##
**Tensor powers of the Carlitz module and zeta values.**
*(English)*
Zbl 0713.11082

Let \(k\) be a number field and \(X\) a smooth, projective variety over \(k\). Let \(H^*(X)\) be the total (graded) cohomology of \(X\) in any one of the standard theories: Betti, \(\ell\)-adic (étale), or de Rham. It is very well-known that such theories satisfy various formal laws such as: finite dimensionality, contrafunctoriality (i.e., cycles of the proper codimension give rise to maps of cohomology which respect the gradation), the Künneth formula (the total cohomology of the product \(X\times Y\) equals \(H^*(X)\otimes H^*(Y))\) and so on. Thus, roughly speaking, by viewing the cohomology of the product as being a “tensor product” one can think of the category of varieties as a “tensor category”.

The prototypical example of a tensor category is the category \(\mathcal{REP}_ G\) of finite-dimensional representations of an affine group scheme \(G\). Clearly, one can form tensor products and duals in this category with \({\hat \rho}^\wedge=\rho\). Let \(\omega_{\text{rep}}\) be the forgetful functor which takes \(\rho\) to the underlying vector space. Then \(\omega_{\text{rep}}\) commutes with tensor product and is faithful. A Tannakian category \(\mathcal C\) is a category with a tensor product and faithful vector space valued “fibre functor” \(\omega\) with enough restriction to make it look like \(\{\mathcal{REP}_ G,\omega_{\text{rep}}\}\). The “Tannakian philosophy” states that the automorphisms of \(\omega\) form a group scheme \(G\) together with an isomorphism of \({\mathcal C}\) and \(\mathcal{REP}_ G\). Standard theory expresses \(G\) as an inverse limit of group schemes of finite type as \(\mathrm{Gal}(\overline{\mathbb Q}/\mathbb Q)\) is the inverse limit of finite Galois groups.

Since the cohomology of \(X\) is graded, Grothendieck asked if there is a category \(\mathcal{MOT}_ k\), derived from the category of varieties, such that in \(\mathcal{MOT}_ k\) we have

(1) The decomposition \(X=\oplus h^ i(X)\) with \(H^*(h^ i(X))=H^ i(X),\)

(2) Each object \(X\in\mathcal{MOT}_ k\) has a “dual” \(\hat X\) with \(\hat X^\wedge=X\); etc.

If certain “Standard Conjectures” on cycles (e.g., the Künneth components of the diagonal are algebraic) are true then \(\mathcal{MOT}_ k\) exists and is called the category of motives. (To obtain duals one formally inverts the “Tate motive” \(\mathcal T\) defined by \(P^ 1=1\oplus\mathcal T.)\) By tweaking the commutativity requirements of \(\otimes\) in \(\mathcal{MOT}_ k\) (which arise from the Künneth formula), one proves that \(\mathcal{MOT}_ k\) is Tannakian with Betti cohomology \(=\omega\). The group scheme \(G\) is called “the geometric Galois group”.

One attaches to a motif an \(L\)-series in the standard way using cohomology. If one knew the above conjectures, then the motif would be determined up to isomorphism by its \(L\)-series; a vast generalization of Faltings’ Isogeny Theorem.

Let \(k\) now be a function field over a finite field and let \(\infty\) be a fixed place. Let \(A\) be the ring of functions holomorphic away from \(\infty\) and \(K=k_{\infty}\). In this situation there is an analog of elliptic curves called “elliptic modules” or “Drinfeld modules”: Let \(L\) be a finite extension of \(k\). A Drinfeld module over \(L\) is an injection \(\phi\) of the ring \(A\) into the ring of \(L\)-endomorphisms of the additive group \(G_ a\) such that \( a=D\circ \phi_ a\); \(D=\)differentiation. For instance, when \(A=F_ q[T]\) the simplest Drinfeld module is called the “Carlitz module” (after L. Carlitz who introduced it in the 1930’s) and is defined by \( C_ T(z)=Tz+z^ r. \)

A foundational result of Drinfeld establishes an analytic equivalence between Drinfeld modules and “\(A\)-lattices” (i.e., discrete \(A\)-submodules of finite rank of the algebraic closure \(\overline K\) of \(K\)). Morphisms of Drinfeld modules (i.e., endomorphisms of \(G_ a\) which commute with the \(A\)-action) correspond to morphisms of lattices as for elliptic curves. The rank of the Drinfeld module is the rank of its lattice, and, like elliptic curves, one views the dual of the lattice (\(\otimes\) field of fractions) as the (first) Betti cohomology. (For Drinfeld modules, as for elliptic curves, one does not need higher groups.)

Based on the above analogy, the first author constructed “motives” for Drinfeld modules in his seminal paper “t-motives” [Duke Math. J. 53, 457–502 (1986; Zbl 0679.14001)]: Instead of \(G_ a\) one uses \(G_ a^ n\). Moreover, one weakens the condition that \(a=D\circ \phi_ a\) by allowing the equation to be true up to a nilpotent endomorphism; thus one obtains “\(t\)-modules”. The “\(t\)-motives” are anti-duals of the \(t\)-modules; in a natural way, one can tensor \(t\)-motives (and thus the \(t\)-modules). Not all t-modules arise from \(n\)-dimensional lattices; those that do are called “uniformizable”. The first author (loc. cit.) gave a criterion for uniformizability - the analog of the classical criterion for a lattice to be algebraizable. Moreover he showed:

(1) Tensor product preserves uniformizability.

(2) Tensor product commutes with lattice formation (almost - the module of Kähler differentials of \(A\) intervenes; a cohomology functor commuting exactly with tensor products can be defined).

The second statement leads to duals: Take the highest exterior power of a motive; this is the motive of \(E^{\otimes n}\) for \(E\) a rank one Drinfeld module. Inverting \(E\) (as with the Tate motif) gives the dual. Thus the isogeny classes of uniformizable t-motives gives rise to an (honest – no unproven conjectures!) Tannakian category \(\mathcal{MOT}_ L\); the fibre functor is the dual of the lattice \(\otimes k\) generation functor! (To obtain good \(L\)-series, one restricts to the Tannakian subcategory of “pure” motives – these have the “correct” eigenvalues of Frobenius.) Very little is known about the geometric Galois group \(G\) of \(\mathcal{MOT}_ L\). For instance, is it reduced? Since \(\text{char}(k)>0\), this is not automatic.

The authors are concerned with the subcategory of \(\mathcal{MOT}_ k\) generated by \(C^{\otimes n}\) in the polynomial case; in fact, they present an elementary, self-contained approach to this category. Moreover, they show the following elegant fact: For each \(n>0\) there exists an algebraic point on \(C^{\otimes n}\) which is essentially the exponential of \(\zeta(n)=\sideset\and {'}\to\sum a^{-n}\) \((\sideset\and {'}\to\sum=\) sum over monics in \(A\)). Using this and his analogues of the theorems of Hermite-Lindemann, Jing Yu [Ann. Math. (2) 134, 1–23 (1991; Zbl 0734.11040)] establishes:

(1) \(\zeta(n)\) is transcendental for all positive \(n\).

(2) Let \(n\not\equiv 0\pmod{q-1}\). Then \(\zeta(n)/{\bar \pi}^ n\) is also transcendental (\({\bar \pi}=\) the period of \(C\)).

Yu and the authors also extended their results to the \(v\)-adic interpolations of \(\zeta(n)\) for \(v\in\text{Spec}(F_ q[T]).\)

The results of the authors were inspired by motivic computations of P. Deligne giving extensions of \(\mathbb Z(0)\) by \(\mathbb Z(n)\). They are broadly in the spirit of recent conjectures of D. Zagier relating multilogs and \(\zeta\)-functions of number fields.

The prototypical example of a tensor category is the category \(\mathcal{REP}_ G\) of finite-dimensional representations of an affine group scheme \(G\). Clearly, one can form tensor products and duals in this category with \({\hat \rho}^\wedge=\rho\). Let \(\omega_{\text{rep}}\) be the forgetful functor which takes \(\rho\) to the underlying vector space. Then \(\omega_{\text{rep}}\) commutes with tensor product and is faithful. A Tannakian category \(\mathcal C\) is a category with a tensor product and faithful vector space valued “fibre functor” \(\omega\) with enough restriction to make it look like \(\{\mathcal{REP}_ G,\omega_{\text{rep}}\}\). The “Tannakian philosophy” states that the automorphisms of \(\omega\) form a group scheme \(G\) together with an isomorphism of \({\mathcal C}\) and \(\mathcal{REP}_ G\). Standard theory expresses \(G\) as an inverse limit of group schemes of finite type as \(\mathrm{Gal}(\overline{\mathbb Q}/\mathbb Q)\) is the inverse limit of finite Galois groups.

Since the cohomology of \(X\) is graded, Grothendieck asked if there is a category \(\mathcal{MOT}_ k\), derived from the category of varieties, such that in \(\mathcal{MOT}_ k\) we have

(1) The decomposition \(X=\oplus h^ i(X)\) with \(H^*(h^ i(X))=H^ i(X),\)

(2) Each object \(X\in\mathcal{MOT}_ k\) has a “dual” \(\hat X\) with \(\hat X^\wedge=X\); etc.

If certain “Standard Conjectures” on cycles (e.g., the Künneth components of the diagonal are algebraic) are true then \(\mathcal{MOT}_ k\) exists and is called the category of motives. (To obtain duals one formally inverts the “Tate motive” \(\mathcal T\) defined by \(P^ 1=1\oplus\mathcal T.)\) By tweaking the commutativity requirements of \(\otimes\) in \(\mathcal{MOT}_ k\) (which arise from the Künneth formula), one proves that \(\mathcal{MOT}_ k\) is Tannakian with Betti cohomology \(=\omega\). The group scheme \(G\) is called “the geometric Galois group”.

One attaches to a motif an \(L\)-series in the standard way using cohomology. If one knew the above conjectures, then the motif would be determined up to isomorphism by its \(L\)-series; a vast generalization of Faltings’ Isogeny Theorem.

Let \(k\) now be a function field over a finite field and let \(\infty\) be a fixed place. Let \(A\) be the ring of functions holomorphic away from \(\infty\) and \(K=k_{\infty}\). In this situation there is an analog of elliptic curves called “elliptic modules” or “Drinfeld modules”: Let \(L\) be a finite extension of \(k\). A Drinfeld module over \(L\) is an injection \(\phi\) of the ring \(A\) into the ring of \(L\)-endomorphisms of the additive group \(G_ a\) such that \( a=D\circ \phi_ a\); \(D=\)differentiation. For instance, when \(A=F_ q[T]\) the simplest Drinfeld module is called the “Carlitz module” (after L. Carlitz who introduced it in the 1930’s) and is defined by \( C_ T(z)=Tz+z^ r. \)

A foundational result of Drinfeld establishes an analytic equivalence between Drinfeld modules and “\(A\)-lattices” (i.e., discrete \(A\)-submodules of finite rank of the algebraic closure \(\overline K\) of \(K\)). Morphisms of Drinfeld modules (i.e., endomorphisms of \(G_ a\) which commute with the \(A\)-action) correspond to morphisms of lattices as for elliptic curves. The rank of the Drinfeld module is the rank of its lattice, and, like elliptic curves, one views the dual of the lattice (\(\otimes\) field of fractions) as the (first) Betti cohomology. (For Drinfeld modules, as for elliptic curves, one does not need higher groups.)

Based on the above analogy, the first author constructed “motives” for Drinfeld modules in his seminal paper “t-motives” [Duke Math. J. 53, 457–502 (1986; Zbl 0679.14001)]: Instead of \(G_ a\) one uses \(G_ a^ n\). Moreover, one weakens the condition that \(a=D\circ \phi_ a\) by allowing the equation to be true up to a nilpotent endomorphism; thus one obtains “\(t\)-modules”. The “\(t\)-motives” are anti-duals of the \(t\)-modules; in a natural way, one can tensor \(t\)-motives (and thus the \(t\)-modules). Not all t-modules arise from \(n\)-dimensional lattices; those that do are called “uniformizable”. The first author (loc. cit.) gave a criterion for uniformizability - the analog of the classical criterion for a lattice to be algebraizable. Moreover he showed:

(1) Tensor product preserves uniformizability.

(2) Tensor product commutes with lattice formation (almost - the module of Kähler differentials of \(A\) intervenes; a cohomology functor commuting exactly with tensor products can be defined).

The second statement leads to duals: Take the highest exterior power of a motive; this is the motive of \(E^{\otimes n}\) for \(E\) a rank one Drinfeld module. Inverting \(E\) (as with the Tate motif) gives the dual. Thus the isogeny classes of uniformizable t-motives gives rise to an (honest – no unproven conjectures!) Tannakian category \(\mathcal{MOT}_ L\); the fibre functor is the dual of the lattice \(\otimes k\) generation functor! (To obtain good \(L\)-series, one restricts to the Tannakian subcategory of “pure” motives – these have the “correct” eigenvalues of Frobenius.) Very little is known about the geometric Galois group \(G\) of \(\mathcal{MOT}_ L\). For instance, is it reduced? Since \(\text{char}(k)>0\), this is not automatic.

The authors are concerned with the subcategory of \(\mathcal{MOT}_ k\) generated by \(C^{\otimes n}\) in the polynomial case; in fact, they present an elementary, self-contained approach to this category. Moreover, they show the following elegant fact: For each \(n>0\) there exists an algebraic point on \(C^{\otimes n}\) which is essentially the exponential of \(\zeta(n)=\sideset\and {'}\to\sum a^{-n}\) \((\sideset\and {'}\to\sum=\) sum over monics in \(A\)). Using this and his analogues of the theorems of Hermite-Lindemann, Jing Yu [Ann. Math. (2) 134, 1–23 (1991; Zbl 0734.11040)] establishes:

(1) \(\zeta(n)\) is transcendental for all positive \(n\).

(2) Let \(n\not\equiv 0\pmod{q-1}\). Then \(\zeta(n)/{\bar \pi}^ n\) is also transcendental (\({\bar \pi}=\) the period of \(C\)).

Yu and the authors also extended their results to the \(v\)-adic interpolations of \(\zeta(n)\) for \(v\in\text{Spec}(F_ q[T]).\)

The results of the authors were inspired by motivic computations of P. Deligne giving extensions of \(\mathbb Z(0)\) by \(\mathbb Z(n)\). They are broadly in the spirit of recent conjectures of D. Zagier relating multilogs and \(\zeta\)-functions of number fields.

Reviewer: David Goss (Columbus/Ohio)

### MSC:

11R58 | Arithmetic theory of algebraic function fields |

11G09 | Drinfel’d modules; higher-dimensional motives, etc. |

14A20 | Generalizations (algebraic spaces, stacks) |

14G25 | Global ground fields in algebraic geometry |

14L05 | Formal groups, \(p\)-divisible groups |

14C35 | Applications of methods of algebraic \(K\)-theory in algebraic geometry |