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.

This is an expanded note prepared for a STAGE talk, Fall 2014. Our main references are [1],[2],[3], [4] and [5].

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 [6]). 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 [7] and [8]) 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 .

Definition 1

- are abelian invariants of "vector spaces" over . The isomorphism classes of finitely generated projective modules over form a semigroup under the direct sum operation. is defined to be the group completion (aka. Grothendieck group) of this semigroup. The analogy with topological -theory is evident.
- are abelian invariants of "matrices" over . Let , where via . Then is defined to be . In fact the commutator , where is generated by elementary matrices (1's on the diagonal, on the -entry).
- is defined to be the group homology , which measures the relations between the elementary matrices . Since is perfect, i.e., , we know that is isomorphic to the kernel of the universal central extension of . Here is known as the
*Steinberg group*of .

Example 1

The definition of these lower -groups may seem a bit random at first glance. The good thing is that they fit nicely in long exact sequence (like in usual cohomology theory): where is an ideal and are suitably defined - (= dimension of vector spaces over a field), (each finitely generated projective -module is isomorphic to a fractional ideal; ).
- , (a matrix over a field or can be transformed via elementary matrices into ; notice, however, this is not true for a general Dedekind domain ).
- can be identified as Milnor's -group (defined for fields) is known to a finite abelian group but harder to compute (it is known that ).

- , where is the -construction applied to the classifying space of . The -construction of a topological space modifies its fundamental group but does not change its homology;
- . Here is a category whose objects are finitely generated projective -modules and is the set of equivalence classes of diagrams such that and are also finitely generated -modules. is the loop space of the classifying space of the category : so .

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 .

Remark 2
Higher -groups are generally very difficult to compute. A useful consequence of Quillen's -construction is the *localization theorem* for computing -groups. For example, for any Dedekind domain with fraction field , there is a long exact sequence where runs over all maximal ideals of . For , more is true: Soule proved that the map is injective, therefore by Example 2for each , we have and for each an exact sequence In particular, by Borel's theorem, is an infinite torsion group.

Remark 3
Quillen's construction can be extended to any scheme . The Chern classes of algebraic -theory induce an isomorphism Here is the weight eigenspace of the Adams operators acting on and is an object in . Theorem 1 is a special case when , and since by a result of Soule. Also, for , the extension group doesn't change when restricting to . Indeed we have when as in the previous remark.

Remark 4
For , the Dedekind zeta function has order of vanishing equal to at . More generally, Beilinson's first conjecture predicts the relation between order of vanishing of motivic -function and motivic cohomology groups when : From this point of view, Borel's theorem verifies the special case , .

Remark 5
The torsion part of is harder to compute. People knew the case when is odd. People also knew the order of and conjectured that should be a cyclic group. The order is related to the -th Bernoulli number: no surprise they should relate to the Riemann zeta function as well (Lichtenbaum's conjecture)! People also conjectured that — this is in fact equivalent to Vandiver's conjecture on class groups of cyclotomic fields and seems to be extremely difficult.

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 :

Theorem 3
is a free pro-unipotent group generated by elements in degree . In particular, when , it follows from Borel's theorem that is a free pro-unipotent group with one generator in each odd degree .

Let us review a bit background on pro-unipotent completions (see [9]). 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.

Definition 2
A topological vector space is *linearly compact* if it is homeomorphic to , where are discrete and finite dimensional. If is linearly compact and is its topological dual, then the linear dual .

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).

Example 3
Suppose is a free abstract group on two generators .

- is the non-commutative polynomial ring in .
- is the non-commutative power series ring in . As a graded vector space, is isomorphic to the , where is a 2-dimensional vector space with basis .
- is the free pro-unipotent group in two generators. As a graded vector space, . The algebra structure is given by the shuffle product and the coalgebra structure is given by de-concatenation.
- consists of group-like power series in .

Remark 6
Analogous to free groups, free pro-unipotent groups can be characterized by the *lifting property*: if is an exact sequence of pro-unipotent groups, for a free pro-unipotent group , any homomorphism lifts to a homomorphism . Also analogous to usual group cohomology, for a pro-unipotent group , (resp., ) has an interpretation of generators (resp., relations). In particular, if , then is a free pro-unipotent group, with generators given by a basis of (see [10]).

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.

Proof
For any -representation , the Hochschild-Serre spectral sequence implies that Since is reductive, we have for . Therefore, Now take , we obtain that Since , we obtain that as desired.
¡õ

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 [11]). We are interested in finding the dimension of the -linear span of multiple zeta values of weight .

Example 4
The multiple zeta value can be viewed as an iterated integral as follows. Let be a word in 0 and 1 starting with 1 and ending with 0. Let Then it is easy to see that From this integral formula one can deduce that the product of two multiple zeta values with words and can be written as an -linear combinations of other multiple zeta values (by shuffling and ). For example, , , . The shuffle product of with is which implies the relation These are known as *shuffle relations*.

Example 5
From the original definition one can also write the product of two multiple zeta values as an -linear combination by reordering the summation. For example, These relations are known as *stuffle relations*.

Example 6
Another more striking relation is given by the *sum theorem* (proved by Granville and Zagier independently). It says that the sum of multiple zeta values with the same weight and depth are the same when varying the depth . For example,

The following remarkable conjecture of Zagier predicts that though there are multiple zeta values with weight , the -span of them is much smaller.

Example 7
The shuffle relation, the stuffle relation and the sum relation together implies that In particular, as predicted (these are all rational multiples of !) Zagier's conjecture can be checked for small values of (this is done at least for ), which probably serves as a good reason for making such a conjecture.

Using the concrete description of pro-unipotent group , we are now able to prove the following upper bound.

Proof (Idea of the proof)
Mimicking the iterated integrals defining the multiple zeta values, one can define motivic iterated integral using the theory of motivic fundamental group of , and then define a graded algebra consisting of motivic multiple zeta values whose image under a period map are the classical multiple zeta values .

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

[1]Mixed Tate motives over $\Bbb Z$, Ann. of Math. (2) 175 (2012), no.2, 949--976.

[2]Motivic periods and the projective line minus three points, ArXiv e-prints (2014).

[3]On multiple zeta values, www.ihes.fr/~brown/Arbeitstatung.pdf.

[4]Groupes fondamentaux motiviques de Tate mixte, Ann. Sci. &EACUTE;cole Norm. Sup. (4) 38 (2005), no.1, 1--56.

[5]Tannakian fundamental groups associated to Galois groups, Galois groups and fundamental groups, Math. Sci. Res. Inst. Publ., 41 Cambridge Univ. Press, Cambridge, 2003, 183--216.

[6]Tate motives and the vanishing conjectures for algebraic $K$-theory, Algebraic $K$-theory and algebraic topology (Lake Louise, AB, 1991), NATO Adv. Sci. Inst. Ser. C Math. Phys. Sci., 407 Kluwer Acad. Publ., Dordrecht, 1993, 167--188.

[7]Algebraic $K$-theory and its applications, Springer-Verlag, New York, 1994.

[8]Algebraic $K$-theory of rings of integers in local and global fields, Handbook of $K$-theory. Vol. 1, 2, Springer, Berlin, 2005, 139--190.

[9]The pro-unipotent completion, http://perso.univ-rennes1.fr/alberto.vezzani/Files/Research/prounipotent.pdf.

[10]Cohomology of unipotent and prounipotent groups, J. Algebra 74 (1982), no.1, 76--95.

[11]Relations among multiple zeta values, http://www2.mathematik.hu-berlin.de/~mzv/mzv2013.pdf.