mathrm{run}}_{\mathbb{Q}_{l}}X\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{v^{*}}$$\textstyle{\operatorname{\bf{Vect}}\mathbb{Q}_{l}}$$\textstyle{\operatorname{Lisse}_{\mathbb{Q}_{l}}\operatorname{Spec}\overline{\mathbb{Q}}_{p}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\sim}$$\textstyle{\operatorname{Lisse}_{\mathbb{Q}_{l}}\operatorname{Spec}{\mathbb{Q}_{p}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\operatorname{Lisse}_{\mathbb{Q}_{l}}Z_{p},\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$

which is filled in by canonical natural $\otimes$-isomorphisms. The functor to
$\operatorname{\bf{Vect}}\mathbb{Q}_{l}$ endows each of the Tannakian
categories appearing in the diagram with a fiber functor which we use as base-
point for Tannakian fundamental groups and gives rise to a morphism of split
short exact sequences of Tannakian fundamental groups

$\textstyle{1\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\pi_{1}\big{(}\operatorname{Lisse}^{\mathrm{un}}_{\mathbb{Q}_{l}}X_{\overline{\mathbb{Q}}_{p}}\big{)}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\pi_{1}\big{(}\operatorname{Lisse}^{\mathrm{run}}_{\mathbb{Q}_{l}}X\big{)}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\pi_{1}\big{(}\operatorname{Lisse}_{\mathbb{Q}_{l}}Z_{p}\big{)}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{1}$$\textstyle{1\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\pi_{1}\big{(}\operatorname{Lisse}^{\mathrm{un}}_{\mathbb{Q}_{l}}X_{\overline{\mathbb{Q}}_{p}}\big{)}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\pi_{1}\big{(}\operatorname{Lisse}^{\mathrm{run}}_{\mathbb{Q}_{l}}X_{{\mathbb{Q}_{p}}}\big{)}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\pi_{1}\big{(}\operatorname{Lisse}_{\mathbb{Q}_{l}}\operatorname{Spec}{\mathbb{Q}_{p}}\big{)}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{1.}$

On the other hand, the natural transformation from profinite étale fundamental
groups to Tannakian $l$-adic fundamental groups provides a commuting square

$\textstyle{\pi_{1}(\operatorname{Lisse}_{\mathbb{Q}_{l}}Z_{p})(\mathbb{Q}_{l})}$$\textstyle{\pi_{1}^{\mathrm{\acute{e}t}}(Z_{p},a)\simeq\operatorname{Gal}(\overline{\mathbb{F}}_{p}/\mathbb{F}_{p})\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\pi_{1}(\operatorname{Lisse}_{\mathbb{Q}_{l}}\operatorname{Spec}{\mathbb{Q}_{p}})(\mathbb{Q}_{l})\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\pi_{1}^{\mathrm{\acute{e}t}}(\operatorname{Spec}{\mathbb{Q}_{p}},a)\simeq\operatorname{Gal}(\overline{\mathbb{Q}}_{p}/\mathbb{Q}_{p}).\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$

where $a$ denotes the base-point associated with our choice of algebraic
closure. This shows that the action of
$\operatorname{Gal}(\overline{\mathbb{Q}}_{p}/\mathbb{Q}_{p})$ on
$\pi_{1}\big{(}\operatorname{Lisse}^{\mathrm{un}}_{\mathbb{Q}_{l}}X_{\overline{\mathbb{Q}}_{p}}\big{)}$
factors through $\operatorname{Gal}(\overline{\mathbb{F}}_{p}/\mathbb{F}_{p})$
as claimed. ∎

### 6.5.

Similar constructions to the one outlined above provide Betti and de Rham
versions of the unipotent fundamental group at a tangential base-point; see,
for instance, §6.12 below. A mixed Hodge structure on the Betti unipotent
fundamental group at a tangential base-point is constructed in works of Hain
and collaborators. For instance, Definition 4.21(ii) of Hain-Zucker [HZ]
provides a structure of pro-variation of mixed Hodge structures on the bundle
whose fiber at a point $x$ is the prounipotent completion of the fundamental
group at $x$; restricting to an appropriate analytic disk and taking a limit
mixed Hodge structure, one obtains a mixed Hodge structure at a tangential
base-point.

Together with natural comparison isomorphisms, this provides a prounipotent
group object $\pi_{1}^{H+l}(X_{\mathbb{Q}},v)$ in the $\mathbb{Q}$-Tannakian
category $\mathcal{R}^{H+l}$ of systems of realizations of type $H+l$
considered in Deligne-Goncharov [DG]. The methods of §4 of loc. cit. provide a
unipotent group object $\pi_{1}^{\mathrm{un}}(X_{\mathbb{Q}},v)$ in the
Tannakian category of mixed Tate motives over $\operatorname{Spec}\mathbb{Q}$
which realizes to $\pi_{1}^{H+l}(X_{\mathbb{Q}},v)$. This is an elaboration on
Remark 4.14 of loc. cit.

By proposition 1.8 of loc. cit., to show that
$\pi_{1}^{\mathrm{un}}(X_{\mathbb{Q}},v)$ belongs to the full subcategory of
mixed Tate motives over $Z$, it’s enough to check that at each $p\in Z$, an
$l$-adic realization ($l\neq p$) is unramified, as was done in Proposition
6.4. This amounts to an elaboration on Remark 4.18 of loc. cit. We write
$\pi_{1}^{\mathrm{un}}(X,v)$ when we regard the unipotent fundamental group as
a prounipotent group object of $\operatorname{\mathbf{MT}}(Z)$.

### 6.6.

For a pair of $Z$-integral base-points $a,b$, similar constructions to the
ones outlined above provide a
$\pi_{1}^{\mathrm{un}}(X,a)$-$\pi_{1}^{\mathrm{un}}(X,b)$-bitorsor
${{}_{b}P_{a}}$ of _motivic paths from $a$ to $b$_. There are _path
composition morphisms_

${{}_{c}P_{b}}\times{{}_{b}P_{a}}\to{{}_{c}P_{a}};$

if $\mathcal{B}$ is a set of $Z$-integral base-points, then the collection

$\\{{{}_{b}P_{a}}\\}_{a,b\in\mathcal{B}}$

has the structure of a _groupoid in $\operatorname{\mathbf{MT}}(Z)$_ in an
obvious sense.

### 6.7.

If $E$ is an object of $\operatorname{\mathbf{MT}}(Z)$, we denote by
$\mathbb{V}^{\lor}E$ the associated vector group object — its image under any
fiber functor $\omega$ is given by

$\omega(\mathbb{V}^{\lor}E)=\operatorname{Spec}\operatorname{Sym}^{*}\omega(E)^{\lor}.$

If $y$ is a tangential base-point whose support is contained in a boundary-
divisor, then there’s an associated _local monodromy morphism_

$\mathbb{V}^{\lor}\mathbb{Q}(1)\to\pi_{1}^{\mathrm{un}}(X,y).$

This is evident in realizations, and the construction of Deligne-Goncharov
[DG] shows that the local monodromy morphisms are indeed motivic.

Suppose $Y\to Z$ satisfies the same conditions as $X\to Z$, $f:X\to Y$ is a
morphism over $Z$, and $x_{1},x_{2}$ are $Z$-integral base-points of $X$ whose
images $y_{1},y_{2}$ are $Z$-integral base-points of $Y$. Then there’s an
induced morphism of affine groupoids in mixed Tate motives from the groupoid
formed by the base-points $x_{1},x_{2}$ to the groupoid formed by the base-
points $y_{1},y_{2}$.

The local monodromy morphisms and the functoriality are compatible in the
following sense: if $f:X\to Y$ sends the tangential base-point $x$ to the
tangential base-point $y$ then the local monodromy morphism associated to $y$
is equal to the composite

$\mathbb{V}^{\lor}\mathbb{Q}(1)\to\pi_{1}^{\mathrm{un}}(X,x)\to\pi_{1}^{\mathrm{un}}(Y,y).$

Indeed, this may be checked in any realization, where it becomes evident.

### 6.8.

Let $Y$ denote the complement of the 5 divisors

$D_{1}=\\{z_{1}=0\\},\quad D_{11}=\\{z_{1}=1\\},\quad
D_{2}=\\{z_{2}=0\\},\quad D_{22}=\\{z_{2}=1\\},$

and

$D_{12}=\\{z_{1}z_{2}=1\\}$

inside $\mathbb{A}^{2}_{\mathbb{Z}}$. Let $j$ denote the natural inclusion

$Y\hookrightarrow\mathbb{A}^{2}\setminus(D_{1}\cup
D_{2})={\mathbb{G}_{m}}\times{\mathbb{G}_{m}}$

in the complement of the cross-hairs $+$. Let $(1,1)_{0}$ denote the tangent
vector $(1,1)$ at $0$.

If $b$ is any base-point, we let $K(b)$ denote the kernel of the induced map
of unipotent fundamental groups

$\pi_{1}^{\mathrm{un}}\big{(}Y,b\big{)}\to\pi_{1}^{\mathrm{un}}\big{(}\mathbb{A}^{2}\setminus(D_{1}\cup
D_{2}),b\big{)}.$

Let

$\pi^{\mathrm{PL}}(Y,b):=\pi_{1}^{\mathrm{un}}\big{(}Y,b\big{)}/[K(b),K(b)]$

and let

$\mathfrak{n}^{\mathrm{PL}}(Y,b):=\operatorname{Lie}\pi^{\mathrm{PL}}(Y,b).$

When $b=(1,1)_{0}$ we simply write $\pi^{\mathrm{PL}}(Y)$,
$\mathfrak{n}^{\mathrm{PL}}(Y)$. Our goal for this section is to establish the
following

The pro-object 363636temp. We’ve added a definition of the Lie algebra
structure.

$\big{(}\mathbb{Q}(1)\big{)}^{2}\ltimes\left(\prod_{i=1}^{\infty}\mathbb{Q}(i)\right)^{3}$

in mixed Tate motives has a natural structure of Lie algebra: the factors on
both sides of the semidirect product are abelian, and the bracket between
factors on the left and factors on the right is induced by the canonical
isomorphisms

$\mathbb{Q}(1)\otimes\mathbb{Q}(i)\xrightarrow{\simeq}\mathbb{Q}(i+1).$

######  Proposition 6.9.

In the situation and the notation above, there is an isomorphism of Lie
algebra objects in the category of mixed Tate motives over $\mathbb{Z}$

$\mathfrak{n}^{\mathrm{PL}}(Y)=\big{(}\mathbb{Q}(1)\big{)}^{2}\ltimes\left(\prod_{i=1}^{\infty}\mathbb{Q}(i)\right)^{3}.$

### 6.10.

We begin by recalling well-known facts about $M_{0,5}$. There’s an isomorphism
$M_{0,5}=Y$, and hence an open immersion

$\kappa:M_{0,5}\hookrightarrow\mathbb{P}^{1}\times\mathbb{P}^{1}$

with complement the 7 divisors

$D_{1}=\\{z_{1}=0\\},\quad D_{11}=\\{z_{1}=1\\},\quad
D_{2}=\\{z_{2}=0\\},\quad D_{22}=\\{z_{2}=1\\},$
$D_{12}=\\{z_{1}z_{2}=1\\},\quad\\{z_{1}=\infty\\},\quad\\{z_{2}=\infty\\}.$

We let

$M_{0,5}\hookrightarrow\overline{M}_{0,5}$

denote the Deligne-Mumford compactification. The map $\kappa$ extends to a map
$\overline{\kappa}$

$\textstyle{\overline{M}_{0,5}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\overline{\kappa}}$$\textstyle{M_{0,5}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\kappa}$$\textstyle{\mathbb{P}^{1}\times\mathbb{P}^{1}}$

which identifies $\overline{M}_{0,5}$ with the blowup of
$\mathbb{P}^{1}\times\mathbb{P}^{1}$ at the three points $(1,1)$,
$(0,\infty)$, $(\infty,0)$. In particular, $\overline{M}_{0,5}$ has 3
exceptional divisors in addition to the 7 boundary divisors listed above.
These are all isomorphic to $\mathbb{P}^{1}$ over
$\operatorname{Spec}\mathbb{Z}$ and have strict normal crossings so that the
formal neighborhood of each intersection is isomorphic to
$\operatorname{Spec}\mathbb{Z}\llbracket t,u\rrbracket$ with divisors given by
$t=0$ and $u=0$. In particular, there are 4 $\mathbb{Z}$-integral tangential
base-points associated to each point of intersection
($(\partial_{t},\partial_{u})=(\pm 1,\pm 1)$).

### 6.11.

The 1-forms

$\xi_{1}=\frac{dz_{1}}{z_{1}},\;\xi_{11}=\frac{dz_{1}}{1-z_{1}},\;\xi_{2}=\frac{dz_{2}}{z_{2}},\;\xi_{22}=\frac{dz_{2}}{1-z_{2}},\;\xi_{12}=\frac{d(z_{1}z_{2})}{1-z_{1}z_{2}}$

form a basis of $H^{1}_{\rm{dR}}(Y_{\mathbb{Q}})$. The construction of Deligne
[Del, §12] provides a canonical splitting of the natural surjection

$\pi_{1}^{\rm{dR}}(Y,y)\twoheadrightarrow H_{1}^{\rm{dR}}(Y_{\mathbb{Q}})$

for any base-point $y$. Together, the basis and the splitting give rise to a
canonical surjection

$\pi(e_{1},e_{2},e_{11},e_{22},e_{12})\twoheadrightarrow\pi_{1}^{\rm{dR}}(Y,y)$

from the free prounipotent group on the set of generators $\Gamma$.

### 6.12.

In the de Rham setting, as in the $l$-adic setting, tangential fiber functors
and local monodromy morphisms may be obtained directly from the 1-dimensional
construction. In [Del, §15.28-36] Deligne constructs a functor from the
category of vector bundles with integrable connection on
$\hat{D}_{\mathbb{Q}}^{*}:=\operatorname{Spec}\mathbb{Q}((t))$ with regular
singularity at $t=0$ to the category of vector bundles with integrable
connection on $\mathbb{G}_{m,\mathbb{Q}}$. Let us denote this functor by
$\delta$.

Let $\operatorname{VIC}^{[\infty]}$ denote the category of unipotent vector
bundles with integrable connection. Recall that a unipotent vector bundle with
integrable connection on the complement of a simple normal
crossings373737temp: changed snc to simple normal crossings divisor inside a
smooth scheme automatically has regular singularities along the divisor. The
same holds for the divisor $t=0$ inside the formally smooth
$\mathbb{Q}$-scheme
$\hat{D}_{\mathbb{Q}}=\operatorname{Spec}\mathbb{Q}\llbracket t\rrbracket$.

We let

$\omega_{1}:\operatorname{VIC}^{[\infty]}(\mathbb{G}_{m,\mathbb{Q}})\to\operatorname{\bf{Vect}}(\mathbb{Q})$

denote the fiber functor

$(E,\nabla)\mapsto E(1)$

associated to the point $1\in{\mathbb{G}_{m}}$. If $y$ is a tangential base
point of $M_{0,5}$ supported at $\overline{y}\in\overline{M}_{0,5}$, we let

$h_{y}:\hat{D}_{\mathbb{Q}}\to\overline{M}_{0,5}$

be a map sending the closed point to $\overline{y}$ and whose derivative sends
$1$ to $y$. We let $h_{y}^{o}$ denote the induced map

$\hat{D}_{\mathbb{Q}}^{*}\to M_{0,5}=Y.$

In terms of the maps and functors defined above, we define

$\omega_{y}:\operatorname{VIC}^{[\infty]}(Y)\to\operatorname{\bf{Vect}}(\mathbb{Q})$

to be the composite

$\operatorname{VIC}^{[\infty]}(Y)\xrightarrow{(h_{y}^{o})^{*}}\operatorname{VIC}^{[\infty]}(\hat{D}^{*}_{\mathbb{Q}})\xrightarrow{\delta}\operatorname{VIC}^{[\infty]}({\mathbb{G}_{m}}_{\mathbb{Q}})\xrightarrow{\omega_{1}}\operatorname{\bf{Vect}}(\mathbb{Q}).$

Then $\pi_{1}^{\rm{dR}}(Y,y)=\mathrm{Aut}\,^{\otimes}(\omega_{y})$ is the de
Rham realization of $\pi_{1}^{\mathrm{un}}(Y,y)$.

If $y$ is a tangential base-point supported at a point contained in the
divisor associated to the generator
$e\in\\{e_{1},e_{2},e_{11},e_{22},e_{12}\\}$ then the associated local
monodromy morphism in mixed Tate motives realizes to the composite

${\mathbb{G}_{a}}=\pi(e)\subset\pi(e_{1},e_{2},e_{11},e_{22},e_{12})\twoheadrightarrow\pi_{1}^{\rm{dR}}(Y,y)$

In terms of the presentation (*), $\pi^{\mathrm{PL}}$ is the prounipotent
group associated to the “abstract polylogarithmic Lie algebra” considered in
§2 above.

### 6.13.

By construction, the kernel $K$ of the projection

$\phi:\pi^{\mathrm{PL}}\twoheadrightarrow\mathbb{V}^{\lor}\mathbb{Q}(1)^{2}$

is commutative. 383838temp: Indeed, this does not require an argument. The
local monodromy morphisms associated to the divisors $D_{1}$, $D_{2}$ induce a
splitting $\mu=\mu_{1}\oplus\mu_{2}$ of $\phi$. Our next goal is to construct
a morphism

$\mu^{\prime}:\mathbb{V}^{\lor}\mathbb{Q}(1)^{3}\to K$

associated to monodromy about the divisors $D_{11}$, $D_{22}$, $D_{12}$.

### 6.14.

We will construct a map

$\mu_{12}:\mathbb{V}^{\lor}\mathbb{Q}(1)\to K\subset\pi^{\mathrm{PL}}(Y)$

corresponding to monodromy about the divisor $D_{12}$. The same construction,
mutatis mutandis, provides similar maps $\mu_{11}$, $\mu_{22}$ corresponding
to monodromy about the divisors $D_{11}$, $D_{22}$, respectively. The map
$\mu^{\prime}$ is then the direct sum

$\mu^{\prime}=\mu_{11}\oplus\mu_{22}\oplus\mu_{12}.$

Let $y$ be a $\mathbb{Z}$-integral tangential base-point supported along the
intersection of $D_{12}$ with the exceptional divisor $E$ over the point
$(1,1)\in Y$ (recall from segment 6.10 above that there are precisely 4 such).
Since $y$ is nowhere tangent to $E$, its image $w$ in the relative tangent
bundle to $\mathbb{A}^{2}\setminus(D_{1}\cup D_{2})$ along the
$\mathbb{Z}$-point $(1,1)$ is again a $\mathbb{Z}$-integral tangential base-
point. The composite of the associated local monodromy map

$\mathbb{V}^{\lor}\mathbb{Q}(1)\to\pi^{\mathrm{PL}}(Y,y)$

with the map

$\pi^{\mathrm{PL}}(Y,y)\to\pi_{1}^{\mathrm{un}}(\mathbb{A}^{2}\setminus(D_{1}\cup
D_{2}),y)=\mathbb{V}^{\lor}\mathbb{Q}(1)^{2}$

is zero, so the local monodromy map factors through a map

$\mathbb{V}^{\lor}\mathbb{Q}(1)\to K(y).$

The action of $\pi^{\mathrm{PL}}(Y)$ on $K$ factors through
$\pi_{1}^{\mathrm{un}}(\mathbb{A}^{2}\setminus(D_{1}\cup D_{2}),(1,1))$. Thus,
$K(y)$ is equal to $K$ twisted by the torsor

$\pi_{1}^{\mathrm{un}}({\mathbb{G}_{m}}\times{\mathbb{G}_{m}},(1,1)_{0},y).$

Since there are no nontrivial Kummer motives over
$\operatorname{Spec}\mathbb{Z}$, this torsor is trivial. Hence there’s a
canonical isomorphism of commutative unipotent group objects

$K(y)=K.$

Composing with (**) we obtain the map $\mu_{12}$.

### 6.15.

The Lie bracket is a morphism of pro-mixed Tate motives

$[\cdot,\cdot]:\mathfrak{n}^{\mathrm{PL}}(Y)\otimes\mathfrak{n}^{\mathrm{PL}}(Y)\to\mathfrak{n}^{\mathrm{PL}}(Y).$

Let $\nu_{1}$, $\nu_{11}$, etc. be the maps of Lie algebras associated to the
morphisms $\mu_{?}$ constructed above. For $n\geq 1$ we let

$\nu_{11,n}:=(\operatorname{ad}\nu_{1})^{n-1}(\nu_{11}),$

that is, $\nu_{11,n}$ is the map

$\mathbb{Q}(n)\xrightarrow{\nu_{1}\otimes\cdots\otimes\nu_{1}\otimes\nu_{11}}\mathfrak{n}^{\mathrm{PL}}(Y)^{\otimes
n}\xrightarrow{[\cdot,\cdots[\cdot,[\cdot,\cdot]]\cdots]}\mathfrak{n}^{\mathrm{PL}}(Y).$

Similarly, we let

$\nu_{22,n}:=(\operatorname{ad}\nu_{2})^{n-1}(\nu_{22}),\quad\nu_{12,n}:=(\operatorname{ad}\nu_{1})^{n-1}(\nu_{12}).$

We could also define

$\nu^{\prime}_{12,n}:=(\operatorname{ad}\nu_{2})^{n-1}(\nu_{12}).$

That $\nu_{12,n}=\nu^{\prime}_{12,n}$ may be checked after passage to de Rham
realization, where it’s 2.2 above. Together, the maps $\nu_{?}$ define a
morphism of pro-mixed Tate motives

$\mathfrak{n}^{\mathrm{PL}}(Y)\xleftarrow{\nu}\big{(}\mathbb{Q}(1)\big{)}^{2}\ltimes\left(\prod_{i=1}^{\infty}\mathbb{Q}(i)\right)^{3}.$

We may check that $\nu$ is an isomorphism of Lie algebra objects after passage
to de Rham realization where it follows from 6.11(*) and 2.3, in view of the
known computation of the de Rham fundamental group of $M_{0,5}$.

This last computation may be extracted from the literature for instance as
follows. 393939temp: We’ve clarified this point somewhat, as this is
admittedly not fully explained in [OU]. Ünver [Ü2, §5] constructs generators
$E_{i,j}$ ($0\leq i,j\leq 4$) for the Lie algebra of the de Rham fundamental
group, and proves that the latter is free pronilpotent on these generators
modulo the relations

$\displaystyle E_{ii}$ $\displaystyle=0,$ $\displaystyle E_{ji}$
$\displaystyle=-E_{ij},$ $\displaystyle\Sigma_{i}E_{ij}$ $\displaystyle=0,$
$\displaystyle[E_{ij},E_{kl}]$ $\displaystyle=0\mbox{ whenever
}\\{i,j\\}\cap\\{k,l\\}=\emptyset.$

(We have capitalized Ünver’s “$E_{ij}$” in order to avoid a conflict with our
notation.) The generators are determined by their action on the universal
prounipotent connection (the “KZ” connection). A presentation of the latter
which makes the action evident is given by Oi-Ueno in §2.1 of [OU] (where the
generators $E_{ij}$ of Ünver are denoted by $\Omega_{ij}$). In terms of these
generators, ours are given by

$\displaystyle e_{1}$ $\displaystyle=E_{12}+E_{13}+E_{14},$ $\displaystyle
e_{2}$ $\displaystyle=E_{23},$ $\displaystyle e_{11}$ $\displaystyle=-E_{14},$
$\displaystyle e_{22}$ $\displaystyle=-E_{12},$ $\displaystyle e_{12}$
$\displaystyle=-E_{24},$

as may be seen, for instance, by computing their action on the KZ-connection.
The implied relations 2.1(R) are listed in [OU, §4.1]. This completes the
proof of Proposition 6.9.

######  Remark 6.16.

Recall that $Y$ denotes the moduli space $M_{0,5}$ in its guise as

$\operatorname{Spec}\mathbb{Z}[z_{1},z_{2},z_{1}^{-1},(1-z_{1})^{-1},z_{2}^{-1},(1-z_{2})^{-1},(1-z_{1}z_{2})^{-1}].$

Let $X=M_{0,4}=\operatorname{Spec}\mathbb{Z}[x,x^{-1},(1-x)^{-1}]$. The map

$\iota:Y\to X^{3}$ $(z_{1},z_{2})\mapsto(z_{1},z_{2},z_{1}z_{2})$

is a closed immersion with image the closed subscheme defined by the equation

$x_{3}=x_{1}x_{2}.$

The de Rham first cohomology vector space
$H^{1}_{\rm{dR}}(X^{3}_{\mathbb{Q}})=H^{1}_{\rm{dR}}(X)^{\otimes 3}$ has basis
the six 1-forms

$\frac{dx_{i}}{x_{i}},\quad\frac{dx_{i}}{1-x_{i}}\quad(i=1,2,3).$

Their pullbacks along $\iota$ span $H^{1}_{\rm{dR}}(Y_{\mathbb{Q}})$ and are
linearly independent modulo the one relation

$\iota^{*}\frac{dx_{3}}{x_{3}}=\iota^{*}\frac{dx_{1}}{x_{1}}+\iota^{*}\frac{dx_{2}}{x_{2}}.$

If we label the six generators of the de Rham unipotent fundamental group
$\pi_{1}^{\rm{dR}}(X^{3})$ of $X^{3}$ at the tangential base-point
$(\vec{1_{0}},\vec{1_{0}},\vec{1_{0}})$ associated to the 1-forms (*) as
follows:

$d_{0}^{1},d_{1}^{1},d_{0}^{2},d_{1}^{2},d_{0}^{3},d_{1}^{3},$

then the map of de Rham unipotent fundamental groups

$\pi_{1}^{\rm{dR}}(Y)\to\pi_{1}^{\rm{dR}}(X^{3})$

sends

$\displaystyle e_{1}$ $\displaystyle\mapsto d^{1}_{0}+d^{3}_{0}$
$\displaystyle e_{11}$ $\displaystyle\mapsto d^{1}_{1}$ $\displaystyle e_{2}$
$\displaystyle\mapsto d^{2}_{0}+d^{3}_{0}$ $\displaystyle e_{22}$
$\displaystyle\mapsto d^{2}_{1}$ $\displaystyle e_{12}$ $\displaystyle\mapsto
d^{3}_{1}.$

In terms of the associated map of Hopf algebras (with dual elements in the
Hopf algebra denoted by $f_{?}$ as usual) equation (*) reads

$\iota^{*}f_{d^{3}_{0}}=\iota^{*}f_{d^{1}_{0}}+\iota^{*}f_{d^{2}_{0}}.$

This gives geometric meaning to the equation
“$f_{e_{3}}=f_{e_{1}}+f_{e_{2}}$”.

Let $\pi_{1}^{\mathrm{PL}}(X^{3})=\pi_{1}^{\mathrm{PL}}(X)^{3}$ denote the
quotient of $\pi_{1}^{\rm{dR}}(X^{3})$ associated to the polylogarithmic
quotient of $\pi_{1}^{\rm{dR}}(X)$ (or a quotient thereof by some step of the
descending central series). Let $Z=\operatorname{Spec}\mathbb{Z}[1/6]$ as
usual, and let $K(Z)$ denote the fraction field of the prounipotent mixed Tate
Galois group $\pi_{1}^{\mathrm{un}}(Z)$. Let
$\mathcal{A}(X^{3})=\mathcal{A}(X)^{\otimes 3}$ denote the coordinate ring of
$\pi^{\mathrm{PL}}(X^{3})_{K(Z)}$. Let $\mathcal{S}(X)$ denote the coordinate
ring of the base-change to $K(Z)$ of the Selmer scheme

${\bf
H}^{1}\big{(}\pi_{1}^{\operatorname{\mathbf{MT}}}(Z),\pi^{\mathrm{PL}}(X)\big{)}={\bf
Z}^{1}\big{(}\pi_{1}^{\mathrm{un}}(Z),\pi^{\mathrm{PL}}(X)\big{)}^{{\mathbb{G}_{m}}}$

and similarly for $X^{3}$; we have

$\mathcal{S}(X^{3})=\mathcal{S}(X)^{\otimes 3}.$

Then the universal cocycle evaluation maps of $Y$ and of $X^{3}$, together
with the maps $\iota_{\mathcal{A}}$, $\iota_{\mathcal{S}}$ induced by the
embedding $\iota$, form a commuting square of $K(Z)$-algebras

$\textstyle{\mathcal{S}(Y)}$$\textstyle{\ignorespaces\ignorespaces\ignorespaces\ignorespaces\mathcal{A}(Y)}$$\scriptstyle{\theta_{Y}}$$\textstyle{\mathcal{S}(X^{3})\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\iota_{\mathcal{S}}}$$\textstyle{\ignorespaces\ignorespaces\ignorespaces\ignorespaces\mathcal{A}(X^{3})\ignorespaces\ignorespaces\ignorespaces\ignorespaces.}$$\scriptstyle{\theta_{X^{3}}}$$\scriptstyle{\iota_{\mathcal{A}}}$

Let $d_{0}$, $d_{1}$ denote the standard generators of the de Rham unipotent
fundamental group of $X$. In Proposition §4.11 we essentially constructed a
certain polynomial $p(y)\in\mathcal{A}(X)[y]$ such that (after translating
along the universal cocycle evaluation map $\theta_{X}$)

$p(\Phi^{\tau}_{d_{0}})=0$

where $\Phi^{\tau}_{d_{0}}$ denotes the function on cocycles

$\Phi^{\tau}_{d_{0}}(c)=\langle\tau,c^{\sharp}d_{0}\rangle.$

Applied to the three copies of $X$, this gives us three polynomials
$p_{1},p_{2},p_{3}$ such that

$p_{1}(\Phi^{\tau}_{d^{1}_{0}})=0,\quad
p_{2}(\Phi^{\tau}_{d^{2}_{0}})=0,\quad\mbox{and}\quad
p_{1}(\Phi^{\tau}_{d^{3}_{0}})=0.$

The images of the three roots in $\mathcal{S}(Y)$ obey the algebraic relation

$\Phi^{\tau}_{e_{3}}=\Phi^{\tau}_{e_{1}}+\Phi^{\tau}_{e_{2}}$

which is again an immediate consequence of $(**)$. This puts the double-
resultant construction 4.10(5) on a geometric footing. (Our construction of
the relation (R) obeyed by $\Phi^{\tau}_{d_{0}}$ over $\mathcal{A}(X)$ remains
ad hoc.)

##  7\. The $p$-adic unipotent Albanese map

Throughout this section, Tannakian fundamental groups are endowed with the
usual _functorial_ product.

### 7.1.

Let

$Y=\operatorname{Spec}\mathbb{Z}[z_{1},z_{2},z_{1}^{-1},z_{2}^{-1},(1-z_{1})^{-1},(1-z_{2})^{-1},(1-z_{1}z_{2})^{-1}],$

let $p$ be a prime, let $\pi^{\mathrm{PL},{\rm{dR}}}(Y_{\mathbb{Q}_{p}})$
denote the polylogarithmic quotient of the de Rham unipotent fundamental group
of $Y$ at the tangential base-point “$(1,1)$ at $(0,0)$” with respect to
functorial composition of paths (§6). Let

$S=\\{e_{1},e_{11},e_{2},e_{22},e_{12}\\}.$

In §6.11 we outlined the construction of the standard presentation

$\pi(S)_{{\mathbb{Q}_{p}}}\twoheadrightarrow\pi^{\mathrm{PL},{\rm{dR}}}(Y_{\mathbb{Q}_{p}}).$

We define

$\Lambda_{\geq-\infty}^{\mathrm{fun}}=\bigcup_{i=1}^{\infty}\Lambda^{\mathrm{fun}}_{-i},\mbox{
where }\Lambda_{-1}^{\mathrm{fun}}=S,$ $\mbox{ and
}\Lambda_{-i}^{\mathrm{fun}}=\\{e_{1}^{i-1}e_{11},e_{2}^{i-1}e_{22},(e_{1}+e_{2})^{i-1}e_{12}\\}\mbox{
for }i\geq 2.$

If $\omega$ is a finite linear combination of words in the alphabet $S$, we
let $f_{\omega}$ denote the linear functional on the completed universal
enveloping algebra $\mathcal{U}(S)$ dual to $\omega$ with respect to the
standard (topological) basis. According to Lemma 3.6 (applied to the opposite
group), the functions $f_{\lambda}$ (for
$\lambda\in\Lambda^{\mathrm{fun}}_{\geq-\infty}$) on $\pi(S)_{\mathbb{Q}_{p}}$
factor through $\pi^{\mathrm{PL},{\rm{dR}}}(Y_{\mathbb{Q}_{p}})$ and form an
algebra basis for its coordinate ring

$A^{\mathrm{PL},{\rm{dR}}}(Y_{\mathbb{Q}_{p}}):=\mathcal{O}(\pi^{\mathrm{PL},{\rm{dR}}}(Y_{\mathbb{Q}_{p}})).$

Let $f_{\lambda}^{\mathrm{BC}}$ denote the Besser-Coleman function on
$Y({\mathbb{Z}_{p}})$ obtained by composing $f_{\lambda}$ with the unipotent
Albanese map

$\alpha:Y({\mathbb{Z}_{p}})\to\pi^{\mathrm{PL},{\rm{dR}}}(Y_{\mathbb{Q}_{p}}).$

Let $e_{3}:=e_{1}+e_{2}$.

######  Proposition 7.2.

In the situation and the notation of §7.1, we have (for $i\geq 1$)

$\displaystyle f_{e_{1}}^{\mathrm{BC}}(z_{1},z_{2})$
$\displaystyle=\log(z_{1})$ $\displaystyle
f_{e_{2}}^{\mathrm{BC}}(z_{1},z_{2})$ $\displaystyle=\log(z_{2})$
$\displaystyle f_{e_{1}^{i-1}e_{11}}^{\mathrm{BC}}(z_{1},z_{2})$
$\displaystyle=\operatorname{Li}_{i}(z_{1})$ $\displaystyle
f_{e_{2}^{i-1}e_{22}}^{\mathrm{BC}}(z_{1},z_{2})$
$\displaystyle=\operatorname{Li}_{i}(z_{2})$ $\displaystyle
f_{e_{3}^{i-1}e_{12}}^{\mathrm{BC}}(z_{1},z_{2})$
$\displaystyle=\operatorname{Li}_{i}(z_{1}z_{2}).$

###### Proof.

Let $X=\operatorname{Spec}\mathbb{Z}[x,x^{-1},(1-x)^{-1}]$ and consider the
maps $p,q:Y\to X$ given by

$\displaystyle(z_{1},z_{2})\overset{p}{\mapsto}z_{1}$
$\displaystyle(z_{1},z_{2})\overset{q}{\mapsto}z_{2}.$

These maps 404040temp: replaced “extend to a formal neighborhood of the
origin” by “induce maps of punctured tangent spaces” induce maps of punctured
tangent spaces at $(z_{1},z_{2})=(0,0)$ and send the tangential base-point
“$(1,1)$ at $(0,0)$” to the tangential base-point “$1$ at $0$”. The unipotent
de Rham fundamental group $\pi_{1}^{\rm{dR}}(X_{\mathbb{Q}},\vec{1_{0}})$ is
freely generated by two elements $d_{0}$ (monodromy about $x=0$) and $d_{1}$
(monodromy about $x=1$). The maps induced by $p,q$ on $\pi_{1}$ send

$\displaystyle e_{1}\overset{p}{\mapsto}d_{0}$ $\displaystyle
e_{1}\overset{q}{\mapsto}0$ $\displaystyle e_{11}\mapsto d_{1}$ $\displaystyle
e_{11}\mapsto 0$ $\displaystyle e_{2}\mapsto 0$ $\displaystyle e_{2}\mapsto
d_{0}$ $\displaystyle e_{22}\mapsto 0$ $\displaystyle e_{22}\mapsto d_{1}$
$\displaystyle e_{12}\mapsto 0$ $\displaystyle e_{12}\mapsto 0.$

We now focus on the map $p$ and the functions
$f^{\mathrm{BC}}_{e^{i-1}_{1}e_{11}}$. By the formula given above for the
induced map on fundamental groups, $f_{d_{0}^{i-1}d_{1}}$ pulls back along $p$
to $f_{e^{i-1}_{1}e_{11}}$. On the other hand $f_{d_{0}^{i-1}d_{1}}$ pulls
back along the $p$-adic unipotent Albanese to $\operatorname{Li}_{i}(x)$. The
$p$-adic unipotent Albanese maps fit into a commuting square

$\textstyle{Y({\mathbb{Z}_{p}})\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{p}$$\textstyle{\pi_{1}^{\rm{dR}}(Y_{\mathbb{Q}_{p}},(1,1)_{(0,0)})\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{X({\mathbb{Z}_{p}})\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\pi_{1}^{\rm{dR}}(X_{\mathbb{Q}_{p}},1_{0}).}$

Combining these facts we find that

$f_{e_{1}^{i-1}e_{11}}^{\mathrm{BC}}(z_{1},z_{2})=\operatorname{Li}_{i}(z_{1}),$

and similarly for $f_{e_{2}^{i-1}e_{22}}^{\mathrm{BC}}$.

We turn to the function $f_{e_{3}^{i-1}e_{12}}^{\mathrm{BC}}$. Let

$\mathcal{U}=\mathbb{Q}\langle\langle S\rangle\rangle/I$

where $I$ is the two-sided ideal generated by the Lie relations 2.1(R). Let
$\mathcal{E}=\mathcal{U}\otimes\mathcal{O}_{Y}$ with connection

$\nabla:\mathcal{E}\to\mathcal{E}\otimes\Omega^{1}_{Y}$

given on a word $W$ in the alphabet $S$ regarded as a section of the trivial
pro-vector bundle $\mathcal{E}$ by (notation as in §6.11)

$\nabla(W)=-e_{1}W\xi_{1}-e_{11}W\xi_{11}-e_{2}W\xi_{2}-e_{22}W\xi_{22}-e_{12}W\xi_{12}.$

Then $(\mathcal{E},\nabla)$ is isomorphic to the universal unipotent
connection on $Y$ (at any base-point), equipped with its de Rham
trivialization. Hence, $f_{e_{3}^{i-1}e_{12}}^{\mathrm{BC}}$ may be
represented by the abstract Coleman function given by the connection
$(\mathcal{E},\nabla)$, the projection
$f_{e_{3}^{i-1}e_{12}}:\mathcal{E}\to\mathcal{O}$, and the Frobenius-
compatible family of horizontal sections on residue polydisks with _constant
term_ $0$ at the tangential base-point $(1,1)_{(0,0)}$ [BF]; this is the same,
mutatis mutandis, as the case of $\mathbb{P}^{1}\setminus\\{0,1,\infty\\}$
treated, for instance, in Theorem 2.3 of Furusho [Fur1]. It follows that the
functions $f^{\mathrm{BC}}_{W}$ obey

$d\sum_{W}f^{\mathrm{BC}}_{W}W=\sum_{V}f^{\mathrm{BC}}_{V}(e_{1}W\xi_{1}+e_{11}W\xi_{11}+e_{2}W\xi_{2}+e_{22}W\xi_{22}+e_{12}W\xi_{12}).$

Hence $f^{\mathrm{BC}}_{e_{12}}$ satisfies the differential equation

$df^{\mathrm{BC}}_{e_{12}}=\frac{d(z_{1}z_{2})}{1-z_{1}z_{2}}.$

Since $\operatorname{Li}_{1}(z_{1}z_{2})$ satisfies the same differential
equation and has constant term $0$ at the base-point $(1,1)_{(0,0)}$, it
follows that

$f^{\mathrm{BC}}_{e_{12}}=\operatorname{Li}_{1}(z_{1}z_{2}).$

Similarly, for $i\geq 1$, $f_{e_{3}^{i}e_{12}}^{\mathrm{BC}}$ satisfies the
differential equation

$df_{e_{3}^{i}e_{12}}^{\mathrm{BC}}=f_{e_{3}^{i-1}e_{12}}^{\mathrm{BC}}\frac{d(z_{1}z_{2})}{z_{1}z_{2}},$

and (by induction), $\operatorname{Li}_{i+1}(z_{1}z_{2})$ satisfies the same
differential equation. Since $\operatorname{Li}_{i+1}(z_{1}z_{2})$ too has
constant term $0$, it follows that

$f_{e_{3}^{i-1}e_{12}}^{\mathrm{BC}}(z_{1},z_{2})=\operatorname{Li}_{i}(z_{1}z_{2})$

as claimed. ∎

##  8\. Summary and construction of $F^{\mathrm{BC}}$

### 8.1.

Fix a prime $p$ not dividing $6$. In §4.10 we constructed a polynomial $F$ in
the 14 variables

$f_{e_{1}},f_{e_{11}},\dots$

listed in §4.1 whose coefficients are rational functions in the 11 symbols

$f_{\tau},f_{\upsilon},\dots$

(also listed in §4.1) over the rationals. Using the equations obtained in
§5.23 and replacing motivic polylogarithms by their $p$-adic periods, we
obtain a polynomial with coefficients in ${\mathbb{Q}_{p}}$. In terms of the
coordinates $z_{1},z_{2}$ on $Y=M_{0,5}$ (§7.1), we replace the indeterminates
by Besser-Coleman functions on $Y({\mathbb{Z}_{p}})$ as follows:

$\displaystyle f_{e_{1}}$ $\displaystyle\mapsto\log(z_{1})$ $\displaystyle
f_{e_{2}}$ $\displaystyle\mapsto\log(z_{2})$ $\displaystyle
f_{e_{11}e_{1}^{i-1}}$ $\displaystyle\mapsto\operatorname{Li}_{i}(z_{1})$
$\displaystyle f_{e_{22}e_{2}^{i-1}}$
$\displaystyle\mapsto\operatorname{Li}_{i}(z_{2})$ $\displaystyle
f_{e_{12}e_{3}^{i-1}}$
$\displaystyle\mapsto\operatorname{Li}_{i}(z_{1}z_{2}).$

This gives us a Besser-Coleman function $F^{\mathrm{BC}}$ on
$M_{0,5}({\mathbb{Z}_{p}})$. Our goal for this section is to explain how the
results obtained above show that $F^{\mathrm{BC}}$ is within $\epsilon$ of a
Kim function for $M_{0,5}$ in half-weight $4$ over
$Z=\operatorname{Spec}\mathbb{Z}[1/6]$ (Theorem 1.1) while clarifying
$\epsilon$ and indicating how to apply our lexicographic computations to
functorial fundamental groups.

### 8.2.

Let
$\pi_{1}^{\operatorname{\mathbf{MT}}}(Z)=\pi_{1}^{\mathrm{un}}(Z)\rtimes{\mathbb{G}_{m}}$
denote the _functorial_ fundamental group of the category of mixed Tate
motives over $Z$ at the de Rham fiber functor and let
$A(Z)=\mathcal{O}(\pi_{1}^{\mathrm{un}}(Z))$ be the associated graded Hopf
algebra. Let $X=\mathbb{P}^{1}\setminus\\{0,1,\infty\\}$, let
$\pi_{1}^{\mathrm{un}}(X,1_{0})$ denote the _functorial_ unipotent fundamental
group of $X$ at the standard $\mathbb{Z}$-integral base point $1_{0}$. Let
${\rm{dR}}^{*}\pi_{1}^{\mathrm{un}}(X,1_{0})$ denote its de Rham realization.
Let $d_{0}$, $d_{1}$ denote the standard generators on the latter. In view of
the canonical trivializations of de Rham path torsors, a word $\omega$ in
$d_{0},d_{1}$ gives rise to a function $f_{\omega}$ on any path torsor. Recall
that given $a\in X(Z)$ and $n\geq 1$ we define the _(functorial, unipotent)
motivic polylogarithm_ $\operatorname{Li}^{\mathfrak{u}}_{n}(a)\in A_{n}(Z)$
to be the function

$\pi_{1}^{\mathrm{un}}(Z)\xrightarrow{o(p^{\rm{dR}})}\pi_{1}^{\mathrm{un}}(X,1_{0},a)\xrightarrow{f_{d_{1}d_{0}^{n-1}}}\mathbb{A}^{1}_{\mathbb{Q}}.$

as in (§5.1)op.

### 8.3.

Let $A(Z)_{[\leq 4]}\subset A(Z)$ denote the subalgebra generated in half-
weights $\leq 4$ and by $\pi_{1}^{\mathrm{un}}(Z)_{\geq-4}$ the associated
quotient of $\pi_{1}^{\mathrm{un}}(Z)$. We let $K(Z)_{[\leq 4]}$ denote the
fraction field of $A(Z)_{[\leq 4]}$ and let

$\eta(Z)_{\geq-4}=\operatorname{Spec}K(Z)_{[\leq 4]}.$

Let $K^{\prime}(Z)_{[\leq 4]}\subset K(Z)_{[\leq 4]}$ denote the maximal
localization of $A(Z)_{[\leq 4]}$ to which the $p$-adic period map
$\mathrm{per}:A(Z)\to{\mathbb{Q}_{p}}$ extends (conjecturally $K^{\prime}=K$)
and let

$\eta^{\prime}(Z)_{\geq-4}=\operatorname{Spec}K^{\prime}(Z)_{[\leq 4]}.$

We denote the map of schemes

$\operatorname{Spec}{\mathbb{Q}_{p}}\to\eta^{\prime}(Z)_{\geq-4}$

induced by the period map by $I_{BC}$.

### 8.4.

The nonabelian cohomology variety

$\mathbf{H}^{1}(\pi_{1}^{\mathrm{un}}(Z,\omega),\pi^{\mathrm{PL}}_{\geq-4}(Y)_{\omega})$

is independent of the choice of fiber functor $\omega$, which we therefore
omit from the notation, and similarly for its filtered $\phi$ variant. Let
$Z_{p}=\operatorname{Spec}{\mathbb{Z}_{p}}$ and let

$\pi_{1}^{\mathrm{MT}}(Z_{p})=\pi_{1}^{\mathrm{un}}(Z_{p})\rtimes{\mathbb{G}_{m}}$

denote the fundamental group of the category of mixed Tate filtered
$\phi$-modules over ${\mathbb{Q}_{p}}$ [DCW2] at the de Rham fiber functor. We
denote the realization of a mixed Tate motive $M$ in mixed Tate filtered
$\phi$-modules by $\operatorname{F\phi^{*}}(M)$, and we denote de Rham
realization by $\mathrm{dR}^{*}(M)$. In the diagrams below, we let $\bf{RL}$
denote the map of nonabelian cohomology varieties obtained by realization and
localization. We let $\kappa,\kappa_{p}$ denote the unipotent motivic and
filtered $\phi$ Kummer maps. We replace $\bf{RL}$ by $RL$ and $\bf{H}$ by $H$
to denote the induced map of ${\mathbb{Q}_{p}}$-points. This completes the
definition of the objects and morphisms in the first diagram:

$\textstyle{Y(Z)\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\kappa}$$\textstyle{Y(Z_{p})\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\kappa_{p}}$$\textstyle{H^{1}\big{(}\pi_{1}^{\mathrm{MT}}(Z)_{\mathbb{Q}_{p}},\pi^{\mathrm{PL}}_{\geq-4}(Y)_{\mathbb{Q}_{p}}\big{)}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{RL}$$\textstyle{H^{1}\big{(}\pi_{1}^{\mathrm{MT}}(Z_{p}),F\phi^{*}\pi^{\mathrm{PL}}_{\geq-4}(Y)\big{)}.}$

For a fuller discussion of a direct analog of this diagram: its commutativity
and its (close) relationship to Kim [Kim1, Kim2], we refer the reader for
instance to [DCW2].

### 8.5.

By Proposition 6.9, the unipotent radical of $\pi_{1}^{\mathrm{MT}}(Z)$ acts
trivially on $\pi^{\mathrm{PL}}_{\geq-4}(Y)$. Consequently, Proposition 5.2.1
of [DCW2] applies to show that the natural map

$r:{\bf
H}^{1}\big{(}\pi_{1}^{\mathrm{MT}}(Z),\pi^{\mathrm{PL}}_{\geq-4}(Y)\big{)}\to{\bf
Z}^{1}\big{(}\pi_{1}^{\mathrm{un}}(Z),\pi^{\mathrm{PL}}_{\geq-4}(Y)\big{)}^{\mathbb{G}_{m}}$

to the $\mathbb{Q}$-scheme (or functor) parametrizing
${\mathbb{G}_{m}}$-equivariant 1-cocycles is an isomorphism414141temp: changed
is iso to is an isomorphism. We also have the usual isomorphism

$c:{\bf
H}^{1}\big{(}\pi_{1}^{\mathrm{MT}}(Z_{p}),F\phi^{*}\pi^{\mathrm{PL}}_{\geq-4}(Y)\big{)}\to\mathrm{dR}^{*}\pi^{\mathrm{PL}}_{\geq-4}(Y)_{\mathbb{Q}_{p}}$

[Kim2, DCW2], which follows from the fact that every
$\pi_{1}^{\mathrm{MT}}(Z_{p})$-equivariant
$F\phi^{*}\pi^{\mathrm{PL}}_{\geq-4}(Y)$-torsor possesses a unique Frobenius-
fixed point and a unique point in filtered degree 0.

Since every ${\mathbb{G}_{m}}$-equivariant cocycle 424242temp. We’ve added
detailed definitions of the various evaluation maps.

$c:\pi_{1}^{\mathrm{un}}(Z)\to\pi^{\mathrm{PL}}_{\geq-4}(Y)$

factors (uniquely) through $\pi_{1}^{\mathrm{un}}(Z)_{\geq-4}$, we have a
canonical isomorphism

${\bf
Z}^{1}\big{(}\pi_{1}^{\mathrm{un}}(Z),\pi^{\mathrm{PL}}_{\geq-4}(Y)\big{)}^{\mathbb{G}_{m}}={\bf
Z}^{1}\big{(}\pi_{1}^{\mathrm{un}}(Z)_{\geq-4},\pi^{\mathrm{PL}}_{\geq-4}(Y)\big{)}^{\mathbb{G}_{m}}$