# Weakly Holomorphic Vector Valued Modular Forms

## Transcript Of Weakly Holomorphic Vector Valued Modular Forms

Weakly Holomorphic Vector Valued Modular Forms

Jitendra Bajpai University of Alberta

Presented at: Atkin Memorial Lecture and Workshop

On Noncongruence modular forms and Galois representations

University of Illinois at Chicago April 29 - May 1, 2011

May 1, 2011

Jitendra Bajpai (U of A)

Vector Valued Modular Forms

May 1, 2011 1 / 32

Notation

Introduction

H∗ = H ∪ Q ∪ {∞} - extended upper Half plane.

Γ(1) = PSL2(Z) = t, s , where

t = ± 1 1 , s = ± 0 −1

01

10

Jitendra Bajpai (U of A)

Vector Valued Modular Forms

May 1, 2011 2 / 32

Notation

Introduction

H∗ = H ∪ Q ∪ {∞} - extended upper Half plane.

Γ(1) = PSL2(Z) = t, s , where

t = ± 1 1 , s = ± 0 −1

01

10

Γ be any genus-0 ﬁnite index subgroup.

CΓ is the set of all inequivalent cusps of Γ.

Jitendra Bajpai (U of A)

Vector Valued Modular Forms

May 1, 2011 2 / 32

Multiplier System Introduction

Let ρ : Γ(1) −→ GLd(C) be rank d representation of Γ(1). We say that ρ is an admissible multiplier of Γ(1) if ρ(t) is a diagonal matrix, i.e. for some diagonal matrix Λ ∈ Md(C), ρ(t) = e2πiΛ.

Jitendra Bajpai (U of A)

Vector Valued Modular Forms

May 1, 2011 3 / 32

Remark

Introduction

For any Γ of Γ(1), admissible multiplier ρ will require that ρ(tc) is a diagonalizable matrix for every cusp c ∈ CΓ.

Jitendra Bajpai (U of A)

Vector Valued Modular Forms

May 1, 2011 4 / 32

Remark

Introduction

For any Γ of Γ(1), admissible multiplier ρ will require that ρ(tc) is a diagonalizable matrix for every cusp c ∈ CΓ.

Here tc denote the generator of the stabilizer subgroup of cusp c.

Jitendra Bajpai (U of A)

Vector Valued Modular Forms

May 1, 2011 4 / 32

Weakly Holomorphic Vector Valued Modular Form

Weakly Holomorphic Vector Valued Modular Form

Let ρ be an admissible multiplier for Γ(1) of rank d. A map X : H −→ Cd is said to be weakly holomorphic vector valued modular form for Γ(1) of weight w and multiplier ρ, if X is holomorphic throughout H and may have poles only at the cusps with following functional and cuspidal behaviour:

Jitendra Bajpai (U of A)

Vector Valued Modular Forms

May 1, 2011 5 / 32

Weakly Holomorphic Vector Valued Modular Form

Weakly Holomorphic Vector Valued Modular Form

Let ρ be an admissible multiplier for Γ(1) of rank d. A map X : H −→ Cd is said to be weakly holomorphic vector valued modular form for Γ(1) of weight w and multiplier ρ, if X is holomorphic throughout H and may have poles only at the cusps with following functional and cuspidal behaviour:

Functional behaviour

X(γτ ) = ρ(γ)(cτ + d)w X(τ ), ∀γ ∈ Γ & ∀τ ∈ H.

Jitendra Bajpai (U of A)

Vector Valued Modular Forms

May 1, 2011 5 / 32

Weakly Holomorphic Vector Valued Modular Form

Weakly Holomorphic Vector Valued Modular Form

Cuspidal behaviour

Since Γ(1) has only one cusp ∞, q−ΛX(τ ) has periodicity 1 therefore it has Fourier expansion of the following form,

q−ΛX(τ ) = anqn,

n∈Z

where

q = e2πiτ

which has at most ﬁnitely many nonzero an ∈ Cd with n < 0.

Jitendra Bajpai (U of A)

Vector Valued Modular Forms

May 1, 2011 6 / 32

Weakly Holomorphic Vector Valued Modular Form

For any weight w ∈ 2Z and multiplier ρ of any Γ, Mw (Γ, ρ, d ) denotes the C-Vector Space of all Weakly Holomorphic Vector Valued Modular Forms.

Jitendra Bajpai (U of A)

Vector Valued Modular Forms

May 1, 2011 7 / 32

Jitendra Bajpai University of Alberta

Presented at: Atkin Memorial Lecture and Workshop

On Noncongruence modular forms and Galois representations

University of Illinois at Chicago April 29 - May 1, 2011

May 1, 2011

Jitendra Bajpai (U of A)

Vector Valued Modular Forms

May 1, 2011 1 / 32

Notation

Introduction

H∗ = H ∪ Q ∪ {∞} - extended upper Half plane.

Γ(1) = PSL2(Z) = t, s , where

t = ± 1 1 , s = ± 0 −1

01

10

Jitendra Bajpai (U of A)

Vector Valued Modular Forms

May 1, 2011 2 / 32

Notation

Introduction

H∗ = H ∪ Q ∪ {∞} - extended upper Half plane.

Γ(1) = PSL2(Z) = t, s , where

t = ± 1 1 , s = ± 0 −1

01

10

Γ be any genus-0 ﬁnite index subgroup.

CΓ is the set of all inequivalent cusps of Γ.

Jitendra Bajpai (U of A)

Vector Valued Modular Forms

May 1, 2011 2 / 32

Multiplier System Introduction

Let ρ : Γ(1) −→ GLd(C) be rank d representation of Γ(1). We say that ρ is an admissible multiplier of Γ(1) if ρ(t) is a diagonal matrix, i.e. for some diagonal matrix Λ ∈ Md(C), ρ(t) = e2πiΛ.

Jitendra Bajpai (U of A)

Vector Valued Modular Forms

May 1, 2011 3 / 32

Remark

Introduction

For any Γ of Γ(1), admissible multiplier ρ will require that ρ(tc) is a diagonalizable matrix for every cusp c ∈ CΓ.

Jitendra Bajpai (U of A)

Vector Valued Modular Forms

May 1, 2011 4 / 32

Remark

Introduction

For any Γ of Γ(1), admissible multiplier ρ will require that ρ(tc) is a diagonalizable matrix for every cusp c ∈ CΓ.

Here tc denote the generator of the stabilizer subgroup of cusp c.

Jitendra Bajpai (U of A)

Vector Valued Modular Forms

May 1, 2011 4 / 32

Weakly Holomorphic Vector Valued Modular Form

Weakly Holomorphic Vector Valued Modular Form

Let ρ be an admissible multiplier for Γ(1) of rank d. A map X : H −→ Cd is said to be weakly holomorphic vector valued modular form for Γ(1) of weight w and multiplier ρ, if X is holomorphic throughout H and may have poles only at the cusps with following functional and cuspidal behaviour:

Jitendra Bajpai (U of A)

Vector Valued Modular Forms

May 1, 2011 5 / 32

Weakly Holomorphic Vector Valued Modular Form

Weakly Holomorphic Vector Valued Modular Form

Let ρ be an admissible multiplier for Γ(1) of rank d. A map X : H −→ Cd is said to be weakly holomorphic vector valued modular form for Γ(1) of weight w and multiplier ρ, if X is holomorphic throughout H and may have poles only at the cusps with following functional and cuspidal behaviour:

Functional behaviour

X(γτ ) = ρ(γ)(cτ + d)w X(τ ), ∀γ ∈ Γ & ∀τ ∈ H.

Jitendra Bajpai (U of A)

Vector Valued Modular Forms

May 1, 2011 5 / 32

Weakly Holomorphic Vector Valued Modular Form

Weakly Holomorphic Vector Valued Modular Form

Cuspidal behaviour

Since Γ(1) has only one cusp ∞, q−ΛX(τ ) has periodicity 1 therefore it has Fourier expansion of the following form,

q−ΛX(τ ) = anqn,

n∈Z

where

q = e2πiτ

which has at most ﬁnitely many nonzero an ∈ Cd with n < 0.

Jitendra Bajpai (U of A)

Vector Valued Modular Forms

May 1, 2011 6 / 32

Weakly Holomorphic Vector Valued Modular Form

For any weight w ∈ 2Z and multiplier ρ of any Γ, Mw (Γ, ρ, d ) denotes the C-Vector Space of all Weakly Holomorphic Vector Valued Modular Forms.

Jitendra Bajpai (U of A)

Vector Valued Modular Forms

May 1, 2011 7 / 32