On Algebraic Independence of Special Zeta Values in Characteristic p

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^1q.

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^1q.

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^1q. Then extend this operator to C^∞((t)) as follows, for f =P

ia_itⁱ∈ C∞((t)) define f⁽⁻¹⁾ :=P

ia^q_i⁻¹tⁱ. 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^1q. Then extend this operator to C^∞((t)) as follows, for f =P

ia_itⁱ∈ C∞((t)) define f⁽⁻¹⁾ :=P

ia^q_i⁻¹tⁱ. 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^1q. Then extend this operator to C^∞((t)) as follows, for f =P

ia_itⁱ∈ C∞((t)) define f⁽⁻¹⁾ :=P

ia^q_i⁻¹tⁱ. 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 Ψ⁰∈ Mat_r(C∞((t))) is also a solution of the Frobenius system from Φ, then Ψ⁰⁻¹Ψ ∈ GLr(Fq(t)).

A power series f =P∞

i=0aitⁱ ∈ C∞[[t]] that converges everywhere and satisfies

[k∞(a₀, a₁, a₂, . . . ) : 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 Ψ⁰∈ Mat_r(C∞((t))) is also a solution of the Frobenius system from Φ, then Ψ⁰⁻¹Ψ ∈ GLr(Fq(t)).

A power series f =P∞

i=0aitⁱ ∈ C∞[[t]] that converges everywhere and satisfies

[k∞(a₀, a₁, a₂, . . . ) : 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^qi).

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 M₁ (M₂) with GP property for set of values S₁ (S₂ respectively), and we are able to determine the Galois group ΓM1 (ΓM2 respectively). To handle the set S1∪ S₂, we form the direct sum of pre-t-motive M = M₁⊕ M₂. Then the dimension of the Galois group ΓM equals to the transcendence degree over ¯k of the compositum of the field K_S1 and K_S2 which is K_S1∪S₂. We have surjective morphisms from Γ_M onto both Γ_M1 and ΓM2. On many occasions this makes it possible to deduce the dimension of Γ_M from the algebraic group structures of Γ_M1 and Γ_M2. As an example

The End. Thank You.