### On Algebraic Independence of Special Zeta Values in Characteristic p

Jing Yu

National Taiwan University and TIMS, Taipei

July, 2010, Workshop on Arithmetic Geometry

### t-motives

Let t, σ be variables independent of θ.

Let ¯k(t)[σ, σ^{−1}] be noncommutative ring of Laurent polynomials
in σ with coefficients in ¯k(t), subject to the relation

σf := f^{(−1)}σ for all f ∈ ¯k(t).

Here f^{(−1)} is the rational function obtained from f ∈ ¯k(t) by
twisting all its coefficients a ∈ ¯k to a^{1}^{q}.

A pre-t-motive M over Fq is a left ¯k(t)[σ, σ^{−1}]-module
which is finite-dimensional over ¯k(t).

Let m ∈ Matr×1(M ) be a ¯k(t)-basis of M .

Multiplying by σ on M is represented by σ(m) = Φm for some
matrix Φ ∈ GL_{r}(¯k(t)).

The category of pre-t-motives over Fq forms a rigid abelian Fq(t)-linear tensor category.

### t-motives

Let t, σ be variables independent of θ.

Let ¯k(t)[σ, σ^{−1}] be noncommutative ring of Laurent polynomials
in σ with coefficients in ¯k(t), subject to the relation

σf := f^{(−1)}σ for all f ∈ ¯k(t).

Here f^{(−1)} is the rational function obtained from f ∈ ¯k(t) by
twisting all its coefficients a ∈ ¯k to a^{1}^{q}.

A pre-t-motive M over Fq is a left ¯k(t)[σ, σ^{−1}]-module
which is finite-dimensional over ¯k(t).

Let m ∈ Matr×1(M ) be a ¯k(t)-basis of M .

Multiplying by σ on M is represented by σ(m) = Φm for some
matrix Φ ∈ GL_{r}(¯k(t)).

The category of pre-t-motives over Fq forms a rigid abelian Fq(t)-linear tensor category.

### Frobenius difference equations

From a pre-t-motive, one associates a “system of Frobenius difference equation”which has solutions in series of t.

Consider the operator on C∞ by x 7→ x^{1}^{q}.
Then extend this operator to C^{∞}((t)) as follows,
for f =P

ia_{i}t^{i}∈ C∞((t)) define f^{(−1)} :=P

ia^{q}_{i}^{−1}t^{i}.
More generally, for matrix B with entries in C∞((t)) define
twisting B^{(−1)} by the rule B^{(−1)}ij = Bij(−1).

Let m ∈ Matr×1(M ) be a ¯k(t)-basis of M .

Multiplying by σ on M is represented by σ(m) = Φm for some
matrix Φ ∈ GL_{r}(¯k(t)).

Theequationto be solved in Ψ ∈ Matr(C^{∞}((t)) is :
Ψ^{(−1)}= ΦΨ.

### Frobenius difference equations

From a pre-t-motive, one associates a “system of Frobenius difference equation”which has solutions in series of t.

Consider the operator on C∞ by x 7→ x^{1}^{q}.
Then extend this operator to C^{∞}((t)) as follows,
for f =P

ia_{i}t^{i}∈ C∞((t)) define f^{(−1)} :=P

ia^{q}_{i}^{−1}t^{i}.
More generally, for matrix B with entries in C∞((t)) define
twisting B^{(−1)} by the rule B^{(−1)}ij = Bij(−1).

Let m ∈ Matr×1(M ) be a ¯k(t)-basis of M .

Multiplying by σ on M is represented by σ(m) = Φm for some
matrix Φ ∈ GL_{r}(¯k(t)).

Theequationto be solved in Ψ ∈ Matr(C^{∞}((t)) is :
Ψ^{(−1)}= ΦΨ.

### Frobenius difference equations

From a pre-t-motive, one associates a “system of Frobenius difference equation”which has solutions in series of t.

Consider the operator on C∞ by x 7→ x^{1}^{q}.
Then extend this operator to C^{∞}((t)) as follows,
for f =P

ia_{i}t^{i}∈ C∞((t)) define f^{(−1)} :=P

ia^{q}_{i}^{−1}t^{i}.
More generally, for matrix B with entries in C∞((t)) define
twisting B^{(−1)} by the rule B^{(−1)}ij = Bij(−1).

Let m ∈ Matr×1(M ) be a ¯k(t)-basis of M .

Multiplying by σ on M is represented by σ(m) = Φm for some
matrix Φ ∈ GL_{r}(¯k(t)).

Theequationto be solved in Ψ ∈ Matr(C^{∞}((t)) is :
Ψ^{(−1)}= ΦΨ.

### Let t = θ

We view Ψ as giving a “fundamental”solution of the system of Frobeniusdifference equations described by the algebraic matrix Φ coming from M .

Note that if Ψ^{0}∈ Mat_{r}(C∞((t))) is also a solution of the
Frobenius system from Φ, then Ψ^{0−1}Ψ ∈ GLr(Fq(t)).

A power series f =P∞

i=0ait^{i} ∈ C∞[[t]] that converges everywhere
and satisfies

[k∞(a_{0}, a_{1}, a_{2}, . . . ) : k∞] < ∞

is called an entire power series. As a function of t it takes values in k∞, when restricted to k∞. The ring of the entire power series is denoted by E.

If all entries of a solution Ψ of the Frobenius system in question are in E, one canspecializ Ψ to Ψ(θ).

### Let t = θ

We view Ψ as giving a “fundamental”solution of the system of Frobeniusdifference equations described by the algebraic matrix Φ coming from M .

Note that if Ψ^{0}∈ Mat_{r}(C∞((t))) is also a solution of the
Frobenius system from Φ, then Ψ^{0−1}Ψ ∈ GLr(Fq(t)).

A power series f =P∞

i=0ait^{i} ∈ C∞[[t]] that converges everywhere
and satisfies

[k∞(a_{0}, a_{1}, a_{2}, . . . ) : k∞] < ∞

is called an entire power series. As a function of t it takes values in k∞, when restricted to k∞. The ring of the entire power series is denoted by E.

If all entries of a solution Ψ of the Frobenius system in question are in E, one canspecializ Ψ to Ψ(θ).

### Rigid analytic trivialization

| · |∞:= a fixed absolute value for the completed field C∞. T := {f ∈ C∞[[t]] | f converges on |t|∞≤ 1}.

L := the fraction field of T.

Pre t-motive M is called rigid analytically trivial if there exists Ψ ∈ GLr(L) such that

Ψ^{(−1)}= ΦΨ.

Such matrix Ψ is called a rigid analytic trivialization of the pre t-motive in question.

The category R of rigid analytically trivial pre-t-motives over Fq

forms a neutral Tannakian category over Fq(t).

### Rigid analytic trivialization

| · |∞:= a fixed absolute value for the completed field C∞. T := {f ∈ C∞[[t]] | f converges on |t|∞≤ 1}.

L := the fraction field of T.

Pre t-motive M is called rigid analytically trivial if there exists Ψ ∈ GLr(L) such that

Ψ^{(−1)}= ΦΨ.

Such matrix Ψ is called a rigid analytic trivialization of the pre t-motive in question.

The category R of rigid analytically trivial pre-t-motives over Fq

forms a neutral Tannakian category over Fq(t).

### Tannakian duality

Given object M in R and let T_{M} be the strictly full Tannakian
subcategory of R generated by M . That is, TM consists of all
objects of R isomorphic to subquotients of finite direct sums of

M^{⊗u}⊗ (M^{∨})^{⊗v} for various u, v,

where M^{∨} is the dual of M . By Tannakianduality, TM is
representable by an affine algebraic group scheme Γ_{M} over Fq(t).

Such Γ_{M} is called the motivic Galois groupof M .

Given rigid analytically trivial pre-t-motive M , the motivic Galois
group Γ_{M} is isomorphic over Fq(t) to the linear algebraic Galois
group Γ_{Ψ} of the associated Frobenius difference equation.

### Tannakian duality

Given object M in R and let T_{M} be the strictly full Tannakian
subcategory of R generated by M . That is, TM consists of all
objects of R isomorphic to subquotients of finite direct sums of

M^{⊗u}⊗ (M^{∨})^{⊗v} for various u, v,

where M^{∨} is the dual of M . By Tannakianduality, TM is
representable by an affine algebraic group scheme Γ_{M} over Fq(t).

Such Γ_{M} is called the motivic Galois groupof M .

Given rigid analytically trivial pre-t-motive M , the motivic Galois
group Γ_{M} is isomorphic over Fq(t) to the linear algebraic Galois
group Γ_{Ψ} of the associated Frobenius difference equation.

### Tannakian duality

Given object M in R and let T_{M} be the strictly full Tannakian
subcategory of R generated by M . That is, TM consists of all
objects of R isomorphic to subquotients of finite direct sums of

M^{⊗u}⊗ (M^{∨})^{⊗v} for various u, v,

where M^{∨} is the dual of M . By Tannakianduality, TM is
representable by an affine algebraic group scheme Γ_{M} over Fq(t).

Such Γ_{M} is called the motivic Galois groupof M .

Given rigid analytically trivial pre-t-motive M , the motivic Galois
group Γ_{M} is isomorphic over Fq(t) to the linear algebraic Galois
group Γ_{Ψ} of the associated Frobenius difference equation.

### Papanikolas theory 2008

This algebraic Galois group Γ_{Ψ} from solution Ψ has the key
property

dim Γ_{Ψ} = tr.deg_{k(t)} k(t)(Ψ).

If furthermore Ψ ∈ Mat_{r}(E) and satisfies

tr.deg¯k(t)¯k(t)(Ψ) = tr.deg¯k¯k(Ψ(θ)), then we say that M has the GP property. It follows that

dim ΓM = tr.deg¯k ¯k(Ψ(θ)).

Pre-t-motives having the GP property first come from

Anderson-Brownawell-Papanikolas 2004, through reformulating the submodule theorem of Yu 1997 which plays the role of W¨ustholz subgroup theorem (1989).

### Papanikolas theory 2008

This algebraic Galois group Γ_{Ψ} from solution Ψ has the key
property

dim Γ_{Ψ} = tr.deg_{k(t)} k(t)(Ψ).

If furthermore Ψ ∈ Mat_{r}(E) and satisfies

tr.deg¯k(t)¯k(t)(Ψ) = tr.deg¯k¯k(Ψ(θ)), then we say that M has the GP property. It follows that

dim ΓM = tr.deg¯k ¯k(Ψ(θ)).

Pre-t-motives having the GP property first come from

Anderson-Brownawell-Papanikolas 2004, through reformulating the submodule theorem of Yu 1997 which plays the role of W¨ustholz subgroup theorem (1989).

### The Carlitz motive

The Carlitz motive C. Let C = ¯k(t) with σ-action:

σf = (t − θ)f^{(−1)}, f ∈ C.

Here Φ = (t − θ). Analytic solution Ψ of the equation
Ψ^{(−1)} = (t − θ)Ψ is given by

ΨC(t) = (−θ)^{−q/(q−1)}

∞

Y

i=1

(1 − t/θ^{q}^{i}).

Note Galois group here is Γ_{C} = Gm which has dimension 1.

Therefore ΨC(θ) = ^{−1}_{π}_{˜} is transcendental over ¯k,

### Motivic Galois theory

We are interested in finitely generated extension of ¯k = Fq(θ)
generated by a set S of special values, denoted by K_{S}. In particular
we want to determine all algebraic relations among elements of S.

From known algebraic relations, we can guess the transcendence
degree of K_{S} over ¯k, and the goal is to prove that is indeed the
specific degree in question.

To proceed, we construct a t-motive MS for this purpose, so that
it has the GP property and its “periods”Ψ_{S}(θ) from rigid analytic
trivialization generate also the field K_{S}, then computing the
dimension of this motivic Galois group ΓMS.

### Motivic Galois theory

We are interested in finitely generated extension of ¯k = Fq(θ)
generated by a set S of special values, denoted by K_{S}. In particular
we want to determine all algebraic relations among elements of S.

From known algebraic relations, we can guess the transcendence
degree of K_{S} over ¯k, and the goal is to prove that is indeed the
specific degree in question.

To proceed, we construct a t-motive MS for this purpose, so that
it has the GP property and its “periods”Ψ_{S}(θ) from rigid analytic
trivialization generate also the field K_{S}, then computing the
dimension of this motivic Galois group ΓMS.

### Motivic Galois games

Suppose we have pre-t-motive M_{1} (M_{2}) with GP property for set
of values S_{1} (S_{2} respectively), and we are able to determine the
Galois group ΓM1 (ΓM2 respectively). To handle the set S1∪ S_{2},
we form the direct sum of pre-t-motive M = M_{1}⊕ M_{2}. Then the
dimension of the Galois group ΓM equals to the transcendence
degree over ¯k of the compositum of the field K_{S}_{1} and K_{S}_{2} which
is K_{S}_{1}∪S_{2}. We have surjective morphisms from Γ_{M} onto both Γ_{M}_{1}
and ΓM2. On many occasions this makes it possible to deduce the
dimension of Γ_{M} from the algebraic group structures of Γ_{M}_{1} and
Γ_{M}_{2}. As an example