On Galois Theory of Several Variables

55  Download (0)

Full text

(1)

On Galois Theory of Several Variables

Jing Yu

National Taiwan University

August 24, 2009, Nankai Institute

Jing Yu Galois Theory of Several Variables

(2)

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 :

trdegQQ(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.

Jing Yu Galois Theory of Several Variables

(3)

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 :

trdegQQ(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.

Jing Yu Galois Theory of Several Variables

(4)

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 :

trdegQQ(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.

Jing Yu Galois Theory of Several Variables

(5)

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.

Jing Yu Galois Theory of Several Variables

(6)

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 Resk/QGm, where k is the quadratic field of multiplications of E.

Jing Yu Galois Theory of Several Variables

(7)

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 Resk/QGm, where k is the quadratic field of multiplications of E.

Jing Yu Galois Theory of Several Variables

(8)

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 Resk/QGm, where k is the quadratic field of multiplications of E.

Jing Yu Galois Theory of Several Variables

(9)

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 Bm are the Bernoulli numbers:

Z eZ− 1 =

X

m=0

Bm

Zm m !.

Jing Yu Galois Theory of Several Variables

(10)

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 Bm are the Bernoulli numbers:

Z eZ− 1 =

X

m=0

Bm

Zm m !.

Jing Yu Galois Theory of Several Variables

(11)

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

(12)

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

(13)

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

(14)

World of positive characteristic

k:= Fq((1θ)), 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

am ∈ 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

(15)

World of positive characteristic

k:= Fq((1θ)), 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

am ∈ 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

(16)

World of positive characteristic

k:= Fq((1θ)), 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

am ∈ 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

(17)

Euler-Carlitz relations

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

ζC(m)p = ζC(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 − θ1−qi

−1

,

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

Jing Yu Galois Theory of Several Variables

(18)

Euler-Carlitz relations

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

ζC(m)p = ζC(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 − θ1−qi

−1

,

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

Jing Yu Galois Theory of Several Variables

(19)

Bernoulli-Carlitz

The Γm are Carlitz factorials :

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

P

i=0

niqi of n, and let

Γn+1=

Y

i=0

Dnii .

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

expC(z) =

X

m=0

gBm

zm Γm+1. Here Carlitz exponential is the series

expC(z) =

X

h=0

zqh

Dh = z Y

a6=0∈Fq[θ]

(1 − z a˜π).

Jing Yu Galois Theory of Several Variables

(20)

Bernoulli-Carlitz

The Γm are Carlitz factorials :

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

P

i=0

niqi of n, and let

Γn+1=

Y

i=0

Dnii .

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

expC(z) =

X

m=0

gBm

zm Γm+1. Here Carlitz exponential is the series

expC(z) =

X

h=0

zqh

Dh = z Y

a6=0∈Fq[θ]

(1 − z a˜π).

Jing Yu Galois Theory of Several Variables

(21)

Carlitz module

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

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

φC(t) : x 7−→ θx + xq,

C expC

−−−−→ Ga(C) = C θ(·)

 y

 yφC(t) C

expC

−−−−→ Ga(C) = C

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

Jing Yu Galois Theory of Several Variables

(22)

Carlitz module

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

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

φC(t) : x 7−→ θx + xq,

C expC

−−−−→ Ga(C) = C θ(·)

 y

 yφC(t) C

expC

−−−−→ Ga(C) = C

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

Jing Yu Galois Theory of Several Variables

(23)

Toward algebraic independence

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

(24)

Toward algebraic independence

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

(25)

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 a1q.

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 ∈ 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

(26)

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 a1q.

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 ∈ 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

(27)

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→ x1q. Then extend this operator to C((t)) as follows, for f =P

iaiti∈ C((t)) define f(−1) :=P

iaqi−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 ∈ GLr(¯k(t)).

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

Jing Yu Galois Theory of Several Variables

(28)

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→ x1q. Then extend this operator to C((t)) as follows, for f =P

iaiti∈ C((t)) define f(−1) :=P

iaqi−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 ∈ GLr(¯k(t)).

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

Jing Yu Galois Theory of Several Variables

(29)

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→ x1q. Then extend this operator to C((t)) as follows, for f =P

iaiti∈ C((t)) define f(−1) :=P

iaqi−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 ∈ GLr(¯k(t)).

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

Jing Yu Galois Theory of Several Variables

(30)

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∈ Matr(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).

Jing Yu Galois Theory of Several Variables

(31)

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∈ Matr(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).

Jing Yu Galois Theory of Several Variables

(32)

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

(33)

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

(34)

Tannakian duality

Given object M in R and let TM 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 ΓM over Fq(t).

Such ΓM is called the motivic Galois groupof M .

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

Jing Yu Galois Theory of Several Variables

(35)

Tannakian duality

Given object M in R and let TM 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 ΓM over Fq(t).

Such ΓM is called the motivic Galois groupof M .

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

Jing Yu Galois Theory of Several Variables

(36)

Tannakian duality

Given object M in R and let TM 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 ΓM over Fq(t).

Such ΓM is called the motivic Galois groupof M .

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

Jing Yu Galois Theory of Several Variables

(37)

Papanikolas theory 2008

This algebraic Galois group ΓΨ from solution Ψ has the key property

dim ΓΨ = tr.degk(t) k(t)(Ψ).

If furthermore Ψ ∈ Matr(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

(38)

Papanikolas theory 2008

This algebraic Galois group ΓΨ from solution Ψ has the key property

dim ΓΨ = tr.degk(t) k(t)(Ψ).

If furthermore Ψ ∈ Matr(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

(39)

Galois theory

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

From known algebraic relations, we can guess the transcendence degree of KS over ¯k, and the goal is to prove that specific degree.

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

trivialization generate also the field KS, then computing the dimension of the motivic Galois group ΓMS.

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

(40)

Galois theory

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

From known algebraic relations, we can guess the transcendence degree of KS over ¯k, and the goal is to prove that specific degree.

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

trivialization generate also the field KS, then computing the dimension of the motivic Galois group ΓMS.

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

(41)

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

(42)

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

(43)

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

(44)

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/θqi).

Note Galois group here is ΓC = Gm which has dimension 1.

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

Jing Yu Galois Theory of Several Variables

(45)

Drinfeld modules

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

τ c = cqτ, 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)

ρt= θ + g1τ + · · · + gr−1τr−1+ ∆τr, Drinfeld exponential expρ(z) =P

h=0chzqh, ch∈ ¯k, on C

linearizes this t-action : C

expρ

−−−−→ Ga(C) = C θ(·)

 y

 y

ρt

C expρ

−−−−→ Ga(C) = C

Jing Yu Galois Theory of Several Variables

(46)

Drinfeld modules

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

τ c = cqτ, 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)

ρt= θ + g1τ + · · · + gr−1τr−1+ ∆τr, Drinfeld exponential expρ(z) =P

h=0chzqh, ch∈ ¯k, on C

linearizes this t-action : C

expρ

−−−−→ Ga(C) = C θ(·)

 y

 y

ρt

C expρ

−−−−→ Ga(C) = C

Jing Yu Galois Theory of Several Variables

(47)

Periods of Drinfeld modules

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

Moreover

expρ(z) = z Y

λ6=0∈Λρ

(1 − z λ).

The nonzero elements in Λρare 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⊂ Λρ}.

Jing Yu Galois Theory of Several Variables

(48)

Periods of Drinfeld modules

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

Moreover

expρ(z) = z Y

λ6=0∈Λρ

(1 − z λ).

The nonzero elements in Λρare 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⊂ Λρ}.

Jing Yu Galois Theory of Several Variables

(49)

Periods of Drinfeld modules

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

Moreover

expρ(z) = z Y

λ6=0∈Λρ

(1 − z λ).

The nonzero elements in Λρare 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⊂ Λρ}.

Jing Yu Galois Theory of Several Variables

(50)

Algebraic relations among periods

The field of fractions of Rρ, denoted by Kρ, 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

(51)

Algebraic relations among periods

The field of fractions of Rρ, denoted by Kρ, 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

(52)

The period matrix

Let ρ be a Drinfeld module of rank r, and let {δ1, . . . , δr−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, . . . , λr} be a fixed basis of Λρ. Thenperiod matrix of ρ corresponding to this choice of basis is

Pρ=

λ1 F11) · · · Fr−11) λ2 F12) · · · Fr−12)

... ... . .. ... λr F1r) · · · Fr−1r)

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

Jing Yu Galois Theory of Several Variables

(53)

Drinfeld motives

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

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

Φρ=

0 1 0 · · · 0

0 0 1 · · · 0

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

0 0 · · · 0 1

(t − θ) −g1(−1) · · · −gr−1(−1)

To solve the Frobenius difference equation, let {λ1, . . . , λr} be a fixed basis of Λρ, and δi : 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.

Jing Yu Galois Theory of Several Variables

(54)

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 ΓM1M2 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 KS1 and KS2 which is KS1∪S2. We have surjective morphisms from ΓM onto both ΓM1

and ΓM2. On many occasions this makes it possible to deduce the dimension of ΓM from the algebraic group structures of ΓM1 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

(55)

The End. Thank You.

Jing Yu Galois Theory of Several Variables

Figure

Updating...

References

Related subjects :