M0, 5: Toward the Chabauty–Kim method in higher dimensions

If Z is an open subscheme of SpecZ$\operatorname{Spec}\mathbb {Z}$ , X is a sufficiently nice Z‐model of a smooth curve over Q$\mathbb {Q}$ , and p is a closed point of Z, the Chabauty–Kim method leads to the construction of locally analytic functions on X(Zp)$X({\mathbb {Z}_p})$ which vanish on X(Z)$X(Z)$ ; 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 M0, 5 should be easier than the previously studied curve M0,4=P1∖{0,1,∞}$M_{0,4} = \mathbb {P}^1 \setminus \lbrace 0,1,\infty \rbrace$ since its points are closely related to those of M0, 4, yet they face a further condition to integrality. This is mirrored by a certain weight advantage we encounter, because of which, M0, 5 possesses new Kim functions not coming from M0, 4. Here we focus on the case “ Z[1/6]$\mathbb {Z}[1/6]$ 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 Wewers, 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.


INTRODUCTION
The Chabauty-Kim method, introduced in [32,33], 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 -adic analytic functions ("Kim functions") that 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 -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 [8], 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 [2, 4-8, 10, 34] 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 [9] 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 [17,19,20] and Corwin-Dan-Cohen [15,16], while so far limited to the simplest of all cases ( = ℙ 1 ⧵ {0, 1, ∞}), 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, [12,14,22,28]).A key point is the possibility of extracting the geometric aspects of the computation from their natural arithmetic surroundings.The result is an algorithm that 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  =  0,4 over  ⊂ Spec ℤ implies Kim's conjecture for  =  0,5 (the moduli space of genus 0 curves with 5 marked points) over .Nevertheless, as we go up from  0,4 to  0,5 , we encounter a weight advantage, which allows us to construct Kim functions on  0,5 (ℤ  ) not coming from  0,4 (see Subsection 8.7).Our first opportunity to take advantage of this weight advantage occurs for 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  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 [8] for a general introduction, and to [20] for our mixed Tate version.The Chabauty-Kim method applied to a smooth mixed-Tate variety  over  ⊂ Spec ℤ, a prime of good reduction  ∈ , and a finite type   -equivariant quotient  un 1 (, ) ↠  ′ of the unipotent fundamental group of  at the -integral base-point , revolves around a commuting diagram which we refer to as Kim's cutter.Here   1 (, dR) is the mixed Tate Galois group of  at the de Rham fiber functor, the decoration "dR" denotes de Rham realization,  is the unipotent Kummer map,  is the unipotent Albanese map-a morphism of -adic analytic spaces, and LR, which is a map of finite-type ℚ  -varieties (in our case, affine spaces) is obtained by a combination of localization and realization.The coordinate ring  ′ of  ′ is graded by the   -action hiding inside the action of    1 (, dR) =   ⋉  un 1 (, dR) on  ′ dR and we refer to the graded degree of a function as its half-weight.If  ∈  ′ ℚ  , is a function of half-weight  such that LR ♯ () = 0, then  BC ∶=  ♯  is a Besser-Coleman function on (ℤ  ) that vanishes on ().We refer to such a function as a -adic Kim function for  over  in halfweight .Let  be any prime not equal to 2 or 3.
Unlike the unipotent fundamental group of ℙ 1 ⧵ {0, 1, ∞}, which has been studied to death, working with the unipotent fundamental group of  0,5 requires that we address some basic issues ourselves.This mostly concerns a certain analog of the polylogarithmic quotient.For instance, in Section 6, we revisit (and generalize in a straightforward way) the beautiful proof due to Deligne [21] and Deligne-Goncharov [22] that the polylogarithmic quotient is semisimple.We also give an explicit algebra-basis for the algebra of functions on our polylogarithmic quotient in Section 3.
Since the polylogarithmic quotient is not fixed under automorphisms of  0,5 , translation by automorphisms gives rise to inequivalent Kim functions.We may then ask if the vanishing locus of  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 [3,23,30].Unfortunately, the large size of  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  0,4 and  0,5 , plays a central role in Grothendieck's vision for anabelian geometry, and, relatedly, in motivating relations between multiple zeta values (complex and -adic) [11,25,26,35] (see also [1, chapter 25] and [27], 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 -integral refinements of Goncharov's conjectures.As explained in [15][16][17]20], our algorithms for ℙ 1 ⧵ {0, 1, ∞} rely on such statements for halting, and a better understanding will lead to faster and more elegant algorithms.As explained in forthcoming work by David Jarossay, David Lilienfeldt, Francesco M. Saettone, Ariel Weiss, and Sa'ar Zehavi, our methods with resultants also help to clarify and simplify the geometric step for ℙ 1 ⧵ {0, 1, ∞} and for punctured lines in general.
This article does not include introductory material on the moduli space  0,5 ; the facts we use, which we learned for instance from [13,29] as well as the references given above, are summarized in Section 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  0,5 .Most readers will want to start with Section 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  denote first , then .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 [15], 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.To prevent any confusion, we systematically denote objects stemming from the lexicographic fundamental groupoid with a superscript "×," which stands for the "x" in lexicographic.
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 -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 that associates to every  > 0 a Besser-Coleman function  BC  on  0,5 (ℤ  ) that is within  of a Kim function.The particular function we construct is then an example with  fixed.As explained for instance in [17], 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  BC  in terms of polylogarithms to change as  shrinks further.Proving this would require proving that the formulae for decomposition of certain motivic polylogarithms in terms of shuffle coordinates on the mixed Tate Galois group obtained via computations of certain -adic periods in Section 5 hold precisely (and not only to within ).Some methods for doing so are demonstrated, for instance, in [15,20] (along with attribution to those who taught us these methods).But these are not needed for the application to integral points.modulo the relations

THE POLYLOGARITHMIC LIE ALGEBRA IN ABSTRACTION
Let  be the Lie ideal generated by  11 ,  22 ,  12 .We define the polylogarithmic Lie algebra (for  0,5 ) by Note that in  PL we have is immediate.This completes the verification that  spans  PL .
Consequently, the alternate grading descends to a grading on  PL .There is an obvious homomorphism from  PL to the free abelian Lie algebra on the five generators, hence the set of generators is linearly independent.As the higher degree elements and the remaining generators to 0, we find that (ad  1 )  ( 12 ) ≠ 0.

2.6.
Finally, the statement regarding the brackets among the basis elements is immediate from the defining relations, from the vanishing 2.4(*) and from its symmetrical twin This completes the proof of Lemma 2.3.

THE POLYLOGARITHMIC HOPF ALGEBRA IN ABSTRACTION
3.1.We fix a base field  of characteristic 0. We refer the reader to [20, section 2] for an efficient review of the basics of free prounipotent groups in notation similar to ours.If  is a set, we denote by () the free Lie algebra on  and by () the free pronilpotent Lie algebra on .We denote the associated free prounipotent -group by ().We let  () denote the completed universal enveloping algebra of () and we let () denote the coordinate ring of ().We recall that there is a nondegenerate pairing We In turn, some simple but tedious combinatorics show that these vanish as well.

The bijections
induced by the basis of associative words in Γ endow with a second product * which satisfies The operation * also induces left and right actions of (Γ) on (Γ) ⊗ (Γ) in an obvious way., and so on, factor through  PL so they can be applied to this equality.This yields Thus, the elements   ∈ ( PL ) form a complete set of coordinates on  PL .as above, we consider the set of elements of the noncommutative polynomial ring ℚ⟨Γ⟩

THE GEOMETRIC STEP
and Since we will be working primarily with Λ ⩾−4 , we abbreviate We declare ,  to have weight −1 and  to have weight −3; we declare the elements of Γ to have weight −1.We consider the polynomial rings ℚ[{  }  ] and ℚ[{Φ   } , ] where  ranges over Λ and  ranges over the three generators {, , } and is required to have weight equal to that of .Explicitly, the complete list of algebra generators of where

4.2.
Let (, , ) be the graded free prounipotent group on three generators in weights −1, −1, −3.Let  PL denote the prounipotent ℚ-group associated to the Lie algebra  PL of Subsection 2.1.Let  PL denote the coordinate ring of  PL .Let  PL [⩽] denote the subalgebra generated by elements in graded degree ⩽ .This is a Hopf-subalgebra and we let be the associated quotient of  PL .Let denote the functor from ℚ-algebras to sets sending the pointed set of   -equivariant cocycles for the trivial group-action.We refer to an -valued point of  1 ((, , ),  PL ⩾−4 )   as an -family of cocycles or an -cocycle for short; we omit the repeating phrase "for  an arbitrary ℚ-algebra".Let  denote the map given on -points by (, ) ↦ (, ()).
Let   denote the map obtained from  by base-change.
Proposition 4.3.In the situation and the notation of segments 4.1-4.2,there is a commuting square of functors in which the vertical maps are isomorphisms.

4.4.
Let  be a prounipotent ℚ-group with coordinate ring  and let  be the completed universal enveloping algebra of Lie .Then there is an isomorphism between  and the topological dual  ∨ .Given  ∈  and  ∈  we denote the action of the linear functional associated to  on  by ⟨, ⟩.
If  is a linear combination of elements of Λ and  is an associative word in the alphabet {, , }, we define on -valued points.Note that for any   -equivariant -cocycle  and  ∈ Λ, is homogeneous of weight equal to the weight of .Thus,    = 0 unless wt() = wt().Note also that    is linear in the subscript, that is For future use, we formulate and prove the following proposition in the slightly more general setting of a free graded prounipotent group (Σ) on the graded set (i.e., ℤ-indexed family of sets) and if  is any word not occurring in the above equations then

4.6.
We begin with a formal calculation, in which Σ −1 may be an arbitrary finite set, and {  } ∈Σ −1 a family of commuting coefficients.In this abstract setting, we claim that Indeed, the left side of the equation 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  =  12 ( 1 +  2 ) −1 of Equation 4.5(4).We have, tautologically We wish to compute the coproduct Δ of both sides, remembering that  ♯ , since it corresponds to a cocycle for the trivial group action, preserves the coproduct.On the right side, we have On the left, we have for  ∈ Σ an individual letter.Alternatively, this is just an immediate consequence of the fact that the elements   for  ∈ Λ ⩾−∞ form an algebra basis of  PL on the one hand, and that the elements  ∈ Σ form a free set of generators for the prounipotent group (Σ) 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    ∈  as in the proposition, we define  ∶ (Σ)  →  PL  to be the unique homomorphism such that whenever the half-weights of  and  are equal, and = 0 otherwise.Here, as usual, we regard (Lie )() as belonging to the universal enveloping algebra of  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.for all words  in the generators {, , }.Let  be the set of words that occur in Equations ( 1)-( 3) of Proposition 4.5 and let   ⊂  denote the words of half-weight .We use the same equations to define new functions Φ   on the space of equivariant cocycles (replace each lower case  by an upper case Φ).We then have for (, ) an -valued point of
Proposition 4.11.The element  of  defined above is a nonzero element of the kernel of  ∶  → .
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  ≠ 0 [18].We nevertheless give a somewhat more nuanced account of the second claim, highlighting those steps in the construction that require something more than direct manipulation.We denote by   1 (, ) the image of  1 (, ) in [, ], and similarly for   1 (),   1 (),   1 ().Direct manipulation shows on the one hand that and on the other hand that and denote by  1 the equivalence class of ; denote by [, and let   1 denote the equivalence class of .We denote the induced homomorphism as well as the homomorphism of localizations simply by .We sometimes write " 1 ()" in place of " 1 " in order to emphasize that it is contained in an algebra over a polynomial algebra in  and can be specialized to particular values of  in any -algebra.We then denote by   1 (?) the specialization of   1 at  =?.Similarly, we sometimes write "  1 ()" and "  1 (?)".In this notation, we have the equation ( )( in the ring )), where the subscript denotes localization and   1 (Φ   1 ) corresponds to .This ring is integral.This follows from the following general fact.If  is an integral domain with function field  and  ∈ [] is monic and irreducible over , then []∕() is again integral.Indeed, Consequently, we have for some  ∈ {1, −1}.

THE ARITHMETIC STEP
For  an open subscheme of Spec(ℤ), we let denote the lexicographic mixed Tate Galois group of .We will focus primarily on our special case  = Spec(ℤ[1∕6]), yet along the way will have occasion to make statements that hold equally for arbitrary .We also let  = ℙ 1 ⧵ {0, 1, ∞} and we let  un 1 (, ⃗ 1 0 ) × denote the lexicographic unipotent fundamental group of  at the tangent vector 1 based at the missing point 0 [22, section 4].We let  PL () × denote its polylogarithmic quotient.We let  () × denote the completed universal enveloping algebra of  un 1 () × .In this section we recall from [15] the construction of generators  2 ,  3 ,  ∈  (ℤ[1∕6]) × and write the ensuing shuffle coordinates as polynomials in unipotent motivic -logarithms.We find it helpful to have several different notations available: we denote the generators of half-weight −1 by  2 ,  3 when we wish to emphasize the associated primes, by ,  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  un 1 (, 1 0 ) ×,dR of the unipotent fundamental group of  is free prounipotent on two generators, determined by the choice of 1-forms   ,  1 −  which define a basis of  1 dR ( ℚ ).We here denote the corresponding generators by  0 ,  1 .
Moreover, the torsor  un 1 (, 1 0 , ) ×,dR of unipotent de Rham paths from 1 0 to  is canonically trivialized by a special "de Rham" path which we denote by  dR .In our lexicographic ordering, the motivic polylogarithm Li   () for  ∈ () and  ⩾ 1 is defined to be the composite where ( dR ) denotes the orbit map  ↦  dR .
The motivic logarithm log  () is defined similarly with   0 in place of   −1 0  1 .

Remarks concerning functoriality
We wish to import computations carried out for (Spec ℤ[1∕2]) × to (Spec ℤ[1∕6]) × .For this purpose, we temporarily allow  to vary among the open subschemes of Spec ℤ.The structures discussed above ( un 1 () × ,  () × − , () ×  , ()  , ()  ) are functorial with respect to .An inclusion  ∶  ′ ⊂  of open subschemes of Spec ℤ (corresponding to an inclusion of finite sets of primes  ′ ⊃ ) gives rise to a surjection and an injection In terms of any choice of homogeneous free generators of  un 1 ( ′ ) × , (*) corresponds to the quotient by the normal subgroup generated by   for  ∈  ⧵  ′ , and so, () × is the corresponding shuffle subalgebra.In particular, a set of generators Σ ′ for  un 1 ( ′ ) × gives rise to a set of generators Σ for  un 1 () × .If  is a generator such that  *  ≠ 0, we denote  *  again by .With this notational convention, Σ is obtained from Σ ′ simply by removing the generators   ,  ∈  ⧵  ′ , and  ♯ (  ) (for  any word in the generators Σ) is equal to   .
→  1 ⊗  1 , and the above table, we find
From this and the exact sequence we find that outputs the number 1. Hence, at least up to the chosen precision, we have We have We have 5.18.By ( 7) and ( 8), we can regard ( 5) and ( 6) as a linear system of equations in ( 3322 ,  2333 ): The shuffle coordinates appearing in the above expressions for  and  have all been expanded in motivic polylogarithms above.It remains to compute the coefficient ⟨Li  3 (4∕3),  3 ⟩.

5.19.
We have By Proposition 5.10, and Thus, Additionally, So,

We deduce
We use the table of Subsection 5.16 and the fact that ker Δ 2,1 is generated by (3).Let us denote by , , ,  the four last lines of the table: We have found According to the table, Thus, Comparing ( 9) and ( 10), we deduce Thus, the above element is a multiple of (3).The coefficient is precisely ⟨Li 3 (4∕3),  3 ⟩.Computation using a computer algebra system shows that the -adic period of is equal to − 1 3 for several primes (in particular, up to high -adic precision).

5.23.
Summarizing the results of our computations, we have the following equalities up to high precision.In longer equations, we abbreviate  = log  ,  = Li  .

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 section 4 of Deligne-Goncharov [22].-Adic aspects are discussed in Ünver [36].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 -adic realization.

6.2.
Let  be an open subscheme of Spec ℤ,  →  a smooth proper morphism whose generic fiber is unirational,  ⊂  a relative simple normal crossings divisor whose irreducible components are smooth and absolutely irreducible, and let  denote its complement in .By a -integral base-point we mean either a section of  →  or a nonvanishing -family of tangent vectors along a stratum of  which are not tangent to any boundary divisor.

6.3.
Let us be more explicit about our assumptions on a -integral tangent vector .We are provided with a -point  ∶  →  of the compactification.Let denote the normal bundle to  in .We use   both for the structure sheaf and for the coordinate ring of .We may equivalently think of   as a quasi-coherent sheaf on  or as a module over the coordinate ring   , and we do not distinguish between these notationally.In this notation,  ∈ Hom  (,   ) =   is a section that is nowhere tangent to the boundary divisors.This last phrase may be interpreted in several equivalent ways; what we need is the following.If  is a ring, we denote by (()) the ring ⟦⟧[ −1 ] of Laurant series with coefficients in .Let D = Spec   ⟦⟧, let and let  0 denote the normal bundle to D along the zero section { = 0}.Then there is a map ℎ ∶ D → X that maps the zero section { = 0} to , such that D•  maps to  and such that the induced map of normal bundles Fix a prime  of ℤ.Let  denote the natural map Pullback along  induces an equivalence of categories of finite étale coverings.Consequently,  * induces a monoidal equivalence of categories of lisse ℚ  -sheaves.We let Lisse ℚ  (… ) denote the category of lisse sheaves.Let g denote the map Spec ℚ →  induced by the choice of an algebraic closure ℚ of ℚ.The composite (diagonal solid arrow below) defines a "tangential" fiber functor on the category of Lisse ℚ  -sheaves associated to the -integral tangent vector .We note the intermediate composite, denoted  * , for future use.We also note that the same construction defines tangential fiber functors over various base-extensions of  ( ℚ  ,  ℚ , …) and we continue to use the same notation   .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 is an obvious compatibility between the two constructions.We claim that is unramified at .By this, we mean the following.Proposition 6.4.Fix arbitrarily an embedding ℚ ⊂ ℚ  .There is an associated decomposition group Then the induced action of   on   1 ( Q, ) factors through Proof.Our construction of the -adic tangential fiber functor   applies equally over   = Spec ℤ  and the verification of the above statement may take place over   .For this purpose, we temporarily replace  by   .We let  denote the structure morphism  →   .
We define a lisse ℚ  -sheaf  on  to be relatively unipotent if  admits a filtration by lisse subsheaves such that gr  ≃  *  for some lisse sheaf  on   .We define relatively lisse sheaves on  ℚ  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 ℚ  -Tannakian categories which is filled in by canonical natural ⊗-isomorphisms.The functor to ℚ  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 On the other hand, the natural transformation from profinite étale fundamental groups to Tannakian -adic fundamental groups provides a commuting square where  denotes the base-point associated with our choice of algebraic closure.This shows that the action of Gal(ℚ  ∕ℚ  ) on  1 (Lisse un ℚ   ℚ  ) factors through Gal(  ∕  ) 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, Subsection 6.12.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 [31] provides a structure of pro-variation of mixed Hodge structures on the bundle whose fiber at a point  is the prounipotent completion of the fundamental group at ; restricting to an appropriate analytic disk and taking a limit mixed Hodge structure, one obtains a mixed Hodge structure at a tangential base-point.
By [22,Proposition 1.8], to show that  un 1 ( ℚ , ) belongs to the full subcategory of mixed Tate motives over , it is enough to check that at each  ∈ , an -adic realization ( ≠ ) is unramified, as was done in Proposition 6.4.This amounts to an elaboration on [22,Remark 4.18].We write monodromy morphism associated to  is equal to the composite  ∨ ℚ(1) →  un 1 (, ) →  un 1 (, ).
Indeed, this may be checked in any realization, where it becomes evident.When  = (1, 1) 0 we simply write  PL (),  PL ().Our goal for this section is to establish the following The pro-object via (*) and on the second copy by zero, while the second factor on the left acts on the second and third copies of the product via (*) and on the first copy by zero.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 In view of Proposition 6.9,  PL () may be thought of as the abstract polylogarithmic Lie algebra of Subsection 3.2 equipped with a (quite trivial) motivic Galois action.From now on, we allow ourselves to write  PL ,  PL , and so on, in place of  PL (),  PL (), and so on.We let  0,5 ↪  0,5 denote the Deligne-Mumford compactification.The map  extends to a map  which identifies  0,5 with the blowup of ℙ 1 × ℙ 1 at the three points (1,1), (0, ∞), (∞, 0).In particular,  0,5 has three exceptional divisors in addition to the seven boundary divisors listed above.These are all isomorphic to ℙ 1 over Spec ℤ and have strict normal crossings so that the formal neighborhood of each intersection is isomorphic to Spec ℤ⟦, ⟧ with divisors given by  = 0 and  = 0.In particular, there are four ℤ-integral tangential base-points associated to each point of intersection ((  ,   ) = (±1, ±1)).from the free prounipotent group on the set of generators Γ.

The 1-forms
6.12.In the de Rham setting, as in the -adic setting, tangential fiber functors and local monodromy morphisms may be obtained directly from the 1-dimensional construction.In [21, sections 15.28-36], Deligne constructs a functor from the category of vector bundles with integrable connection on D * ℚ ∶= Spec ℚ(()) with regular singularity at  = 0 to the category of vector bundles with integrable on  ,ℚ .Let us denote this functor by .
Let VIC [∞] 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 crossings divisor inside a smooth scheme automatically has regular singularities along the divisor.The same holds for the divisor  = 0 inside the formally smooth ℚ-scheme Dℚ = Spec ℚ⟦⟧.
We let denote the fiber functor associated to the point 1 ∈   .If  is a tangential base-point of  0,5 supported at  ∈  0,5 , we let ℎ  ∶ Dℚ →  0,5 be a map sending the closed point to  and whose derivative sends 1 to .We let ℎ   denote the induced map D * ℚ →  0,5 = .
In terms of the presentation 6.11(*),  PL is the prounipotent group associated to the "abstract polylogarithmic Lie algebra" considered in Section 2. Let  be a ℤ-integral tangential base-point supported along the intersection of  12 with the exceptional divisor  over the point (1, 1) ∈  (recall from segment 6.10 that there are precisely four such).As  is nowhere tangent to , its image  in the relative tangent bundle to  2 ⧵ ( 1 ∪  2 ) along the ℤ-point (1,1) is again a ℤ-integral tangential base-point.
That  12, =  ′ 12, may be checked after passage to de Rham realization, where it is 2.2.Together, the maps  ?define a morphism of pro-mixed Tate motives . (*) We may check that  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  0,5 .This last computation may be extracted from the literature, for instance, as follows.Ünver [36, section 5] constructs generators  , (0 ⩽ ,  ⩽ 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   = 0,   = −  , Σ    = 0, [  ,   ] = 0 whenever {, } ∩ {, } = ∅.
(We have capitalized Ünver's "  " 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 that makes the action evident is given by Oi-Ueno in [35,  as may be seen, for instance, by computing their action on the KZ-connection.The implied relations 2.1(R) are listed in [35, section 4.1].This completes the proof of Proposition 6.9.Remark 6.16.Recall that  denotes the moduli space  0,5 in its guise as is a closed immersion with image the closed subscheme defined by the equation The de Rham first cohomology vector space  1 dR ( 3 ℚ ) =  1 dR () ⊕3 has basis the six 1-forms Their pullbacks along  span  1 dR ( ℚ ) and are linearly independent modulo the one relation (**) If we label the six generators of the de Rham unipotent fundamental group  dR 1 ( 3 ) of  3 at the tangential base-point ( ⃗ 1 0 , ⃗ 1 0 , ⃗ 1 0 ) associated to the 1-forms (*) as follows: then the map of de Rham unipotent fundamental groups In terms of the associated map of Hopf algebras (with dual elements in the Hopf algebra denoted by  ?as usual), equation (*) reads This gives geometric meaning to the equation "  3 =   1 +   2 ".Let  PL 1 ( 3 ) =  PL 1 () 3 denote the quotient of  dR 1 ( 3 ) associated to the polylogarithmic quotient of  dR 1 () (or a quotient thereof by some step of the descending central series).Let  = Spec ℤ[1∕6] as usual, and let () denote the fraction field of the prounipotent mixed Tate Galois group  un 1 ().Let ( 3 ) = () ⊗3 denote the coordinate ring of  PL ( 3 ) () .Let () denote the coordinate ring of the base-change to () of the Selmer scheme )   and similarly for  3 ; we have Then the universal cocycle evaluation maps of  and of  3 , together with the maps   ,   induced by the embedding , form a commuting square of ()-algebras Let  0 ,  1 denote the standard generators of the de Rham unipotent fundamental group of .In These maps induce maps of punctured tangent spaces at ( 1 ,  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  dR 1 ( ℚ , ⃗ 1 0 ) is freely generated by two elements  0 (monodromy about  = 0) and  1 (monodromy about  = 1).The maps induced by ,  on  1 send Then (, ∇) is isomorphic to the universal unipotent connection on  (at any base-point), equipped with its de Rham trivialization.Hence,  BC  −1 3  12 may be represented by the abstract Coleman function given by the connection (, ∇), the projection   −1 3  12 ∶  → , and the Frobenius-compatible family of horizontal sections on residue polydisks with constant term 0 at the tangential base-point (1, 1) (0,0) [11]; this is the same, mutatis mutandis, as the case of ℙ 1 ⧵ {0, 1, ∞} treated, for instance, in Theorem 2.3 of Furusho [24]

) 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 follows: the first factor on the left acts on the first and third copies of the product ∏ ∞ =1 ℚ()

2 ,(
Proposition 4.11, we essentially constructed a certain polynomial () ∈ ()[] such that (after translating along the universal cocycle evaluation map   ) = ⟨,  ♯  0 ⟩.Applied to the three copies of , this gives us three polynomials  1 ,  2 ,  3 such that The images of the three roots in () obey the algebraic relation 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 Φ   0 over () remains ad hoc.)Proof.Let  = Spec ℤ[,  −1 , (1 − ) −1 ] and consider the maps ,  ∶  →  given by

11 . 1 1 𝑒 11 .
We now focus on the map  and the functions  BC  −1 1  By the formula given above for the induced map on fundamental groups,   −1 0  1 pulls back along  to   −On the other hand,   −1 0  1 pulls back along the -adic unipotent Albanese to Li  ().The -adic unipotent Albanese maps fit into a commuting square Combining these facts, we find that where  is the two-sided ideal generated by the Lie relations 2.1(R).Let  =  ⊗   with connection ∇ ∶  →  ⊗ Ω 1  given on a word  in the alphabet  regarded as a section of the trivial pro-vector bundle  by (notation as in Subsection 6.11) ∇() = − 1  1 −  11  11 −  2  2 −  22  22 −  12  12 .

8. 1 . 8 . 6 .
Fix a prime  not dividing 6.In Subsection 4.10, we constructed a polynomial  in the 14 variables   1 ,  11 , … ( lex ) listed in Subsection 4.1 whose coefficients are rational functions in the 11 symbols   ,   , … ( lex ) (also listed in Subsection 4.1) over the rationals.Using the equations obtained in Subsection 5.23 and replacing motivic polylogarithms by their -adic periods, we obtain a polynomial with coefficients in ℚ  .In terms of the coordinates  1 ,  2 on  =  0,5 (Subsection 7.1), we replace the We may then base-change  along the evident maps (Subsection 8.3).We denote the base-change to  ′ () ⩾−4 by  ′ , we denote the base-change to Spec ℚ  by    , and we denote the base-change to () ⩾−4 simply by .This completes our definitions of the objects and morphisms in the following diagram, whose commutativity is clear.(**)Since Lyndon words provide an algebra-basis for the shuffle algebra, the arithmetic shuffle coordinates (A f un ) obtained by reversing the order of letters in 8.1(A lex ), form an algebra basis of ()[⩽4]  .The morphism 6.11(*) provides a presentation of  un 1 () (at any base-point) with Liealgebra relations given by 2.1(R)[35].According to Lemma 3.3, the geometric shuffle coordinates (G f un ) obtained by reversing the order of letters in 8.1(G lex ), form an algebra basis of PL [⩽4] () = (dR *  PL ⩾−4 ()).In this way,  defines a function on() ⩾−4 × dR *  PL ⩾−4 ().Propositions (4.3) op and (4.11) op show that  vanishes on the image of the evaluation map .The computations of (Subsection 5) op as summarized in (Subsection 5.23) op allow us to replace the coefficients of  by polynomials in motivic polylogarithms that are unramified over , at the cost of a possible -adic error of size determined by the precision of the -adic periods on which these computations depend.Numerical evaluation of the -adic periods of the coefficients then shows that  factors through  ′ () ⩾−4 × dR *  PL ⩾−4 () and it follows that  vanishes on the image of  ′ .Pullback by   corresponds to replacing the coefficients in  by their -adic periods.Further, according to Proposition 7.2, pullback by the J O U R N A L I N F O R M AT I O N Mathematika is owned by University College London and published by the London Mathematical Society.All surplus income from the publication of Mathematika is returned to mathematicians and mathematics research via the Society's research grants, conference grants, prizes, initiatives for early career researchers and the promotion of mathematics.