### On Galois Theory of Several Variables

Jing Yu

National Taiwan University

August 24, 2009, Nankai Institute

Jing Yu Galois Theory of Several Variables

### 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_{Q}Q(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 h_{1}(A) ⊕ Q(1).

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

Jing Yu Galois Theory of Several Variables

### 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_{Q}Q(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 h_{1}(A) ⊕ Q(1).

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

Jing Yu Galois Theory of Several Variables

### 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_{Q}Q(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 h_{1}(A) ⊕ Q(1).

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

Jing Yu Galois Theory of Several Variables

### Elliptic curves

Example. Consider elliptic curve E : y^{2} = 4x^{3}− g_{2}x − g_{3} 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 ∈ H_{1}(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(ω_{1}) η(ω_{2})

!

= ±2π√

−1.

Jing Yu Galois Theory of Several Variables

### CM Elliptic curves

If E has no complex multiplications, one conjectures that
ω_{1}, ω_{2}, η(ω_{1}), η(ω_{2}) 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/Q}Gm, where k is the quadratic field of multiplications of E.

Jing Yu Galois Theory of Several Variables

### CM Elliptic curves

If E has no complex multiplications, one conjectures that
ω_{1}, ω_{2}, η(ω_{1}), η(ω_{2}) 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/Q}Gm, where k is the quadratic field of multiplications of E.

Jing Yu Galois Theory of Several Variables

### CM Elliptic curves

If E has no complex multiplications, one conjectures that
ω_{1}, ω_{2}, η(ω_{1}), η(ω_{2}) 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/Q}Gm, where k is the quadratic field of multiplications of E.

Jing Yu Galois Theory of Several Variables

### 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)^{m}Bm

2m! .

where B_{m} are the Bernoulli numbers:

Z
e^{Z}− 1 =

∞

X

m=0

Bm

Z^{m}
m !.

Jing Yu Galois Theory of Several Variables

### 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)^{m}Bm

2m! .

where B_{m} are the Bernoulli numbers:

Z
e^{Z}− 1 =

∞

X

m=0

Bm

Z^{m}
m !.

Jing Yu Galois Theory of Several Variables

### 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.

Jing Yu Galois Theory of Several Variables

### 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.

Jing Yu Galois Theory of Several Variables

### 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.

Jing Yu Galois Theory of Several Variables

### 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.

Jing Yu Galois Theory of Several Variables

### 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.

Jing Yu Galois Theory of Several Variables

### 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.

Jing Yu Galois Theory of Several Variables

### 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˜^{m}Bem

Γm+1

,

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

˜

π = θ(−θ)^{q−1}^{1}

∞

Y

i=1

1 − θ^{1−q}^{i}

−1

,

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

Jing Yu Galois Theory of Several Variables

### 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˜^{m}Bem

Γm+1

,

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

˜

π = θ(−θ)^{q−1}^{1}

∞

Y

i=1

1 − θ^{1−q}^{i}

−1

,

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

Jing Yu Galois Theory of Several Variables

### Bernoulli-Carlitz

The Γm are Carlitz factorials :

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

∞

P

i=0

niq^{i} of n, and let

Γ_{n+1}=

∞

Y

i=0

D^{n}_{i}^{i} .

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).

Jing Yu Galois Theory of Several Variables

### Bernoulli-Carlitz

The Γm are Carlitz factorials :

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

∞

P

i=0

niq^{i} of n, and let

Γ_{n+1}=

∞

Y

i=0

D^{n}_{i}^{i} .

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).

Jing Yu Galois Theory of Several Variables

### 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}.

Jing Yu Galois Theory of Several Variables

### 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}.

Jing Yu Galois Theory of Several Variables

### 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.

Jing Yu Galois Theory of Several Variables

### 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.

Jing Yu Galois Theory of Several Variables

### 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 ∈ 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.

Jing Yu Galois Theory of Several Variables

### 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 ∈ 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.

Jing Yu Galois Theory of Several Variables

### 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.

Jing Yu Galois Theory of Several Variables

### 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.

Jing Yu Galois Theory of Several Variables

### 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.

Jing Yu Galois Theory of Several Variables

### 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).

Jing Yu Galois Theory of Several Variables

### 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).

Jing Yu Galois Theory of Several Variables

### 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).

Jing Yu Galois Theory of Several Variables

### 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).

Jing Yu Galois Theory of Several Variables

### 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.

Jing Yu Galois Theory of Several Variables

### 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.

Jing Yu Galois Theory of Several Variables

### 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.

Jing Yu Galois Theory of Several Variables

### 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).

Jing Yu Galois Theory of Several Variables

### 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).

Jing Yu Galois Theory of Several Variables

### 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.¯

Jing Yu Galois Theory of Several Variables

### 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.¯

Jing Yu Galois Theory of Several Variables

### 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.

Jing Yu Galois Theory of Several Variables

### 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.

Jing Yu Galois Theory of Several Variables

### 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.

Jing Yu Galois Theory of Several Variables

### 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,

Jing Yu Galois Theory of Several Variables

### 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^{q}τ, 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 + g_{1}τ + · · · + g_{r−1}τ^{r−1}+ ∆τ^{r},
Drinfeld exponential exp_{ρ}(z) =P∞

h=0c_{h}z^{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∞

Jing Yu Galois Theory of Several Variables

### 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^{q}τ, 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 + g_{1}τ + · · · + g_{r−1}τ^{r−1}+ ∆τ^{r},
Drinfeld exponential exp_{ρ}(z) =P∞

h=0c_{h}z^{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∞

Jing Yu Galois Theory of Several Variables

### 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_{ρ}are the periodsof the Drinfeld
module ρ. They are all transcendental over ¯k (Yu 1986).

Morphisms of Drinfeld modules f : ρ1→ ρ_{2} are the twisting
polynomials f ∈ ¯k[τ ] satisfying (ρ_{2})_{t}◦ f = f ◦ (ρ_{1})_{t}.

Isomorphisms from ρ_{1} to ρ_{2} 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}}.

Jing Yu Galois Theory of Several Variables

### 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_{ρ}are the periodsof the Drinfeld
module ρ. They are all transcendental over ¯k (Yu 1986).

Morphisms of Drinfeld modules f : ρ1→ ρ_{2} are the twisting
polynomials f ∈ ¯k[τ ] satisfying (ρ_{2})_{t}◦ f = f ◦ (ρ_{1})_{t}.

Isomorphisms from ρ_{1} to ρ_{2} 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}}.

Jing Yu Galois Theory of Several Variables

### 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_{ρ}are the periodsof the Drinfeld
module ρ. They are all transcendental over ¯k (Yu 1986).

Morphisms of Drinfeld modules f : ρ1→ ρ_{2} are the twisting
polynomials f ∈ ¯k[τ ] satisfying (ρ_{2})_{t}◦ f = f ◦ (ρ_{1})_{t}.

Isomorphisms from ρ_{1} to ρ_{2} 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}}.

Jing Yu Galois Theory of Several Variables

### 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.

Jing Yu Galois Theory of Several Variables

### 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.

Jing Yu Galois Theory of Several Variables

### The period matrix

Let ρ be a Drinfeld module of rank r, and let {δ_{1}, . . . , 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
{λ_{1}, . . . , λ_{r}} be a fixed basis of Λ_{ρ}. Thenperiod matrix of ρ
corresponding to this choice of basis is

Pρ=

λ_{1} F_{1}(λ_{1}) · · · F_{r−1}(λ_{1})
λ2 F1(λ2) · · · F_{r−1}(λ2)

... ... . .. ...
λ_{r} F_{1}(λ_{r}) · · · F_{r−1}(λ_{r})

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

Jing Yu Galois Theory of Several Variables

### 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) −g_{1}^{(−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→ τ^{i}, 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.

Jing Yu Galois Theory of Several Variables

### Galois games

Suppose we have pre-t-motive M_{1} (M_{2}) with GP property for set
of values S1 (S2 respectively), and we are able to determine the
Galois group Γ_{M}_{1} (Γ_{M}_{2} respectively). To handle the set S_{1}∪ S_{2},
we form the direct sum of pre-t-motive M = M1⊕ 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 KS1∪S2. We have surjective morphisms from ΓM onto both ΓM1

and Γ_{M}_{2}. On many occasions this makes it possible to deduce the
dimension of Γ_{M} from the algebraic group structures of Γ_{M}_{1} 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!

Jing Yu Galois Theory of Several Variables

### The End. Thank You.

Jing Yu Galois Theory of Several Variables