Transcendence Theory of Drinfeld Modules

(1)

Transcendence Theory of Drinfeld Modules

Jing Yu

National Taiwan University

BIRS Workshop on t-motives September 28, 2009

Jing Yu Drinfeld modules

(2)

World of positive characteristic

Let p be a fixedprime; q a fixed power of p.

A := Fq[θ] ←→ Z k := Fq(θ) ←→ Q k∞:= Fq((1/θ)) ←→ R k inside k¯ ∞ ←→ Q C∞:= ck∞ ←→ C

|f |_∞:= q^{deg f} ←→ | · |

(3)

Drinfeld F

q

[t]-modules

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

F c = c^qF, for all c ∈ ¯k.

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

ρ_t= θ + g₁F + · · · + g_r−1F^r−1+ ∆F^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∞

(4)

Drinfeld F

q

[t]-modules

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

F c = c^qF, for all c ∈ ¯k.

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

ρ_t= θ + g₁F + · · · + g_r−1F^r−1+ ∆F^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∞

(5)

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 Λρare the periodsof the Drinfeld module ρ. They are all transcendental over ¯k(1986). In fact, any u ∈ C∞ such that exp_ρ(u) ∈ ¯k are transcendental, these are called Drinfeld logarithms(of algebraic points) w.r.t ρ.

Morphisms of Drinfeld modules h : ρ₁ → ρ₂ are the twisting polynomials h ∈ ¯k[F ] satisfying (ρ2)t◦ h = h ◦ (ρ₁)t.

Isomorphisms from ρ1 to ρ2 are given by constant polynomials h ∈ ¯k ⊂ ¯k[F ] such that h Λ_ρ₁ = Λ_ρ₂.

(6)

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 Λρare the periodsof the Drinfeld module ρ. They are all transcendental over ¯k(1986). In fact, any u ∈ C∞ such that exp_ρ(u) ∈ ¯k are transcendental, these are called Drinfeld logarithms(of algebraic points) w.r.t ρ.

Morphisms of Drinfeld modules h : ρ₁ → ρ₂ are the twisting polynomials h ∈ ¯k[F ] satisfying (ρ2)t◦ h = h ◦ (ρ₁)t.

Isomorphisms from ρ1 to ρ2 are given by constant polynomials h ∈ ¯k ⊂ ¯k[F ] such that h Λ_ρ₁ = Λ_ρ₂.

(7)

Algebraic relations among periods

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

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 coming from the endomorphisms.

One goal of transcendence theory for Drinfeld modules is to prove that these are the only source of algebraic relations among periods.

(8)

Algebraic relations among periods

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

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 coming from the endomorphisms.

One goal of transcendence theory for Drinfeld modules is to prove that these are the only source of algebraic relations among periods.

(9)

Detour to Drinfeld A-modules

Now let k be any function field with field of constants Fq. Fix a place and call it ∞.

Take A to be the ring of functions in k regular away from ∞.

A Drinfeld A-module ρ is simply an A-action on Ga defined over ¯k which linearizes to the scalar A-action on Lie Ga.

Take any non-constant “t”in A. Then ρ can be viewed as Drinfeld Fq[t]-module with “complex multiplications”by A.

For the purpose of transcendence theory, the study of Drinfeld A-modules can thus be reduced to the study of Drinfeld Fq[t]-modules.

(10)

Detour to Drinfeld A-modules

Now let k be any function field with field of constants Fq. Fix a place and call it ∞.

Take A to be the ring of functions in k regular away from ∞.

A Drinfeld A-module ρ is simply an A-action on Ga defined over ¯k which linearizes to the scalar A-action on Lie Ga.

Take any non-constant “t”in A. Then ρ can be viewed as Drinfeld Fq[t]-module with “complex multiplications”by A.

For the purpose of transcendence theory, the study of Drinfeld A-modules can thus be reduced to the study of Drinfeld Fq[t]-modules.

(11)

Periods of the 2nd kind

To introduce quasi-periods, we consider certain (bi-)derivations.

A Fq-linear map from δ : Fq[t] → ¯k[F ]F is called a derivation of the Drinfeld module ρ if, for all a, b ∈ Fq[t], the following holds

δ_ab = a(θ)δ_a+ δ_aρ_b.

Given derivation δ of ρ, there is Fq-linear entire function F_δ(z) =P∞

h=1b_hz^q^h, b_h∈ ¯k, on C∞, satisfying the following difference equation :

F_δ(θz) − θF_δ(z) = δ_t(exp_ρ(z)).

This F_δ(z) is quasi-periodic in the sense

Fδ(z + λ) = Fδ(z) + Fδ(λ), for λ ∈ Λρ. Z

λ

δ := Fδ(λ) is Fq[θ] − linear in λ ∈ Λρ.

(12)

Periods of the 2nd kind

To introduce quasi-periods, we consider certain (bi-)derivations.

A Fq-linear map from δ : Fq[t] → ¯k[F ]F is called a derivation of the Drinfeld module ρ if, for all a, b ∈ Fq[t], the following holds

δ_ab = a(θ)δ_a+ δ_aρ_b.

Given derivation δ of ρ, there is Fq-linear entire function F_δ(z) =P∞

h=1b_hz^q^h, b_h∈ ¯k, on C∞, satisfying the following difference equation :

F_δ(θz) − θF_δ(z) = δ_t(exp_ρ(z)).

This F_δ(z) is quasi-periodic in the sense

Fδ(z + λ) = Fδ(z) + Fδ(λ), for λ ∈ Λρ. Z

λ

δ := Fδ(λ) is Fq[θ] − linear in λ ∈ Λρ.

(13)

Periods and quasi-periods

The values F_δ(λ), λ ∈ Λρ, are called the quasi-periods of ρ w.r.t.

the derivation δ. All nonzero quasi-periods are also transcendental over ¯k (1990).

The set of all derivations of ρ modulo “strictly inner”derivations is a ¯k-vector space of dimension r = rank ρ. This gives the de Rham cohomology of the Drinfeld module ρ.

δ is called strictly inner derivation if there exists m ∈ ¯k[F ]F so that δ = δ^(m): a 7−→ mρ_a− a(θ)m, for all a ∈ Fq[t].

Strictly inner derivations only give zero quasi-periods.

Consider the derivation δ⁽¹⁾ : a 7→ a(θ) − ρ_a, then

F_δ(1)(z) = z − exp_ρ(z). Hence periods of ρ are just quasi-periods w.r.t. the 1st kind derivation δ⁽¹⁾.

(14)

Periods and quasi-periods

(15)

Periods and quasi-periods

(16)

The period matrix

The de Rham isomorphism says that δ 7→ (λ 7→R

λδ) gives a natural isomorphism from the de Rham cohomology of ρ onto a

¯k-structure of the space Hom_A(Λ_ρ, C∞).

Let {[δ0 = [δ⁽¹⁾], [δ1], . . . , [δr−1]} be a basis of the de Rham cohomology of ρ. Let {λ₁, . . . , λ_r} be a fixed A-basis of Λ_ρ. Then period matrixof ρ corresponding to this choices of basis is

Pρ=

Z

λi

δj

=







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

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





 ,

where Fi is the quasi-periodic function from the derivation δ_i, i = 1, . . . , r − 1.

(17)

The period matrix

The de Rham isomorphism says that δ 7→ (λ 7→R

λδ) gives a natural isomorphism from the de Rham cohomology of ρ onto a

¯k-structure of the space Hom_A(Λ_ρ, C∞).

Let {[δ0 = [δ⁽¹⁾], [δ1], . . . , [δr−1]} be a basis of the de Rham cohomology of ρ. Let {λ₁, . . . , λ_r} be a fixed A-basis of Λ_ρ. Then period matrixof ρ corresponding to this choices of basis is

Pρ=

Z

λi

δj

=







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

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





 ,

where Fi is the quasi-periodic function from the derivation δ_i, i = 1, . . . , r − 1.

(18)

t-motives

Analogue of Legendre’s relation (Anderson, Gekeler) says det P_ρ= α˜π, with α 6= 0 ∈ ¯k.

Here ˜π is period of the rank one Carlitz module.

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 ∈ 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)).

(19)

t-motives

(20)

t-motives

(21)

t-motives

(22)

Motives associated to Drinfeld modules

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

Let Drinfeld Fq[t]-module ρ of rank r (over ¯k) be given by ρt= θ + g1F + · · · + gr−1F^r−1+ F^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/q₁ · · · −g_r−1^1/q^r−1







(23)

Motives associated to Drinfeld modules

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

Let Drinfeld Fq[t]-module ρ of rank r (over ¯k) be given by ρt= θ + g1F + · · · + gr−1F^r−1+ F^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/q₁ · · · −g_r−1^1/q^r−1







(24)

Anderson generating function

Now fix an A-basis {λ1, . . . , λr} of the period lattice Λ_ρ. For each 1 ≤ i ≤ r, consider the sequence of t-division points:

exp_ρ(λi/θ), exp_ρ(λi/θ²), exp_ρ(λi/θ³), . . . . TheAnderson generating functions is: for 1 ≤ i ≤ r,

f_i(t) :=

∞

X

j=0

exp_ρ(λ_i/θ^j+1)t^j = λ_i/(θ − t) +

∞

X

j=1

c_jλ_i^q^j/(θ^q^j− t).

We observe that

Res_t=θfi = −λi = − Z

λi

δ⁽¹⁾.

Let δ_j be the derivation given by t 7→ F^j for 1 ≤ j ≤ r − 1. For

` ∈ N, let f_i^(`) be the series obtained from f_i by changing all coefficients to its q^`-th roots, then also

f_i^(j)(θ) = Z

λi

δ_j.

(25)

Anderson generating function

Now fix an A-basis {λ1, . . . , λr} of the period lattice Λ_ρ. For each 1 ≤ i ≤ r, consider the sequence of t-division points:

exp_ρ(λi/θ), exp_ρ(λi/θ²), exp_ρ(λi/θ³), . . . . TheAnderson generating functions is: for 1 ≤ i ≤ r,

f_i(t) :=

∞

X

j=0

exp_ρ(λ_i/θ^j+1)t^j = λ_i/(θ − t) +

∞

X

j=1

c_jλ_i^q^j/(θ^q^j− t).

We observe that

Res_t=θfi = −λi = − Z

λi

δ⁽¹⁾.

Let δ_j be the derivation given by t 7→ F^j for 1 ≤ j ≤ r − 1. For

` ∈ N, let f_i^(`) be the series obtained from f_i by changing all coefficients to its q^`-th roots, then also

f_i^(j)(θ) = Z

λi

δ_j.

(26)

A Frobenius difference equation

Ψ :=b







f1 f2 · · · fr

f₁⁽¹⁾ f₂⁽¹⁾ · · · fr⁽¹⁾

... ... ...

f₁^(r−1) f₂^(r−1) · · · f_r^(r−1)





 .

L :=







g1 g₂⁽⁻¹⁾ g₃⁽⁻²⁾ · · · g^(−r+2)_r−1 1 g₂ g₃⁽⁻¹⁾ g₄⁽⁻²⁾ · · · 1

... ... g_r−1 1

1







and set Ψ := (L⁻¹{[ bΨ⁽¹⁾]⁻¹})t. Then Ψ(θ) gives essentially the period matrix P_ρof the Drinfeld module ρ. Moreover

Ψ⁽⁻¹⁾= ΦΨ.

(27)

Linear independence (over ¯ k) theory

Method of Schneider-Langin positive characteristic.

Fq-linear functions as functions satisfying algebraic differential equations:

f (z) =

∞

X

h=0

c_hz^q^h, c_h∈ ¯k.

Method of Baker-W¨ustholzfor t-modules.

Analogue of W¨ustholz subgroup theorem :

Let G = (G^da, φ) be a t-module defined over ¯k. Let u be a point in Lie G(C∞) such that exp_G(u) ∈ G(¯k). Then the smallest vector subspace in Lie G defined over ¯k which is invariant under d(φt) and which contains u must be the tangent space at the origin of a t-submodule of G.

(28)

Linear independence (over ¯ k) theory

Method of Schneider-Langin positive characteristic.

Fq-linear functions as functions satisfying algebraic differential equations:

f (z) =

∞

X

h=0

c_hz^q^h, c_h∈ ¯k.

Method of Baker-W¨ustholzfor t-modules.

Analogue of W¨ustholz subgroup theorem :

Let G = (G^da, φ) be a t-module defined over ¯k. Let u be a point in Lie G(C∞) such that exp_G(u) ∈ G(¯k). Then the smallest vector subspace in Lie G defined over ¯k which is invariant under d(φt) and which contains u must be the tangent space at the origin of a t-submodule of G.

(29)

Anderson’s t-modules

A t-module of dimension d is a pair (G^d_a, φ), consisting of Fq-algebra homomorphism

φ : Fq[t] −→ End_F_q(G^da) ∼= Matd(¯k[F ]).

given by

φt= θI + N + g1F + · · · + grF^r, where N ∈ Mat_d(¯k) is nilpotent.

One also has the exponential map exp_G for t-module G:

C^d∞ exp_G

−−−−→ G^da(C∞) = C^d∞ d(φt)



 y



 y^φ^t C^d∞

exp_G

−−−−→ G^d_a(C∞) = C^d∞

(30)

Anderson’s t-modules

A t-module of dimension d is a pair (G^d_a, φ), consisting of Fq-algebra homomorphism

φ : Fq[t] −→ End_F_q(G^da) ∼= Matd(¯k[F ]).

given by

φt= θI + N + g1F + · · · + grF^r, where N ∈ Mat_d(¯k) is nilpotent.

One also has the exponential map exp_G for t-module G:

C^d∞ exp_G

−−−−→ G^da(C∞) = C^d∞ d(φt)



 y



 y^φ^t C^d∞

exp_G

−−−−→ G^d_a(C∞) = C^d∞

(31)

Linear independence Theorem

The t-submodule theorem says that all linear relations satisfied by a logarithmic vector of an algebraic point on t-module should come from algebraic relations inside the t-module under consideration.

Structure of t-modules is “rigid”. Usually it is possible to analyze the t-submodules in question.

Using the t-submodule theorem, one obtains:

Let ρ be Drinfeld module of rank r with field of multiplications K_ρ. Let [δ₁] = [δ⁽¹⁾, . . . , [δ_r] be a basis of the de Rham cohomology of ρ, with corresponding quasi-periodic functions F_δ₁, . . . , F_δ_r. Let u₁, . . . , u_n, be logarithms with exp_ρ(u_i) ∈ ¯k for each i.

Suppose that these u1, . . . , un are linearly independent over Kρ. Then the rn + 1 elements, 1, ui, F_δ_j(ui), i = 1, . . . , n, j = 2, . . . , r, are linearly independent over ¯k.

(32)

Linear independence Theorem

The t-submodule theorem says that all linear relations satisfied by a logarithmic vector of an algebraic point on t-module should come from algebraic relations inside the t-module under consideration.

Structure of t-modules is “rigid”. Usually it is possible to analyze the t-submodules in question.

Using the t-submodule theorem, one obtains:

Let ρ be Drinfeld module of rank r with field of multiplications K_ρ. Let [δ₁] = [δ⁽¹⁾, . . . , [δ_r] be a basis of the de Rham cohomology of ρ, with corresponding quasi-periodic functions F_δ₁, . . . , F_δ_r. Let u₁, . . . , u_n, be logarithms with exp_ρ(u_i) ∈ ¯k for each i.

Suppose that these u1, . . . , un are linearly independent over Kρ. Then the rn + 1 elements, 1, ui, F_δ_j(ui), i = 1, . . . , n, j = 2, . . . , r, are linearly independent over ¯k.

(33)

Construction of t-modules

First, a t-module Gρ of dimension r = rank ρ :

(φ_ρ)_t:=







ρt 0 0 · · · 0 · · · 0 (δ₂)_t θF⁰ 0 · · · 0

... ... ...

(δ_r)_t 0 · · · θF⁰





 .

This has exponential map :

exp_G_ρ :





 z₁ z2

... zr





 7−→







exp_ρ(z₁) z2+ Fδ2(z1)

... zr+ F_δ_r(z1)







Let G be the dirct sum of the trivial t-module Ga with n copies of this t-moduleGρ. Then apply the t-submodule theorem to the following logarithmic vector :

(34)

Construction of t-modules

First, a t-module Gρ of dimension r = rank ρ :

(φ_ρ)_t:=







ρt 0 0 · · · 0 · · · 0 (δ₂)_t θF⁰ 0 · · · 0

... ... ...

(δ_r)_t 0 · · · θF⁰





 .

This has exponential map :

exp_G_ρ :





 z₁ z2

... zr





 7−→







exp_ρ(z₁) z2+ Fδ2(z1)

... zr+ F_δ_r(z1)







Let G be the dirct sum of the trivial t-module Ga with n copies of this t-moduleGρ. Then apply the t-submodule theorem to the following logarithmic vector :

(35)

From linear independence to algebraic independence

u = (1, u₁, −F_δ₂(u₁), · · · , −F_δ_r(u₁), · · · , u_n, −F_δ₂(u_n), · · · , −F_δ_r(u_n)).

The algebraic point exp_G(u) corresponding to this vector is (1, exp_ρ(u₁), 0, · · · , exp_ρ(u₂), 0, · · · , · · · , exp_ρ(u_n), 0, · · · ).

The hypothesis that u1, . . . , un are linearly independent over Kρ

implies precisely that this algebraic point on G does not fall in any proper t-submodule of G.

Extensive efforts of using the t-submodule theorem to prove linear independence results by many people in the late 1990’s, e.g.

A-B-P concerning the independence of geometric Gamma values, lead to a “motivic”way for attackingalgebraic independence in positive characteristic.

(36)

From linear independence to algebraic independence

u = (1, u₁, −F_δ₂(u₁), · · · , −F_δ_r(u₁), · · · , u_n, −F_δ₂(u_n), · · · , −F_δ_r(u_n)).

The algebraic point exp_G(u) corresponding to this vector is (1, exp_ρ(u₁), 0, · · · , exp_ρ(u₂), 0, · · · , · · · , exp_ρ(u_n), 0, · · · ).

The hypothesis that u1, . . . , un are linearly independent over Kρ

implies precisely that this algebraic point on G does not fall in any proper t-submodule of G.

Extensive efforts of using the t-submodule theorem to prove linear independence results by many people in the late 1990’s, e.g.

A-B-P concerning the independence of geometric Gamma values, lead to a “motivic”way for attackingalgebraic independence in positive characteristic.

(37)

Transcendence Theory of Drinfeld Modules