A relative basis for mixed Tate motives over the projective line minus three points

Soudères, Ismaël

doi:10.4310/cntp.2016.v10.n1.a4

Cited by 2 publications

(5 citation statements)

References 17 publications

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“…This intuition is confirmed in [Sou14] which shows that motives associated to cycles L 0 W | {x=z} generate the Deligne-Goncharov motivic fundamental group. This is a consequence of this paper because their cobracket (corresponding to the differential of cycles) is exactly given by Equation (1) and hence by Ihara's cobracket.…”

Section: Multiple Polylogarithms and Algebraic Cyclessupporting

confidence: 62%

“…The main result of this paper (Theorem 5.8) provides such algebraic cycles denoted by L 0 W ; here the indexing set W consists in Lyndon words. It is shown in [Sou14] that the motives attached to these cycles generate the Lie coalgebra associated to the Deligne-Goncharov fundamental group.…”

Section: Introductionmentioning

confidence: 99%

“…In particular, the cycles L 0 W , built in this paper, induce motivic elements which generate the Lie coalgebra associated to the Deligne-Goncharov motivic fundamental group (see [Sou14]). …”

Section: Introductionmentioning

confidence: 99%

“…The above reasons, together with the integral computed in the previous section, lead the author to believe that cycles L 0 W | x=1 correspond to multiple zeta values. In this direction, the author proved that the bar elements associated to the family L 0 qf, W give a basis ofDeligne-Goncharov motivic fundamental group (see [Sou14]). It remains to explicitly compute the periods of these motives, that is the associated integral.…”

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confidence: 99%

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The cycle complex over P1 minus 3 points: Toward multiple zeta value cycles

Soudères

2016

Journal of Pure and Applied Algebra

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Abstract. In this paper, we construct a family of algebraic cycles in Bloch's cycle complex over P 1 minus three points, which are expected to correspond to multiple polylogarithms in one variable. Elements in this family of weight p belong to the cubical cycle group of codimension p in (P 1 \ {0, 1, ∞}) × (P 1 \ {1}) 2p−1 and in weight greater than or equal to 2, they naturally extend as equidimensional cycles over A 1 .Thus, we can consider their fibers at the point 1. This is one of the main differences with the work of Gangl, Goncharov and Levin. Considering the fiber of our cycles at 1 makes it possible to view these cycles as those corresponding to weight n multiple zeta values which are viewed here as the values at 1 of multiple polylogarithms.After the introduction, we recall some properties of Bloch's cycle complex, and explain the difficulties on a few examples. Then a large section is devoted to the combinatorial situation, essentially involving the combinatorics of trivalent trees in relation with the structure of the free Lie algebra on two generators. In the last section, two families of cycles are constructed as solutions to a "differential system" in Bloch's cycle complex. One of these families contains only cycles with empty fiber at 0; these correspond to multiple polylogarithms. Lie coalgebra generating its 1-minimal model is isomorphic to the above Tannakian Lie coalgebra. ContentsIn order to obtain a motive, that is a comodule, it is enough to have an element in the relevant Tannakian Lie coalgebra. Then the motive is the comodule cogenerated by this element. Since the Tannakian Lie coalgebra is a "H 0 modulo product", we write one of its elements as a class [L B ] in this H 0 . This class can be represented in the bar construction over N • by an element L B . The element L B is essentially determined by its projection onto its tensor degree 1 part, which gives an algebraic cycle L in N • . The cycle L has a decomposable boundary because the element L B leads to a class in the H 0 .Hence a first step toward explicit motives via algebraic cycles is to build algebraic cycles in N • having a decomposable boundary. The main result of this paper (Theorem 5.8) provides such algebraic cycles denoted by L 0 W ; here the indexing set W consists in Lyndon words. It is shown in [Sou14] that the motives attached to these cycles generate the Lie coalgebra associated to the Deligne-Goncharov fundamental group.However lifting algebraic cycles to objects in the bar construction requires, in general, a strong combinatorial or algebraic control of the boundaries involved. In our context, this control is insured (Theorem 5.8) by the algebraic and combinatorial structure of Ihara's special derivations studied at Proposition 4.27.The next steps in order to obtain motivic multiple zeta values are:(1) Our main theorem makes it possible to build a commutative differential graded algebra morphism between the cobar construction over the Lie coalgebra associated to Ihara's special derivations (combinatorial structure...

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Section: Multiple Polylogarithms and Algebraic Cyclessupporting

confidence: 62%

Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

mentioning

confidence: 99%

See 2 more Smart Citations

The cycle complex over P1 minus 3 points: Toward multiple zeta value cycles

Soudères

2016

Journal of Pure and Applied Algebra

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“…Soudéres [22,21] extends the family of algebraic cycles studied by Gangl, Goncharov and Levin to include those over a more general base scheme, in particular giving a rigorous construction of unital values of the multiple polylogarithms ie. multiple zeta values, as periods (and not just non-unital values of the multiple logarithms).…”

Section: Introductionmentioning

confidence: 99%

Rational mixed Tate motivic graphs

Agarwala

Patashnick

2017

Ann. K-Th.

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In this paper, we study the combinatorics of a subcomplex of the Bloch-Kriz cycle complex [4] used to construct the category of mixed Tate motives. The algebraic cycles we consider properly contain the subalgebra of cycles that correspond to multiple logarithms (as defined in [12]). We associate an algebra of graphs to our subalgebra of algebraic cycles. We give a purely graphical criterion for admissibilty. We show that sums of bivalent graphs correspond to coboundary elements of the algebraic cycle complex. Finally, we compute the Hodge realization for an infinite family of algebraic cycles represented by sums of graphs that are not describable in the combinatorial language of [12].

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A relative basis for mixed Tate motives over the projective line minus three points

Cited by 2 publications

References 17 publications

The cycle complex over P1 minus 3 points: Toward multiple zeta value cycles

The cycle complex over P1 minus 3 points: Toward multiple zeta value cycles

Rational mixed Tate motivic graphs

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