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
.
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 [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.
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 [11]). 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
[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. École 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.