On Galois Theory of Several Variables

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On Galois Theory of Several Variables

Jing Yu

National Taiwan University

August 24, 2009, Nankai Institute

Jing Yu Galois Theory of Several Variables

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Algebraic relations

We are interested in understanding transcendental invariants which arise naturally in mathematics. Satisfactory understanding means that we are able to determine all the algebraic relations among these very special values.

Let A be an abelian variety over Q of dimension d, and let P be the period matrix of A. Grothendieck in 1960’s madeconjecture :

trdeg_QQ(P ) = dim MT(A),

where MT(A) is the Mumford-Tate group of A and is an algebraic subgroup of GL2d×Gm. This Mumford-Tate group is the motivic Galois group of the motive h₁(A) ⊕ Q(1).

Extending this conjecture to general motives, one believes that all algebraic relations of the special values in question can beknown.

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Algebraic relations

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Algebraic relations

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Elliptic curves

Example. Consider elliptic curve E : y² = 4x³− g₂x − g₃ over Q ⊂ C. Its transcendental invariants are the periods and

quasi-periods. Periods lattice ΛE = Zω1+ Zω2 ⊂ C consisting of periods of dfk :

Z

c

dx

y , c ∈ H₁(E(C), Z).

From differential of the 2nd kind xdx/y, one has quasi-periods η(ω), which is Z-linear in ω ∈ ΛE. All the non-zero periods and quasi-periods are transcendental, by Siegel-Schneider 1930’s. On the other hand Legendre’s relation says

det ω1 ω2

η(ω₁) η(ω₂)

!

= ±2π√

−1.

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CM Elliptic curves

If E has no complex multiplications, one conjectures that ω₁, ω₂, η(ω₁), η(ω₂) are algebraically independent, i.e.

trdeg

QQ(π, ω, η(ω); ω ∈ ΛE) = 4.

Complex multiplications certainly give rise algebraic relations, thus in 1970’s Chudnovsky showed that if E has CM, then the above finitely generated extension has transcendence degree only two.

In the non-CM case the motivic Galois group for E should be GL2, in the CM case the Galois group is a 2-dim torus, which is just Res_k/QGm, where k is the quadratic field of multiplications of E.

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CM Elliptic curves

trdeg

QQ(π, ω, η(ω); ω ∈ ΛE) = 4.

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CM Elliptic curves

trdeg

QQ(π, ω, η(ω); ω ∈ ΛE) = 4.

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Euler relations

Example. Consider the following values from arithmetic of Q : S = {2π√

−1, ζ(2), ζ(3), · · · , ζ(m) · · · , }, where

ζ(m) :=

∞

X

n=1

n^−m.

The value of Riemann zeta function at positive integer m > 1.

For meven, one knows from Euler the relations:

ζ(m) = −(2π√

−1)^mBm

2m! .

where B_m are the Bernoulli numbers:

Z e^Z− 1 =

∞

X

m=0

Bm

Z^m m !.

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Euler relations

Example. Consider the following values from arithmetic of Q : S = {2π√

−1, ζ(2), ζ(3), · · · , ζ(m) · · · , }, where

ζ(m) :=

∞

X

n=1

n^−m.

The value of Riemann zeta function at positive integer m > 1.

For meven, one knows from Euler the relations:

ζ(m) = −(2π√

−1)^mBm

2m! .

where B_m are the Bernoulli numbers:

Z e^Z− 1 =

∞

X

m=0

Bm

Z^m m !.

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Mixed Tate motives

Oneconjectures that these relations generate all the algebraic relations among numbers from S over the field of algebraic numbers Q. In particular, all the zeta values ζ(m) for odd integer m > 1 should be transcendental, and algebraically independent from each other, as well as algebraically independent from π.

(Presently the transcendence of ζ(3) is still unknown).

This follows from the period conjecture for Mixed Tate motives.

Arithmetic of positive characteristic.

Fq:= the finite field of q elements.

k := Fq(θ) := the rational function field in the variable θ over Fq.

¯k := fixed algebraic closure of k.

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Mixed Tate motives

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Mixed Tate motives

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World of positive characteristic

k∞:= Fq((¹_θ)), completion of k with respect to the infinite place.

k∞:= a fixed algebraic closure of k∞ containing ¯k.

C∞:= completion of k∞ with respect to the canonical extension of the infinite place.

Natural transcendental “numbers”from function field arithmetic : Carlitz zeta values (1935), m ≥ 1,

ζC(m) = X

a∈Fq[θ]+

1

a^m ∈ Fq (1 θ),

where Fq[θ]₊ consists of monic polynomials in A := Fq[θ].

Development of transcendence theory in positive characteristic:

1st stage : Transporting classical (characteristic zero) theory (e.g.

methods of Siegel, Schneider, Lang, Baker, and W¨ustholz) to positive characteristic world, from 1980’s to 1990’s.

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World of positive characteristic

ζC(m) = X

a∈Fq[θ]+

1

a^m ∈ Fq (1 θ),

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World of positive characteristic

ζC(m) = X

a∈Fq[θ]+

1

a^m ∈ Fq (1 θ),

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Euler-Carlitz relations

One has the obvious Frobenius relations among special zeta values in characteristic p :

ζ_C(m)^p = ζ_C(mp).

If m iseven, i.e. m ≡ 0 (mod q − 1) because the ring Fq[θ] has q − 1 signs, one also has the Euler-Carlitz relation.

ζC(m) = π˜^mBem

Γm+1

,

where ˜π is afundamental periodof the Carlitz module for Fq :

˜

π = θ(−θ)^q−1¹

∞

Y

i=1

1 − θ^1−qⁱ

−1

,

which is transcendental over Fq(θ) (Wade 1942).

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Euler-Carlitz relations

One has the obvious Frobenius relations among special zeta values in characteristic p :

ζ_C(m)^p = ζ_C(mp).

If m iseven, i.e. m ≡ 0 (mod q − 1) because the ring Fq[θ] has q − 1 signs, one also has the Euler-Carlitz relation.

ζC(m) = π˜^mBem

Γm+1

,

where ˜π is afundamental periodof the Carlitz module for Fq :

˜

π = θ(−θ)^q−1¹

∞

Y

i=1

1 − θ^1−qⁱ

−1

,

which is transcendental over Fq(θ) (Wade 1942).

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Bernoulli-Carlitz

The Γm are Carlitz factorials :

setting D0= 1, and Di = (θ^qⁱ− θ^qⁱ⁻¹) · · · (θ^qⁱ− θ), for i ≥ 1, writing down the q-adic expansion

∞

P

i=0

niqⁱ of n, and let

Γ_n+1=

∞

Y

i=0

Dⁿ_iⁱ .

The gB_m ∈ Fq(θ) are the Bernoulli-Carlitz “numbers”given by z

exp_C(z) =

∞

X

m=0

gBm

z^m Γ_m+1. Here Carlitz exponential is the series

exp_C(z) =

∞

X

h=0

z^q^h

D_h = z Y

a6=0∈Fq[θ]

(1 − z a˜π).

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Bernoulli-Carlitz

The Γm are Carlitz factorials :

setting D0= 1, and Di = (θ^qⁱ− θ^qⁱ⁻¹) · · · (θ^qⁱ− θ), for i ≥ 1, writing down the q-adic expansion

∞

P

i=0

niqⁱ of n, and let

Γ_n+1=

∞

Y

i=0

Dⁿ_iⁱ .

The gB_m ∈ Fq(θ) are the Bernoulli-Carlitz “numbers”given by z

exp_C(z) =

∞

X

m=0

gBm

z^m Γ_m+1. Here Carlitz exponential is the series

exp_C(z) =

∞

X

h=0

z^q^h

D_h = z Y

a6=0∈Fq[θ]

(1 − z a˜π).

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Carlitz module

Carlitz period ˜π fits in exact sequence of Fq-linear maps : 0 → Fq[θ] ˜π → C^∞ −−−−→ C^exp^C ^∞→ 0.

Carlitz exponential linearizes the Fq[t]-action (Carlitz module) given by

φC(t) : x 7−→ θx + x^q,

C∞ exp_C

−−−−→ Ga(C∞) = C∞ θ(·)



 y



 y^φ^C^(t) C∞

exp_C

−−−−→ Ga(C∞) = C∞

When q = 2, all integers are “even”, Euler-Carlitz says that all ζC(m), m ≥ 1, are rational multiples of ˜π^m.

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Carlitz module

Carlitz period ˜π fits in exact sequence of Fq-linear maps : 0 → Fq[θ] ˜π → C^∞ −−−−→ C^exp^C ^∞→ 0.

Carlitz exponential linearizes the Fq[t]-action (Carlitz module) given by

φC(t) : x 7−→ θx + x^q,

C∞ exp_C

−−−−→ Ga(C∞) = C∞ θ(·)



 y



 y^φ^C^(t) C∞

exp_C

−−−−→ Ga(C∞) = C∞

When q = 2, all integers are “even”, Euler-Carlitz says that all ζC(m), m ≥ 1, are rational multiples of ˜π^m.

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Toward algebraic independence

For arbitrary q, interested in following set of special zeta values : Sq = {˜π, ζ_C(1), ζ_C(2), · · · , ζ_C(m), · · · .}

Yu 1991 proves that all these are transcendental over ¯k = Fq(θ), and in 1997 it is shown that all linear relations among them come from the Euler-Carlitz relations.

2nd stage of positive characteristic transcendence theory in last decade: Go beyond its classical counterparts, i.e. from linear independence to algebraic independence. In particular an analogue of Grothendieck’s motivic design actually works (Anderson,

Brownawell, C.-Y. Chang, Papanikolas, and J. Yu).

Chang-Yu 2005 proves that the Euler-Carlitz relations and the Frobenius relations generate all the algebraic relations among special Carlitz zeta values over the field ¯k.

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Toward algebraic independence

For arbitrary q, interested in following set of special zeta values : Sq = {˜π, ζ_C(1), ζ_C(2), · · · , ζ_C(m), · · · .}

Yu 1991 proves that all these are transcendental over ¯k = Fq(θ), and in 1997 it is shown that all linear relations among them come from the Euler-Carlitz relations.

2nd stage of positive characteristic transcendence theory in last decade: Go beyond its classical counterparts, i.e. from linear independence to algebraic independence. In particular an analogue of Grothendieck’s motivic design actually works (Anderson,

Brownawell, C.-Y. Chang, Papanikolas, and J. Yu).

Chang-Yu 2005 proves that the Euler-Carlitz relations and the Frobenius relations generate all the algebraic relations among special Carlitz zeta values over the field ¯k.

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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 a¹^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 ∈ Mat_r×1(M ) be a ¯k(t)-basis of M .

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

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

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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 a¹^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 ∈ Mat_r×1(M ) be a ¯k(t)-basis of M .

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

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

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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→ x¹^q. 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.

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Frobenius difference equations

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Frobenius difference equations

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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).

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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).

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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).

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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).

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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.

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Tannakian duality

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Tannakian duality

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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).

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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).

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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 specific degree.

We construct a t-motive M_S for this purpose, so that it has the GP property and its “periods”ΨS(θ) from rigid analytic

trivialization generate also the field K_S, then computing the dimension of the motivic Galois group Γ_M_S.

Following transcendental arithmetic values have been tackled:

Periods and quasi-periods of Drinfeld Fq[t]-module defined over ¯k (arbitrary rank), Chang-Papanikolas 2009.

Logarithms at algebraic points of Drinfeld Fq[t]-module over k, rank 1 Papanikolas 2008, rank 2 Chang-Papanikolas 2009.¯

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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 specific degree.

We construct a t-motive M_S for this purpose, so that it has the GP property and its “periods”ΨS(θ) from rigid analytic

trivialization generate also the field K_S, then computing the dimension of the motivic Galois group Γ_M_S.

Following transcendental arithmetic values have been tackled:

Periods and quasi-periods of Drinfeld Fq[t]-module defined over ¯k (arbitrary rank), Chang-Papanikolas 2009.

Logarithms at algebraic points of Drinfeld Fq[t]-module over k, rank 1 Papanikolas 2008, rank 2 Chang-Papanikolas 2009.¯

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Construction of motives

The Carlitz zeta values ζC(m), m ≥ 1, Chang-Yu 2007.

Geometric Gamma values Γ(α), α ∈ Fq(θ) − Fq(θ)₊, Anderson-Brownawell-Papanikolas 2004 (analogue of Lang-Rohrlich conjecture).

Arithmnetic Gamma values r!, r ∈ Q ∩ (Zp− Z) (p is the characteristic), Chang-Thakur-Papanikolas-Yu 2008.

Construction of the t-motives in question rely on the arithmetic-geometric structures in question:

Canonical t-motive associated to Drinfeld module over ¯k of rank r following Anderson. In case the Drinfelds module has full CM, the Galois group is a torus of dim r. In the generic case, the Galois group is GLr.

For logarithms at algebraic points, the Galois group is an extension of the Galois group for the Drinfeld module by a vector group.

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Construction of motives

The Carlitz zeta values ζC(m), m ≥ 1, Chang-Yu 2007.

Geometric Gamma values Γ(α), α ∈ Fq(θ) − Fq(θ)₊, Anderson-Brownawell-Papanikolas 2004 (analogue of Lang-Rohrlich conjecture).

Arithmnetic Gamma values r!, r ∈ Q ∩ (Zp− Z) (p is the characteristic), Chang-Thakur-Papanikolas-Yu 2008.

Construction of the t-motives in question rely on the arithmetic-geometric structures in question:

Canonical t-motive associated to Drinfeld module over ¯k of rank r following Anderson. In case the Drinfelds module has full CM, the Galois group is a torus of dim r. In the generic case, the Galois group is GLr.

For logarithms at algebraic points, the Galois group is an extension of the Galois group for the Drinfeld module by a vector group.

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Galois groups

By a formula of Anderson-Thakur, these special zeta values are linear combinations of polylogarithms at algebraic points.

The Galois group is an extension of Gm by a vector group.

The motive construction for these special geometric Gamma values is by way of geometric cyclotomy, or “solitons”. The Galois groups for these values come from tori which are obtained from Gm via restriction of scalars from the geometric CM field of the motive in question.

The motive here is the one associated to the Carlitz module with CM from a constant field extension. The Galois groups for special arithmetic Gamma values are tori obtained from Gm via restriction of scalars from the constant field extension in question.

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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,

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Drinfeld modules

Let τ : x 7→ x^q be the Frobenius endomorphism of Ga/Fq. Let ¯k[τ ] be the twisted polynomial ring :

τ c = c^qτ, for all c ∈ ¯k.

A Drinfeld Fq[t]-module ρ of rank r (over ¯k) is a Fq-linear ring homomorphism (Drinfeld 1974) ρ : Fq[t] → ¯k[τ ] given by (∆ 6= 0)

ρ_t= θ + g₁τ + · · · + g_r−1τ^r−1+ ∆τ^r, Drinfeld exponential exp_ρ(z) =P∞

h=0c_hz^q^h, c_h∈ ¯k, on C∞

linearizes this t-action : C∞

exp_ρ

−−−−→ Ga(C∞) = C∞ θ(·)



 y



 y

ρt

C∞ exp_ρ

−−−−→ Ga(C∞) = C∞

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Drinfeld modules

Let τ : x 7→ x^q be the Frobenius endomorphism of Ga/Fq. Let ¯k[τ ] be the twisted polynomial ring :

τ c = c^qτ, for all c ∈ ¯k.

A Drinfeld Fq[t]-module ρ of rank r (over ¯k) is a Fq-linear ring homomorphism (Drinfeld 1974) ρ : Fq[t] → ¯k[τ ] given by (∆ 6= 0)

ρ_t= θ + g₁τ + · · · + g_r−1τ^r−1+ ∆τ^r, Drinfeld exponential exp_ρ(z) =P∞

h=0c_hz^q^h, c_h∈ ¯k, on C∞

linearizes this t-action : C∞

exp_ρ

−−−−→ Ga(C∞) = C∞ θ(·)



 y



 y

ρt

C∞ exp_ρ

−−−−→ Ga(C∞) = C∞

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Periods of Drinfeld modules

Kernel of exp_ρ is a discrete free Fq[θ]-module Λρ⊂ C∞ of rank r.

Moreover

exp_ρ(z) = z Y

λ6=0∈Λρ

(1 − z λ).

The nonzero elements in Λ_ρare the periodsof the Drinfeld module ρ. They are all transcendental over ¯k (Yu 1986).

Morphisms of Drinfeld modules f : ρ1→ ρ₂ are the twisting polynomials f ∈ ¯k[τ ] satisfying (ρ₂)_t◦ f = f ◦ (ρ₁)_t.

Isomorphisms from ρ₁ to ρ₂ are given by constant polynomials f ∈ ¯k ⊂ ¯k[τ ] such that f Λρ1 = Λρ2.

The endomorphism ring of Drinfeld module ρ can be identified with Rρ= {α ∈ ¯k| αΛρ⊂ Λ_ρ}.

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Periods of Drinfeld modules

Moreover

exp_ρ(z) = z Y

λ6=0∈Λρ

(1 − z λ).

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Periods of Drinfeld modules

Moreover

exp_ρ(z) = z Y

λ6=0∈Λρ

(1 − z λ).

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Algebraic relations among periods

The field of fractions of R_ρ, denoted by K_ρ, is called the field of multiplications of ρ. One has that [Kρ: k] always divides the rank of the Drinfeld module ρ.

Drinfeld module ρ of rank r is said to be without Complex Multiplications CM, if Kρ= k, and with “full”CM if [Kρ: k] = r.

If ρ has CM, there are non-trivial algebraic relations among its periods.

In late 1980’s, quasi-periods for Drinfeld modules are introduced by Anderson, Deligne, Gekeler, and Yu.

All nonzero quasi-periods are also transcendental over ¯k (Yu 1990), and there are algebraic relations between periods, quasi-periods and the Carlitz period ˜π (Anderson, Gekeler 1989), as analogue of the Legendre relation.

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Algebraic relations among periods

The field of fractions of R_ρ, denoted by K_ρ, is called the field of multiplications of ρ. One has that [Kρ: k] always divides the rank of the Drinfeld module ρ.

Drinfeld module ρ of rank r is said to be without Complex Multiplications CM, if Kρ= k, and with “full”CM if [Kρ: k] = r.

If ρ has CM, there are non-trivial algebraic relations among its periods.

In late 1980’s, quasi-periods for Drinfeld modules are introduced by Anderson, Deligne, Gekeler, and Yu.

All nonzero quasi-periods are also transcendental over ¯k (Yu 1990), and there are algebraic relations between periods, quasi-periods and the Carlitz period ˜π (Anderson, Gekeler 1989), as analogue of the Legendre relation.

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The period matrix

Let ρ be a Drinfeld module of rank r, and let {δ₁, . . . , δ_r−1} be a basis of the de Rham cohomology of ρ. Let Fi(z) be the

quasi-periodic function associated to δi, i = 1, . . . , r − 1, and {λ₁, . . . , λ_r} be a fixed basis of Λ_ρ. Thenperiod matrix of ρ corresponding to this choice of basis is

Pρ=







λ₁ F₁(λ₁) · · · F_r−1(λ₁) λ2 F1(λ2) · · · F_r−1(λ2)

... ... . .. ... λ_r F₁(λ_r) · · · F_r−1(λ_r)







Analogue of Legendre’s relation amounts to det P_ρ= α˜π, with α 6= 0 ∈ ¯k.

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Drinfeld motives

Let Drinfeld Fq[t]-module ρ of rank r (over ¯k) be given by ρt= θ + g1τ + · · · + gr−1τ^r−1+ τ^r,

We associate to ρ a dimension r pre-t-motive Mρ via the matrix

Φ_ρ=







0 1 0 · · · 0

0 0 1 · · · 0

... ... . .. ... ...

0 0 · · · 0 1

(t − θ) −g₁⁽⁻¹⁾ · · · −g_r−1⁽⁻¹⁾







To solve the Frobenius difference equation, let {λ1, . . . , λr} be a fixed basis of Λ_ρ, and δ_i : t 7→ τⁱ, i = 1, . . . , r − 1, be chosen basis of the de Rham cohomology of ρ. Then the solution Ψρcan be explicitly written down which specializes (setting t = θ) to the period matrix P ρ.

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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 Γ_M₁ (Γ_M₂ 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 KS1∪S2. We have surjective morphisms from ΓM onto both ΓM1

and Γ_M₂. On many occasions this makes it possible to deduce the dimension of Γ_M from the algebraic group structures of Γ_M₁ and ΓM2. As an example

Theorem (2008, Chang-Yu) Let ρ be a Drinfeld modules with full CM, then its periods and quasi-periods are algebraically

independent over ¯k from the values ζC(m), (q − 1) - m.

Similar phenomena should hold also in the classical world!

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On Galois Theory of Several Variables