Images of Galois representations with values in mod p Hecke algebras

Images of Galois representations with values in mod p Hecke algebras

Laia Amor´os Caraf´ı Universit´e du Luxembourg

Universitat de Barcelona

Images of Galois representations with values in mod p Hecke algebras

• mod pmodular forms, modpHecke algebras

• Galois representations with values in these algebras

• Computation of the image of these Galois representations

• Application

Images of Galois representations with values in mod p Hecke algebras

• mod pmodular forms, modpHecke algebras

• Galois representations with values in these algebras

• Computation of the image of these Galois representations

• Application

2 / 17

Images of Galois representations with values in mod p Hecke algebras

• mod pmodular forms, modpHecke algebras

• Galois representations with values in these algebras

• Computation of the image of these Galois representations

• Application

Images of Galois representations with values in mod p Hecke algebras

• mod pmodular forms, modpHecke algebras

• Galois representations with values in these algebras

• Computation of the image of these Galois representations

• Application

2 / 17

mod p Hecke algebras

End_C(Sk(N;C))⊃Tk(N)

T:=Tk(N)⊗Fq finite-dimensional commutativeFq-algebra T'Q

mTm, where Tm localisation ofTat a maximal idealm ofT

λf :T→Fq, Tn7→an=an modp mf := kerλf

T^f :=T^mf assumem²_f = 0

Tf 'Fq[X₁, . . . ,X_m]/(X_iX_j)_1≤i,j≤m finite-dimensional local commutative algebra, m=dim_Fqm_f

mod p Hecke algebras

Sk(N, ε;C) space ofmodular formsf(z) =P

n≥0anqⁿ(q=e^2πiz) of levelN≥1, weightk≥2 and Dirichlet characterε: (Z/NZ)^×→C^×. Moreover assumea₀= 0.

End_C(Sk(N;C))⊃Tk(N)

T:=Tk(N)⊗Fq finite-dimensional commutativeFq-algebra T'Q

mTm, where Tm localisation ofTat a maximal idealm ofT

λf :T→Fq, Tn7→an=an modp mf := kerλf

T^f :=Tmf assumem²_f = 0

Tf 'Fq[X₁, . . . ,X_m]/(X_iX_j)_1≤i,j≤m finite-dimensional local commutative algebra, m=dim_Fqm_f

3 / 17

mod p Hecke algebras

Sk(N, ε;C) space ofcuspidal modular formsor cusp forms

End_C(Sk(N;C))⊃Tk(N)

T:=Tk(N)⊗Fq finite-dimensional commutativeFq-algebra T'Q

mTm, where Tm localisation ofTat a maximal idealm ofT

λ_f :T→Fq, T_n7→a_n=a_n modp m_f := kerλ_f Tf :=Tm_f assumem²_f = 0

Tf 'Fq[X1, . . . ,Xm]/(XiXj)1≤i,j≤m finite-dimensional local commutative algebra, m=dim_Fqmf

mod p Hecke algebras

Sk(N;C) space of cusp forms

End_C(Sk(N;C))⊃Tk(N)

T:=Tk(N)⊗Fq finite-dimensional commutativeFq-algebra T'Q

mTm, where Tm localisation ofTat a maximal idealm ofT

λ_f :T→Fq, T_n7→a_n=a_n modp m_f := kerλ_f Tf :=Tm_f assumem²_f = 0

Tf 'Fq[X1, . . . ,Xm]/(XiXj)1≤i,j≤m finite-dimensional local commutative algebra, m=dim_Fqmf

3 / 17

mod p Hecke algebras

Sk(N;C) space of cusp forms

End_C(Sk(N;C))⊃Tk(N) :=hTp Hecke operator :p primei

T:=Tk(N)⊗Fq finite-dimensional commutativeFq-algebra T'Q

mTm, where Tm localisation ofTat a maximal idealm ofT

λ_f :T→Fq, T_n7→a_n=a_n modp m_f := kerλ_f Tf :=Tm_f assumem²_f = 0

Tf 'Fq[X1, . . . ,Xm]/(XiXj)1≤i,j≤m finite-dimensional local commutative algebra, m=dim_Fqmf

mod p Hecke algebras

Sk(N;C) space of cusp forms

End_C(Sk(N;C))⊃Tk(N) finite-dimensional commutativeZ-algebra

T:=Tk(N)⊗Fq finite-dimensional commutativeFq-algebra T'Q

mTm, where Tm localisation ofTat a maximal idealm ofT

λ_f :T→Fq, T_n7→a_n=a_n modp m_f := kerλ_f Tf :=Tm_f assumem²_f = 0

Tf 'Fq[X1, . . . ,Xm]/(XiXj)1≤i,j≤m finite-dimensional local commutative algebra, m=dim_Fqmf

3 / 17

mod p Hecke algebras

S_k(N;C) space of cusp forms

End_C(S_k(N;C))⊃Tk(N) finite-dimensional commutativeZ-algebra Sk(N;Z)

T:=Tk(N)⊗Fq finite-dimensional commutativeFq-algebra T'Q

mTm, where Tm localisation ofTat a maximal idealm ofT

λf :T→F^q, Tn7→an=an modp mf := kerλf

Tf :=Tmf assumem²_f = 0

Tf 'Fq[X1, . . . ,Xm]/(XiXj)1≤i,j≤m finite-dimensional local commutative algebra, m=dim_Fqmf

mod p Hecke algebras

S_k(N;C) space of cusp forms

End_C(S_k(N;C))⊃Tk(N) finite-dimensional commutativeZ-algebra Sk(N;Fq) :=Sk(N;Z)⊗_ZFq

T:=Tk(N)⊗Fq finite-dimensional commutativeFq-algebra T'Q

mTm, where Tm localisation ofTat a maximal idealm ofT

λf :T→F^q, Tn7→an=an modp mf := kerλf

Tf :=Tmf assumem²_f = 0

Tf 'Fq[X1, . . . ,Xm]/(XiXj)1≤i,j≤m finite-dimensional local commutative algebra, m=dim_Fqmf

3 / 17

mod p Hecke algebras

S_k(N;C) space of cusp forms

End_C(S_k(N;C))⊃Tk(N) finite-dimensional commutativeZ-algebra Sk(N;Fq) :=Sk(N;Z)⊗_ZFq

T:=Tk(N)⊗Fq finite-dimensional commutativeFq-algebra

T'Q

mTm, where Tm localisation ofTat a maximal idealm ofT

λf :T→F^q, Tn7→an=an modp mf := kerλf

Tf :=Tmf assumem²_f = 0

Tf 'Fq[X1, . . . ,Xm]/(XiXj)1≤i,j≤m finite-dimensional local commutative algebra, m=dim_Fqmf

mod p Hecke algebras

S_k(N;C) space of cusp forms

End_C(S_k(N;C))⊃Tk(N) finite-dimensional commutativeZ-algebra Sk(N;Fq) :=Sk(N;Z)⊗_ZFq

T:=Tk(N)⊗Fq finite-dimensional commutativeFq-algebra T'Q

mT^m, whereT^m localisation ofTat a maximal idealm ofT

λf :T→F^q, Tn7→an=an modp mf := kerλf

Tf :=Tmf assumem²_f = 0

Tf 'Fq[X1, . . . ,Xm]/(XiXj)1≤i,j≤m finite-dimensional local commutative algebra, m=dim_Fqmf

3 / 17

mod p Hecke algebras

Sk(N;C) space of cusp forms

End_C(Sk(N;C))⊃Tk(N) finite-dimensional commutativeZ-algebra Sk(N;F^q) :=Sk(N;Z)⊗_ZF^q

T:=Tk(N)⊗Fq finite-dimensional commutativeFq-algebra T'Q

mTm, whereTm localisation ofTat a maximal idealm ofT Let us takef(z) =P

n≥0anqⁿ∈Sk(N;C) ,q=e^2πiz, simultaneous eigenvector for all Hecke operators,a1= 1

λf :T→Fq, Tn7→an=an modp mf := kerλf

T^f :=Tmf assumem²_f = 0

Tf 'Fq[X₁, . . . ,X_m]/(X_iX_j)_1≤i,j≤m finite-dimensional local commutative algebra, m=dim_Fqm_f

mod p Hecke algebras

Sk(N;C) space of cusp forms

End_C(Sk(N;C))⊃Tk(N) finite-dimensional commutativeZ-algebra Sk(N;F^q) :=Sk(N;Z)⊗_ZF^q

T:=Tk(N)⊗Fq finite-dimensional commutativeFq-algebra T'Q

mTm, whereTm localisation ofTat a maximal idealm ofT Let us takef(z) =P

n≥0anqⁿ∈Sk(N;C)normalised Hecke eigenform

λf :T→Fq, Tn7→an=an modp mf := kerλf

T^f :=Tmf assumem²_f = 0

Tf 'Fq[X₁, . . . ,X_m]/(X_iX_j)_1≤i,j≤m finite-dimensional local commutative algebra, m=dim_Fqm_f

3 / 17

mod p Hecke algebras

Sk(N;C) space of cusp forms

End_C(Sk(N;C))⊃Tk(N) finite-dimensional commutativeZ-algebra Sk(N;F^q) :=Sk(N;Z)⊗_ZF^q

T:=Tk(N)⊗Fq finite-dimensional commutativeFq-algebra T'Q

mTm, whereTm localisation ofTat a maximal idealm ofT Let us takef(z) =P

n≥0anqⁿ∈Sk(N;Fq) normalised Hecke eigenform modp

λf :T→Fq, Tn7→an=an modp mf := kerλf

T^f :=Tmf assumem²_f = 0

Tf 'Fq[X₁, . . . ,X_m]/(X_iX_j)_1≤i,j≤m finite-dimensional local commutative algebra, m=dim_Fqm_f

mod p Hecke algebras

Sk(N;C) space of cusp forms

End_C(Sk(N;C))⊃Tk(N) finite-dimensional commutativeZ-algebra Sk(N;F^q) :=Sk(N;Z)⊗_ZF^q

T:=Tk(N)⊗Fq finite-dimensional commutativeFq-algebra T'Q

mTm, whereTm localisation ofTat a maximal idealm ofT Let us takef(z) =P

n≥0anqⁿ∈Sk(N;Fq) normalised Hecke eigenform modp

λf :T→Fq, Tn7→an=an modp mf := kerλf

T^f :=Tmf assumem²_f = 0

Tf 'Fq[X₁, . . . ,X_m]/(X_iX_j)_1≤i,j≤m finite-dimensional local commutative algebra, m=dim_Fqm_f

3 / 17

mod p Hecke algebras

Sk(N;C) space of cusp forms

End_C(Sk(N;C))⊃Tk(N) finite-dimensional commutativeZ-algebra Sk(N;F^q) :=Sk(N;Z)⊗_ZF^q

T:=Tk(N)⊗Fq finite-dimensional commutativeFq-algebra T'Q

mTm, whereTm localisation ofTat a maximal idealm ofT Let us takef(z) =P

n≥0anqⁿ∈Sk(N;Fq) normalised Hecke eigenform modp

λf :T→Fq, Tn7→an=an modp mf := kerλf

Tf :=Tmf assumem²_f = 0

Tf 'Fq[X₁, . . . ,X_m]/(X_iX_j)_1≤i,j≤m finite-dimensional local commutative algebra, m=dim_Fqm_f

mod p Hecke algebras

Sk(N;C) space of cusp forms

End_C(Sk(N;C))⊃Tk(N) finite-dimensional commutativeZ-algebra Sk(N;F^q) :=Sk(N;Z)⊗_ZF^q

T:=Tk(N)⊗Fq finite-dimensional commutativeFq-algebra T'Q

mTm, whereTm localisation ofTat a maximal idealm ofT Let us takef(z) =P

n≥0anqⁿ∈Sk(N;Fq) normalised Hecke eigenform modp

λf :T→Fq, Tn7→an=an modp mf := kerλf

Tf :=Tmf assumem²_f = 0

Tf 'Fq[X₁, . . . ,X_m]/(X_iX_j)_1≤i,j≤m finite-dimensional local commutative algebra, m=dim_Fqm_f

3 / 17

Galois representations

Next we want to: attach a Galois representation toTf.

Letf =P

n≥0a_nqⁿ∈S_k(N;Fq) given byλ_f :T→Fq, T_n→a_n Deligne, Shimura: We can attach tof aGalois representation

ρ_f :Gal(Q/Q)→GL2(Fp) unramified outsideNp and, for every`-Np:

tr(ρ_f(Frob`)) =λf(T`) and det(ρ_f(Frob`)) =`^k−1 LetTf :=Tm_f as before, but without the operatorsT` with`|Np. Carayol: Ifρ_f is absolutely irreducible, then there exists a continuous Galois representation

ρf :Gal(Q/Q)→GL2(Tf) unramified outsideNp and, for every`-Np:

tr(ρf(Frob`)) =λf(T`) and det(ρf(Frob`)) =`^k−1 whereλf :T→Tf. This representation is unique up to conjugation.

Galois representations

Next we want to: attach a Galois representation toTf. Letf =P

n≥0a_nqⁿ∈S_k(N;Fq) given byλ_f :T→Fq, T_n→a_n

Deligne, Shimura: We can attach tof aGalois representation ρ_f :Gal(Q/Q)→GL2(Fp)

unramified outsideNp and, for every`-Np:

tr(ρ_f(Frob`)) =λf(T`) and det(ρ_f(Frob`)) =`^k−1 LetTf :=Tm_f as before, but without the operatorsT` with`|Np. Carayol: Ifρ_f is absolutely irreducible, then there exists a continuous Galois representation

ρf :Gal(Q/Q)→GL2(Tf) unramified outsideNp and, for every`-Np:

tr(ρf(Frob`)) =λf(T`) and det(ρf(Frob`)) =`^k−1 whereλf :T→Tf. This representation is unique up to conjugation.

4 / 17

Galois representations

Next we want to: attach a Galois representation toTf. Letf =P

n≥0a_nqⁿ∈S_k(N;Fq) given byλ_f :T→Fq, T_n→a_n Deligne, Shimura: We can attach tof aGalois representation

ρ_f :Gal(Q/Q)→GL2(Fp) unramified outsideNp and, for every`-Np:

tr(ρ_f(Frob`)) =λf(T`) and det(ρ_f(Frob`)) =`^k−1

LetTf :=Tm_f as before, but without the operatorsT` with`|Np. Carayol: Ifρ_f is absolutely irreducible, then there exists a continuous Galois representation

ρf :Gal(Q/Q)→GL2(Tf) unramified outsideNp and, for every`-Np:

tr(ρf(Frob`)) =λf(T`) and det(ρf(Frob`)) =`^k−1 whereλf :T→Tf. This representation is unique up to conjugation.

Galois representations

Next we want to: attach a Galois representation toTf. Letf =P

n≥0a_nqⁿ∈S_k(N;Fq) given byλ_f :T→Fq, T_n→a_n Deligne, Shimura: We can attach tof aGalois representation

ρ_f :Gal(Q/Q)→GL2(Fp) unramified outsideNp and, for every`-Np:

tr(ρ_f(Frob`)) =λf(T`) and det(ρ_f(Frob`)) =`^k−1 LetTf :=Tm_f as before, but without the operatorsT` with`|Np.

Carayol: Ifρ_f is absolutely irreducible, then there exists a continuous Galois representation

ρf :Gal(Q/Q)→GL2(Tf) unramified outsideNp and, for every`-Np:

tr(ρf(Frob`)) =λf(T`) and det(ρf(Frob`)) =`^k−1 whereλf :T→Tf. This representation is unique up to conjugation.

4 / 17

Galois representations

Next we want to: attach a Galois representation toTf. Letf =P

n≥0a_nqⁿ∈S_k(N;Fq) given byλ_f :T→Fq, T_n→a_n Deligne, Shimura: We can attach tof aGalois representation

ρ_f :Gal(Q/Q)→GL2(Fp) unramified outsideNp and, for every`-Np:

tr(ρ_f(Frob`)) =λf(T`) and det(ρ_f(Frob`)) =`^k−1 LetTf :=Tm_f as before, but without the operatorsT` with`|Np.

Carayol: Ifρ_f is absolutely irreducible, then there exists a continuous Galois representation

ρf :Gal(Q/Q)→GL2(Tf) unramified outsideNp and, for every`-Np:

tr(ρf(Frob`)) =λf(T`) and det(ρf(Frob`)) =`^k−1 whereλ :T→T . This representation is unique up to conjugation.

Image of ρ

GOAL:Compute the image ofρf :Gal(Q/Q)→GL2(Tf). LetD=Im(det◦ρ_f)⊆F^×q

GL^D₂(Tf) :={g ∈GL2(Tf) : det(g)∈D} GL^D₂(Fq) :={g ∈GL2(Fq) : det(g)∈D} Assume thatIm(ρf)⊆GL^D₂(Tf).

We have the following commutative diagram

that gives us a short exact sequence:

1→ker(π)→GL^D₂(T^f)→^π GL^D₂(F^q)→1

5 / 17

Image of ρ

GOAL:Compute the image ofρf :Gal(Q/Q)→GL2(T^f).

LetD=Im(det◦ρ_f)⊆F^×q

GL^D₂(Tf) :={g ∈GL2(Tf) : det(g)∈D} GL^D₂(Fq) :={g ∈GL2(Fq) : det(g)∈D} Assume thatIm(ρf)⊆GL^D₂(Tf).

We have the following commutative diagram

that gives us a short exact sequence:

1→ker(π)→GL^D₂(T^f)→^π GL^D₂(F^q)→1

Image of ρ

GOAL:Compute the image ofρf :Gal(Q/Q)→GL2(T^f).

LetD=Im(det◦ρ_f)⊆F^×q

GL^D₂(Tf) :={g ∈GL2(Tf) : det(g)∈D}

GL^D₂(Fq) :={g ∈GL2(Fq) : det(g)∈D}

Assume thatIm(ρf)⊆GL^D₂(Tf).

We have the following commutative diagram

that gives us a short exact sequence:

1→ker(π)→GL^D₂(T^f)→^π GL^D₂(F^q)→1

5 / 17

Image of ρ

GOAL:Compute the image ofρf :Gal(Q/Q)→GL2(T^f).

LetD=Im(det◦ρ_f)⊆F^×q

GL^D₂(Tf) :={g ∈GL2(Tf) : det(g)∈D}

GL^D₂(Fq) :={g ∈GL2(Fq) : det(g)∈D}

Assume thatIm(ρf)⊆GL^D₂(Tf).

We have the following commutative diagram

that gives us a short exact sequence:

1→ker(π)→GL^D₂(T^f)→^π GL^D₂(F^q)→1

Image of ρ

GOAL:Compute the image ofρf :Gal(Q/Q)→GL2(T^f).

LetD=Im(det◦ρ_f)⊆F^×q

GL^D₂(Tf) :={g ∈GL2(Tf) : det(g)∈D}

GL^D₂(Fq) :={g ∈GL2(Fq) : det(g)∈D}

Assume thatIm(ρf)⊆GL^D₂(Tf).

We have the following commutative diagram

that gives us a short exact sequence:

1→ker(π)→GL^D₂(T^f)→^π GL^D₂(F^q)→1

5 / 17

Image of ρ

GOAL:Compute the image ofρf :Gal(Q/Q)→GL2(T^f).

LetD=Im(det◦ρ_f)⊆F^×q

GL^D₂(Tf) :={g ∈GL2(Tf) : det(g)∈D}

GL^D₂(Fq) :={g ∈GL2(Fq) : det(g)∈D}

Assume thatIm(ρf)⊆GL^D₂(Tf).

We have the following commutative diagram

that gives us a short exact sequence:

1→ker(π)→GL^D₂(T^f)→^π GL^D₂(F^q)→1

Image of ρ

as a semi-direct product

Assumptions:

• m²_f = 0

• Im(ρ_f) =GL^D₂(Fq)The residual representation hasbig image 1 → ker(π) → GL^D₂(Tf) →^π GL^D₂(Fq) → 1

_a

1+a₂m_f b₁+b₂m_f c₁+c₂m_f d₁+d₂m_f

7→ ^a_c^1b1

1d₁

Takeg =_a

1+a₂m_f b₁+b₂m_f c₁+c₂m_f d₁+d₂m_f

∈GL^D₂(Tf), withai,bi,ci,di ∈Fq. Then

g ∈ker(π)⇔g =

1+a2mf b2mf

c2mf 1+d2mf

and det(g) = 1+(a2+d2)mf ∈D⊆F^×q.

⇔g = 1 +_a

2m_f b₂m_f c₂m_f d₂m_f

anda2=−d2⇔ker(π) = 1 +M⁰₂(mf)

0 → M⁰₂(m_f) →^ι GL^D₂(Tf) →^π GL^D₂(Fq) → 1

a₂m_f b₂m_f c2mf −a2mf

7→ 1 + ^a_c^{2mf b2mf}

2mf −a2mf

6 / 17

Image of ρ

as a semi-direct product

Assumptions:

• m²_f = 0

• Im(ρ_f) =GL^D₂(Fq)The residual representation hasbig image 1 → ker(π) → GL^D₂(Tf) →^π GL^D₂(Fq) → 1

_a

1+a₂m_f b₁+b₂m_f c₁+c₂m_f d₁+d₂m_f

7→ ^a_c^1b1

1d₁

Takeg =_a

1+a₂m_f b₁+b₂m_f c₁+c₂m_f d₁+d₂m_f

∈GL^D₂(Tf), withai,bi,ci,di ∈Fq. Then

g ∈ker(π)⇔g =

1+a2mf b2mf

c2mf 1+d2mf

and det(g) = 1+(a2+d2)mf ∈D⊆F^×q.

⇔g = 1 +_a

2m_f b₂m_f c₂m_f d₂m_f

anda2=−d2⇔ker(π) = 1 +M⁰₂(mf)

0 → M⁰₂(m_f) →^ι GL^D₂(Tf) →^π GL^D₂(Fq) → 1

a₂m_f b₂m_f c2mf −a2mf

7→ 1 + ^a_c^{2mf b2mf}

2mf −a2mf

Image of ρ

as a semi-direct product

Assumptions:

• m²_f = 0

• Im(ρ_f) =GL^D₂(Fq)

The residual representation hasbig image 1 → ker(π) → GL^D₂(Tf) →^π GL^D₂(Fq) → 1

_a

1+a₂m_f b₁+b₂m_f c₁+c₂m_f d₁+d₂m_f

7→ ^a_c^1b1

1d₁

Takeg =_a

1+a₂m_f b₁+b₂m_f c₁+c₂m_f d₁+d₂m_f

∈GL^D₂(Tf), withai,bi,ci,di ∈Fq. Then

g ∈ker(π)⇔g =

1+a2mf b2mf

c2mf 1+d2mf

and det(g) = 1+(a2+d2)mf ∈D⊆F^×q.

⇔g = 1 +_a

2m_f b₂m_f c₂m_f d₂m_f

anda2=−d2⇔ker(π) = 1 +M⁰₂(mf)

0 → M⁰₂(m_f) →^ι GL^D₂(Tf) →^π GL^D₂(Fq) → 1

a₂m_f b₂m_f c2mf −a2mf

7→ 1 + ^a_c^{2mf b2mf}

2mf −a2mf

6 / 17

Image of ρ

as a semi-direct product

Assumptions:

• m²_f = 0

• Im(ρ_f) =GL^D₂(Fq)The residual representation hasbig image

1 → ker(π) → GL^D₂(Tf) →^π GL^D₂(Fq) → 1 _a

1+a₂m_f b₁+b₂m_f c₁+c₂m_f d₁+d₂m_f

7→ ^a_c^1b1

1d₁

Takeg =_a

1+a₂m_f b₁+b₂m_f c₁+c₂m_f d₁+d₂m_f

∈GL^D₂(Tf), withai,bi,ci,di ∈Fq. Then

g ∈ker(π)⇔g =

1+a2mf b2mf

c2mf 1+d2mf

and det(g) = 1+(a2+d2)mf ∈D⊆F^×q.

⇔g = 1 +_a

2m_f b₂m_f c₂m_f d₂m_f

anda2=−d2⇔ker(π) = 1 +M⁰₂(mf)

0 → M⁰₂(m_f) →^ι GL^D₂(Tf) →^π GL^D₂(Fq) → 1

a₂m_f b₂m_f c2mf −a2mf

7→ 1 + ^a_c^{2mf b2mf}

2mf −a2mf

Image of ρ

as a semi-direct product

Assumptions:

• m²_f = 0

• Im(ρ_f) =GL^D₂(Fq)The residual representation hasbig image 1 → ker(π) → GL^D₂(Tf) →^π GL^D₂(Fq) → 1

_a

1+a₂m_f b₁+b₂m_f c₁+c₂m_f d₁+d₂m_f

7→ ^a_c^1b1

1d₁

Takeg =_a

1+a₂m_f b₁+b₂m_f c₁+c₂m_f d₁+d₂m_f

∈GL^D₂(Tf), withai,bi,ci,di ∈Fq. Then

g ∈ker(π)⇔g =

1+a2mf b2mf

c2mf 1+d2mf

and det(g) = 1+(a2+d2)mf ∈D⊆F^×q.

⇔g = 1 +_a

2m_f b₂m_f c₂m_f d₂m_f

anda2=−d2⇔ker(π) = 1 +M⁰₂(mf)

0 → M⁰₂(m_f) →^ι GL^D₂(Tf) →^π GL^D₂(Fq) → 1

a₂m_f b₂m_f c2mf −a2mf

7→ 1 + ^a_c^{2mf b2mf}

2mf −a2mf

6 / 17

Image of ρ

as a semi-direct product

Assumptions:

• m²_f = 0

• Im(ρ_f) =GL^D₂(Fq)The residual representation hasbig image 1 → ker(π) → GL^D₂(Tf) →^π GL^D₂(Fq) → 1

_a

1+a₂m_f b₁+b₂m_f c₁+c₂m_f d₁+d₂m_f

7→ ^a_c^1b1

1d₁

Takeg =_a

1+a₂m_f b₁+b₂m_f c₁+c₂mf d₁+d₂mf

∈GL^D₂(Tf), withai,bi,ci,di ∈Fq.

Then

g ∈ker(π)⇔g =

1+a2mf b2mf

c2mf 1+d2mf

and det(g) = 1+(a2+d2)mf ∈D⊆F^×q.

⇔g = 1 +_a

2m_f b₂m_f c₂m_f d₂m_f

anda2=−d2⇔ker(π) = 1 +M⁰₂(mf)

0 → M⁰₂(m_f) →^ι GL^D₂(Tf) →^π GL^D₂(Fq) → 1

a₂m_f b₂m_f c2mf −a2mf

7→ 1 + ^a_c^{2mf b2mf}

2mf −a2mf

Image of ρ

as a semi-direct product

Assumptions:

• m²_f = 0

• Im(ρ_f) =GL^D₂(Fq)The residual representation hasbig image 1 → ker(π) → GL^D₂(Tf) →^π GL^D₂(Fq) → 1

_a

1+a₂m_f b₁+b₂m_f c₁+c₂m_f d₁+d₂m_f

7→ ^a_c^1b1

1d₁

Takeg =_a

1+a₂m_f b₁+b₂m_f c₁+c₂mf d₁+d₂mf

∈GL^D₂(Tf), withai,bi,ci,di ∈Fq. Then

g ∈ker(π)⇔g =

1+a2mf b2mf

c2mf 1+d2mf

and det(g) = 1+(a2+d2)mf ∈D⊆F^×q.

⇔g = 1 +_a

2m_f b₂m_f c₂m_f d₂m_f

anda2=−d2⇔ker(π) = 1 +M⁰₂(mf)

0 → M⁰₂(m_f) →^ι GL^D₂(Tf) →^π GL^D₂(Fq) → 1

a₂m_f b₂m_f c2mf −a2mf

7→ 1 + ^a_c^{2mf b2mf}

2mf −a2mf

6 / 17

Image of ρ

as a semi-direct product

Assumptions:

• m²_f = 0

• Im(ρ_f) =GL^D₂(Fq)The residual representation hasbig image 1 → ker(π) → GL^D₂(Tf) →^π GL^D₂(Fq) → 1

_a

1+a₂m_f b₁+b₂m_f c₁+c₂m_f d₁+d₂m_f

7→ ^a_c^1b1

1d₁

Takeg =_a

1+a₂m_f b₁+b₂m_f c₁+c₂mf d₁+d₂mf

∈GL^D₂(Tf), withai,bi,ci,di ∈Fq. Then

g ∈ker(π)⇔g =

1+a2mf b2mf

c2mf 1+d2mf

and det(g) = 1+(a2+d2)mf ∈D⊆F^×q.

⇔g = 1 +_a

2m_f b₂m_f c₂m_f d₂m_f

anda2=−d2

⇔ker(π) = 1 +M⁰₂(mf)

0 → M⁰₂(m_f) →^ι GL^D₂(Tf) →^π GL^D₂(Fq) → 1

a₂m_f b₂m_f c2mf −a2mf

7→ 1 + ^a_c^{2mf b2mf}

2mf −a2mf

Image of ρ

as a semi-direct product

Assumptions:

• m²_f = 0

• Im(ρ_f) =GL^D₂(Fq)The residual representation hasbig image 1 → ker(π) → GL^D₂(Tf) →^π GL^D₂(Fq) → 1

_a

1+a₂m_f b₁+b₂m_f c₁+c₂m_f d₁+d₂m_f

7→ ^a_c^1b1

1d₁

Takeg =_a

1+a₂m_f b₁+b₂m_f c₁+c₂mf d₁+d₂mf

∈GL^D₂(Tf), withai,bi,ci,di ∈Fq. Then

g ∈ker(π)⇔g =

1+a2mf b2mf

c2mf 1+d2mf

and det(g) = 1+(a2+d2)mf ∈D⊆F^×q.

⇔g = 1 +_a

2m_f b₂m_f c₂m_f d₂m_f

anda2=−d2⇔ker(π) = 1 +M⁰₂(mf)

0 → M⁰₂(m_f) →^ι GL^D₂(Tf) →^π GL^D₂(Fq) → 1

a₂m_f b₂m_f c2mf −a2mf

7→ 1 + ^a_c^{2mf b2mf}

2mf −a2mf

6 / 17

Image of ρ

as a semi-direct product

Assumptions:

• m²_f = 0

• Im(ρ_f) =GL^D₂(Fq)The residual representation hasbig image 1 → ker(π) → GL^D₂(Tf) →^π GL^D₂(Fq) → 1

_a

1+a₂m_f b₁+b₂m_f c₁+c₂m_f d₁+d₂m_f

7→ ^a_c^1b1

1d₁

Takeg =_a

1+a₂m_f b₁+b₂m_f c₁+c₂mf d₁+d₂mf

∈GL^D₂(Tf), withai,bi,ci,di ∈Fq. Then

g ∈ker(π)⇔g =

1+a2mf b2mf

c2mf 1+d2mf

and det(g) = 1+(a2+d2)mf ∈D⊆F^×q.

⇔g = 1 +_a

2m_f b₂m_f c₂m_f d₂m_f

anda2=−d2⇔ker(π) = 1 +M⁰₂(mf)

0 → M⁰₂(m_f) →^ι GL^D₂(Tf) →^π GL^D₂(Fq) → 1

a₂m_f b₂m_f c2mf −a2mf

7→ 1 + ^a_c^{2mf b2mf}

2mf −a2mf