After recalling the category of mixed Tate motives over and its relationship with -groups, we discuss the basics of algebraic -theory and Borel's computation of the rational -groups of . We then explain how to use Borel's theorem to give a concrete description of the motivic Galois group of and bound the dimension of the span of multiple zeta values.
Let be a number field with ring of integers . Last time we introduced a neutral Tannakian category of mixed Tate motives over . Intuitively, it consists of mixed motives over which are successive extensions of the Tate objects . Since the category of mixed motives has yet to be constructed, we had to work around to define . People (e.g, Voevodsky) have constructed a triangulated tensor category that behaves like the bounded derived category of the conjectural (even over more general base schemes). We take a triangulated tensor subcategory generated by the Tate objects. The Beilinson-Soule vanishing conjecture is true for and it allows one to define a -structure on using its natural weight structure. The category of mixed Tate motives is then defined as the heart of this -structure (see ). We also introduced a subcategory of mixed Tate motives over . The -adic realization of a mixed Tate motive defined over is unramified at any place of . These categories can be pictured as follows. Recall that the subcategory is defined via restricting the extension group These can be reinterpreted as the first algebraic -group and the restriction can be viewed as a special case of the following theorem, which is a (highly nontrivial) consequence of the construction of .
We now briefly explain the basics of algebraic -theory (see  and ) and state Borel's theorem on these rational -groups of . Algebraic -theory is a sequence of functors which, roughly speaking, extracts abelian invariants from "linear algebra construction" of .
These two constructions look very different on the surface, but they turn out to produce the same -groups.
When is the ring of integers of a number field , Quillen proved that is always finitely generated. A celebrated theorem of Borel further computed the rank of .
Recall that the de Rham realization functor is a fiber functor (i.e., a faithful -linear exact tensor functor). Here is multiplicity space of the weight piece of . This fiber functor makes is neutral Tannakian category. We denote by the corresponding Tannakian fundamental group. Then is a pro-algebraic group defined over and is equivalent to , the category of finite dimensional -representations of . Naturally is called the motivic Galois group of . By definition we have Our next goal is to give a concrete description of using this relation between the group cohomology of and the extension groups computed by Borel's theorem.
For any object , we denote by the full Tannakian subcategory generated by (whose objects are subquotients of ). If , then we have a surjection , which induces an isomorphism The "smallest" nontrivial Tannakian subcategory is nothing but the category of pure Tate motives over . Notice that is determined by its action on the 1-dimensional vector space , hence is . This induces a surjection and we denote its kernel by . Since acts trivially on each graded piece of , is pro-unipotent. Moreover, the exact sequence in fact splits: a splitting is given by multiplication by on . This identifies as the semi-direct product It remains to compute :
Let us review a bit background on pro-unipotent completions (see ). For any abstract group , its group algebra is naturally a Hopf algebra under Moreover, the group can be recovered as the group-like elements of the Hopf algebra , These two functors form an adjoint pair Notice that the Hopf algebra is cocommutative not necessarily commutative (it is commutative if and only if is abelian), hence does not correspond to the coordinate ring of an algebraic group. To obtain a commutative Hopf algebra, the natural idea is to take dual. To ensure taking dual is a reasonable operation, the topology on such Hopf algebras should not be too far away from finite dimensional vector spaces.
For a Hopf algebra with argumentation ideal , the -adic completion is is linearly compact if and only if is finite dimensional. Restricting to such a subcategory we obtain a functor given by . Moreover, for a unipotent algebraic group , gives an adjoint functor (but not an equivalence).
Now let us come back to our original situation , where is the motivic Galois group of . The cohomology can be computed as follows. Theorem 3 then follows from it in view of the previous Remark 6 and Theorem 1.
Finally, we will use the concrete description of in Theorem 3 to give an explicit upper bound for the span of multiple zeta values. Recall that for such that , the multiple zeta values is defined to be where is called its depth and is called its weight. The depth case recovers the classical zeta values , . These are transcendental real numbers satisfying a lot of mysterious relations (see ). We are interested in finding the dimension of the -linear span of multiple zeta values of weight .
The following remarkable conjecture of Zagier predicts that though there are multiple zeta values with weight , the -span of them is much smaller.
Using the concrete description of pro-unipotent group , we are now able to prove the following upper bound.
More precisely, let be the motivic torsor of paths from 0 to 1. Evaluating a regular function on at the straight path (droit chemin in French) from 0 to 1 gives a homomorphism Concretely, is a group-like power series in two letters 0 and 1 (see Example 3) whose coefficient before a word is given by the iterated integral . Recall that is an ind-object in and thus acts on it. Then is defined to be the quotient of by the "motivic relations" between multiple zeta values: namely the quotient by the largest ideal which is stable under the action of .
By Theorem 3, we have as a graded vector space (where has degree ). It turns out that one can find a (non-canonical) injective -comodule morphism which maps to and to . Clearly an upper bound on the dimension of the weight piece of gives an upper bound on , hence on .
Now the point is that we can compute explicitly! It is the coefficients of in the generating series which is equal to This last generating series is nothing but ! ¡õ
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