On values of Modular Forms at Algebraic Points

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On values of Modular Forms at Algebraic Points

Jing Yu

National Taiwan University, Taipei, Taiwan

August 14, 2010, 18th ICFIDCAA, Macau

Jing Yu, NTU, Taiwan Values at Algebraic Points

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Hermite-Lindemann-Weierstrass

In value distribution theory the exponential function e^z is a key.

This fuction also enjoys the following extraordinary properties:

(Hermite-Lindemann-Weierstrass 1880) For α 6= 0 ∈ Q,

e^α is transcendental. Moreover, if algebraic numbers α1, . . . , αn

are linearly independent over Q then e^α¹, . . . , e^αⁿ are algebraically independent, i.e. for any polynomial P 6= 0 ∈ Q(x1, . . . , x_n),

P (e^α¹, . . . , e^αⁿ) 6= 0

Tools for proving this come from Complex Analysis.

Let H be the complex upper half plane.

We are also interested in values of “natural”holomorphic functions taking at “algebraic points”of H.

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Hermite-Lindemann-Weierstrass

In value distribution theory the exponential function e^z is a key.

This fuction also enjoys the following extraordinary properties:

(Hermite-Lindemann-Weierstrass 1880) For α 6= 0 ∈ Q,

e^α is transcendental. Moreover, if algebraic numbers α1, . . . , αn

are linearly independent over Q then e^α¹, . . . , e^αⁿ are algebraically independent, i.e. for any polynomial P 6= 0 ∈ Q(x1, . . . , x_n),

P (e^α¹, . . . , e^αⁿ) 6= 0

Tools for proving this come from Complex Analysis.

Let H be the complex upper half plane.

We are also interested in values of “natural”holomorphic functions taking at “algebraic points”of H.

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Siegel-Schneider

The modular function j : SL2(Z)\H ∼= C which parametrizes isomorphism classes of complex elliptic curves.

This function j can be employed for proving the Picard theorem.

This j also has beautiful transcendence property:

(Siegel-Schneider 1930) If α ∈ Q ∩ H and α is not quadratic, then j(α) is transcendental.

If α is (imaginary) quadratic, then j(α) is actually an algebraic integer, as known to Kronecker.

Call α ∈ H algebraic point if j(α) ∈ Q. Thus unless an algebraic point α ∈ H is imaginary quadratic number, it must be a

transcendental number.

Elliptic curves correspond to algebraic points can all be defined over Q.

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Siegel-Schneider

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Siegel-Schneider

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Arithmetic modular forms

(Meromorphic) Modular form f : H −→ C ∪ {∞}, of weight k, here k is a fixed integer, satisfying for all z ∈ H

f (az + b

cz + d) = (cz + d)^kf (z),

∀ a b

c d

!

∈ SL₂(Z).

Modular forms are required to be meromorphic at ∞, i.e. with Fourier expansion:

f (z) =

∞

X

n=n0

a_ne^2πinz.

Call f arithmetic modular form if all coefficients an∈ Q.

Note one can replace SL2(Z) by its congruence subgroups Γ , and requiring f to be meromorphic at all “cusps”.

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Arithmetic modular forms

(Meromorphic) Modular form f : H −→ C ∪ {∞}, of weight k, here k is a fixed integer, satisfying for all z ∈ H

f (az + b

cz + d) = (cz + d)^kf (z),

∀ a b

c d

!

∈ SL₂(Z).

Modular forms are required to be meromorphic at ∞, i.e. with Fourier expansion:

f (z) =

∞

X

n=n0

a_ne^2πinz.

Call f arithmetic modular form if all coefficients an∈ Q.

Note one can replace SL2(Z) by its congruence subgroups Γ , and requiring f to be meromorphic at all “cusps”.

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Values at algebraic points

Reformulating works of Siegel-Schneider, one has

Theorem. Let f be arithmetic modular form of nonzero weight k.

Let α ∈ H is an algebraic point which is neither zero nor pole of f , then f (α) is transcendental.

Open Problem.Let f as above, α₁, . . . , α_n∈ H be algebraic points which are neither zeros nor poles of f . Suppose that the αi are pairwise non-isogenous, are the values f (α1), . . . , f (αn) algebraically independent?

Here α and β ∈ H are said to be non-isogenous, if the elliptic curves they correspond are not isogenous.

Note that the value f (α) is always an algebraic multiple of the k-th power of a period (of the elliptic curve corresponding to α) dividing by π.

Can prove the linearly independence over Q of these values.

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Values at algebraic points

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Values at algebraic points

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World 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.

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.

Non-archimedean analytic function theory on C∞, and on Drinfeld upper-half space H∞ which is C∞− k_∞.

Natural non-archimedean analytic functions come from Drinfeld modules theory.

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World of Positive characteristic

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World of Positive characteristic

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Drinfeld modules

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

τ c = c^qτ, for all c ∈ C∞.

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

ρt= θ + g1τ + · · · + gr−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^ρ^t 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 C∞[τ ] be the twisted polynomial ring :

τ c = c^qτ, for all c ∈ C∞.

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

ρt= θ + g1τ + · · · + gr−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^ρ^t C∞

exp_ρ

−−−−→ Ga(C∞) = C∞

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Transcendence theory

Analogue of Hermite-Lindemann-Weierstrass, and Siegel-Schneider:

Theorem 1.(Yu 1986) Let ρ be a Drinfeld Fq[t]-module defined over ¯k, with associated exponential map exp_ρ(z) on C∞. If α 6= 0 ∈ ¯k, then exp_ρ(α) is transcendental over k.

Theorem 2.(A. Thiery 1995) Suppose the Drinfeld module ρ is of rank 1. If α1, . . . , αn∈ ¯k are linearly independent over k , then exp_ρ(α₁), . . . , exp_ρ(α_n) are algebraically independent over k.

Drinfeld upper-half space H∞ parametrizes isomorphism classes of rank 2 Drinfeld modules, let j = g₁^q+1/∆ :

j : GL₂(Fq[θ])\H∞∼= C∞.

Call α ∈ H∞ algebraic point if j(α) ∈ ¯k. Drinfeld modules corresponding to algebraic points can be defined over ¯k.

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Transcendence theory

j : GL₂(Fq[θ])\H∞∼= C∞.

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Transcendence theory

j : GL₂(Fq[θ])\H∞∼= C∞.

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The j values and periods

We have the following

Theorem 3.(Yu 1986) If α ∈ ¯k ∩ H∞ and α is not quadratic over k, then j(α) is transcendental over k. Moreover, for those α quadratic over k, j(α) are integral over Fq[θ].

If Drinfeld module ρ is of rank r, 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 by Theorem 1.

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

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The j values and periods

exp_ρ(z) = z Y

λ6=0∈Λρ

(1 − z λ).

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The j values and periods

exp_ρ(z) = z Y

λ6=0∈Λρ

(1 − z λ).

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Algebraic relations among periods

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

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

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

Drinfeld module ρ of rank 2 is said to be without Complex Multiplications if K_ρ= k, and with CM if [K_ρ: k] = 2.

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

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Algebraic relations among periods

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Algebraic relations among periods

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Drinfeld modular forms

Modular form f : H∞−→ C∞∪ {∞}, of weight k and type m, here k is a fixed integer, m ∈ Z/(q − 1)Z, satisfying for all z ∈ H∞

f (az + b

cz + d) = (det γ)^m(cz + d)^kf (z),

∀γ = a b c d

!

∈ GL₂(Fq[θ]).

Modular forms are required to be “rigid”meromorphic functions, and at ∞ with “Fourier”expansion:

f (z) =

∞

X

n=n0

a_nq∞(z)ⁿ, q∞(z) = X

a∈Fq[θ]

1 z − a. Call f arithmetic modular form if all coefficients an∈ ¯k.

Note one can replace GL2(Fq[θ]) by its congruence subgroups Γ , and requiring f to be meromorphic at all “cusps”.

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Values at algebraic points

Here one also proves

Theorem.(Yu) Let f be arithmetic modular form of nonzero weight k. Let α ∈ H∞ is an algebraic point which is neither zero nor pole of f , then f (α) is transcendental over k.

Open Problem.Let f as above, α₁, . . . , α_n∈ H_∞ be algebraic points which are neither zeros nor poles of f . Suppose that the αi are pairwise non-isogenous, are the values f (α1), . . . , f (αn) algebraically independent over k?

Here α and β ∈ H∞ are said to be non-isogenous, if the Drinfeld modules they correspond are not isogenous.

Again the value f (α) is equal to an element of ¯k times k-th power of a period (of the rank 2 Drinfeld modules defined over ¯k

corresponding to α) dividing by the Carlitz period (of rank 1 Drinfeld Fq[t]-module).

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Values at algebraic points

Here one also proves

Theorem.(Yu) Let f be arithmetic modular form of nonzero weight k. Let α ∈ H∞ is an algebraic point which is neither zero nor pole of f , then f (α) is transcendental over k.

Open Problem.Let f as above, α₁, . . . , α_n∈ H_∞ be algebraic points which are neither zeros nor poles of f . Suppose that the αi are pairwise non-isogenous, are the values f (α1), . . . , f (αn) algebraically independent over k?

Here α and β ∈ H∞ are said to be non-isogenous, if the Drinfeld modules they correspond are not isogenous.

Again the value f (α) is equal to an element of ¯k times k-th power of a period (of the rank 2 Drinfeld modules defined over ¯k

corresponding to α) dividing by the Carlitz period (of rank 1 Drinfeld Fq[t]-module).

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Values at CM points

The CM points are those α ∈ H∞ which are quadratic over k, hence correspond to Drinfeld modules with CM.

Theorem.(C.-Y. Chang 2010) Let f be arithmetic modular form of nonzero weight. Let α1, . . . , αn∈ H_∞ be CM points which are neither zeros nor poles of f . Suppose that the α_i are pairwise non-isogenous, then the values f (α1), . . . , f (αn) are algebraically independent over k.

Here CM points α and β ∈ H∞ are non-isogenous, precisely when they belong to different quadratic extension of k.

Method for proving algebraic independence in positive characteristic, developed in the last 10 years, by Anderson, Brownawell, Chang, Papanikolas, and Yu. Crucial step by Papanikolas 2008.

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Motivic transcendence theory

Realizing a program of Grothendieck in positive characteristic.

We are interested in finitely generated extension of ¯k 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, one can guess the transcendence degree of K_S over ¯k, and the goal is to prove that is indeed the specific degree in question.

To proceed, we construct a t-motive MS for this purpose, so that it has the GP property and its “periods”Ψ_S(θ) from “rigid analytic trivialization ”generate also the field KS, then computing the dimension of the motivic Galois (algebraic) group Γ_M_S. GP property of the motive M_S requires:

dim Γ_M_S = tr.deg¯k ¯k(Ψ_S(θ)).

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Motivic transcendence theory

Realizing a program of Grothendieck in positive characteristic.

We are interested in finitely generated extension of ¯k 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, one can guess the transcendence degree of K_S over ¯k, and the goal is to prove that is indeed the specific degree in question.

To proceed, we construct a t-motive MS for this purpose, so that it has the GP property and its “periods”Ψ_S(θ) from “rigid analytic trivialization ”generate also the field KS, then computing the dimension of the motivic Galois (algebraic) group Γ_M_S. GP property of the motive M_S requires:

dim Γ_M_S = tr.deg¯k ¯k(Ψ_S(θ)).

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On values of Modular Forms at Algebraic Points