The Atkin–Lehner operators on Γ₁(N) and the nebentypus #
An Atkin–Lehner matrix W for an exact divisor Q of N normalizes Γ₀(N), and the operators
of TauCeti/NumberTheory/ModularForms/AtkinLehner/Operator.lean act on M_k(Γ₀(N)). It
normalizes Γ₁(N) as well, so the weight-k slash by W is also an operator W_Q on
M_k(Γ₁(N)) and on S_k(Γ₁(N)); that is the operator built here, the carrier on which forms of
nontrivial nebentypus live.
W_Q does not commute with the diamond operators. Moving a representative γ ∈ Γ₀(N) of a label
d ∈ (ZMod N)ˣ across W inverts the residue of the label modulo Q and keeps its residue modulo
N / Q (TauCeti.IsAtkinLehnerMatrix.toHomUnits_gamma0Map_of_mul_eq_mul); writing ι_Q for this
automorphism TauCeti.Nat.IsExactDivisor.unitsInvPart of (ZMod N)ˣ,
W_Q ∘ ⟨d⟩ = ⟨ι_Q d⟩ ∘ W_Q.
Read on a nebentypus space this is Atkin and Li's transport of characters: W_Q carries
M_k(N, χ) into M_k(N, χ ∘ ι_Q), and for χ = χ_Q · χ_{N/Q} split along N = Q · (N / Q),
χ ∘ ι_Q = χ_Q⁻¹ · χ_{N/Q} (TauCeti.Nat.IsExactDivisor.comp_unitsInvPart). So W_Q preserves the
nebentypus space only when the Q-part of χ is quadratic, which is why the Atkin–Lehner theory
of a newform of general nebentypus is a theory of pseudo-eigenvalues rather than eigenvalues.
On Γ₁(N) the operator depends on the chosen Atkin–Lehner matrix, unlike on Γ₀(N): two choices
differ by γ ∈ Γ₀(N) on the left, and replacing W by γ W precomposes W_Q with the diamond
operator of γ. On M_k(N, χ) that is the scalar χ(d_γ), so the operator is determined by Q
up to a scalar there. At Q = N the Fricke matrix gives the Fricke operator of
TauCeti/NumberTheory/ModularForms/Fricke/Operator.lean, and ι_N is inversion.
Main definitions #
TauCeti.atkinLehnerOperatorGamma1,TauCeti.atkinLehnerOperatorGamma1Cusp: the slash by an Atkin–Lehner matrix onM_k(Γ₁(N))and onS_k(Γ₁(N)).TauCeti.atkinLehnerGamma1CharRestrict,TauCeti.atkinLehnerGamma1CharCuspRestrict: those operators restricted to linear mapsM_k(N, χ) → M_k(N, χ ∘ ι_Q)andS_k(N, χ) → S_k(N, χ ∘ ι_Q).
Main results #
TauCeti.Gamma1_map_inv_conjAct_atkinLehnerGL_eq:Wnormalizes the image ofΓ₁(N)inGL (Fin 2) ℝ.TauCeti.atkinLehnerOperatorGamma1_diamondOp,TauCeti.atkinLehnerOperatorGamma1Cusp_diamondOpCusp: the diamond shiftW_Q ∘ ⟨d⟩ = ⟨ι_Q d⟩ ∘ W_Q.TauCeti.atkinLehnerOperatorGamma1_mem_modFormCharSpace,TauCeti.atkinLehnerOperatorGamma1Cusp_mem_cuspFormCharSpace:W_Qcarries the nebentypusχtoχ ∘ ι_Q.TauCeti.atkinLehnerOperatorGamma1_mul_left,TauCeti.atkinLehnerOperatorGamma1_mul_left_of_mem_modFormCharSpace: the dependence on the matrix,W_{γ W} = W_W ∘ ⟨d_γ⟩, a scalarχ(d_γ)onM_k(N, χ); with cusp-form counterpartsTauCeti.atkinLehnerOperatorGamma1Cusp_mul_leftandTauCeti.atkinLehnerOperatorGamma1Cusp_mul_left_of_mem_cuspFormCharSpace.TauCeti.atkinLehnerOperatorGamma1_fricke,TauCeti.atkinLehnerOperatorGamma1Cusp_fricke: at the Fricke matrix the operator isfrickeOperator.TauCeti.atkinLehnerOperatorGamma1_atkinLehnerOperatorGamma1,TauCeti.atkinLehnerOperatorGamma1Cusp_atkinLehnerOperatorGamma1Cusp: the squareW_Q ∘ W_Q = Q ^ (k - 2) ⟨u⟩, withu ≡ -1moduloQandQ u ≡ W₁₁ ^ 2moduloN.TauCeti.atkinLehnerOperatorGamma1_atkinLehnerOperatorGamma1_of_mem_modFormCharSpace,TauCeti.atkinLehnerOperatorGamma1Cusp_atkinLehnerOperatorGamma1Cusp_of_mem_cuspFormCharSpace: forW₁₁ ≡ 1moduloN / Q, the square onM_k(N, χ_Q χ_{N/Q})is the constantQ ^ (k - 2) χ_Q(-1) χ_{N/Q}(Q)⁻¹of Atkin and Li.
References #
- A. O. L. Atkin and W.-C. W. Li, Twists of newforms and pseudo-eigenvalues of
W-operators, Invent. Math. 48 (1978), 221–243, §1. - F. Diamond and J. Shurman, A First Course in Modular Forms, §5.
Moving Γ₀(N) past W in GL (Fin 2) ℝ, with the label shift: g W = W g' for some
g' ∈ Γ₀(N) whose diamond label is the label of g with its residue modulo Q inverted.
Moving W past Γ₀(N) in GL (Fin 2) ℝ, with the label shift: W g = g' W for some
g' ∈ Γ₀(N) whose diamond label is the label of g with its residue modulo Q inverted.
W normalizes Γ₁(N) in GL (Fin 2) ℝ. Moving an element of Γ₁(N), of diamond label
1, across W gives an element of label ι_Q 1 = 1. This is what makes the slash by W an
operator on modular forms of level Γ₁(N).
The Atkin–Lehner slash operator on M_k(Γ₁(N)): f ↦ f ∣[k] W, as a ℂ-linear
endomorphism, for an Atkin–Lehner matrix W of an exact divisor Q of N. Like the operator on
M_k(Γ₀(N)) it carries no normalizing scalar. It depends on W, through a diamond operator
(atkinLehnerOperatorGamma1_mul_left).
Equations
- One or more equations did not get rendered due to their size.
Instances For
On underlying functions the Atkin–Lehner operator on M_k(Γ₁(N)) is ⇑f ∣[k] W.
The Atkin–Lehner slash operator on S_k(Γ₁(N)), the cusp-form counterpart of
atkinLehnerOperatorGamma1.
Equations
- One or more equations did not get rendered due to their size.
Instances For
On underlying functions the Atkin–Lehner operator on S_k(Γ₁(N)) is ⇑f ∣[k] W.
The two Atkin–Lehner operators on Γ₁(N) agree under the coercion
S_k(Γ₁(N)) → M_k(Γ₁(N)): both slash by W.
The diamond shift W_Q ∘ ⟨d⟩ = ⟨ι_Q d⟩ ∘ W_Q on M_k(Γ₁(N)), where ι_Q inverts the
residue of d modulo Q and keeps its residue modulo N / Q.
The diamond shift on cusp forms: the S_k(Γ₁(N)) counterpart of
atkinLehnerOperatorGamma1_diamondOp.
W_Q shifts the nebentypus χ to χ ∘ ι_Q: it carries M_k(N, χ) into
M_k(N, χ ∘ ι_Q). For χ = χ_Q · χ_{N/Q} the new nebentypus is χ_Q⁻¹ · χ_{N/Q}
(Nat.IsExactDivisor.comp_unitsInvPart).
W_Q shifts the nebentypus χ to χ ∘ ι_Q on cusp forms: it carries S_k(N, χ) into
S_k(N, χ ∘ ι_Q).
The Atkin–Lehner operator on Γ₁(N) restricted to a nebentypus space, as a ℂ-linear
map M_k(N, χ) →ₗ[ℂ] M_k(N, χ ∘ ι_Q). This is atkinLehnerOperatorGamma1 cut down by
LinearMap.restrict.
Equations
- TauCeti.atkinLehnerGamma1CharRestrict hQ hQN h k χ = (TauCeti.atkinLehnerOperatorGamma1 hQ hQN h k).restrict ⋯
Instances For
On underlying modular forms, atkinLehnerGamma1CharRestrict is atkinLehnerOperatorGamma1.
The Atkin–Lehner operator on Γ₁(N) restricted to a nebentypus space of cusp forms, as a
ℂ-linear map S_k(N, χ) →ₗ[ℂ] S_k(N, χ ∘ ι_Q). The cusp-form counterpart of
atkinLehnerGamma1CharRestrict.
Equations
- TauCeti.atkinLehnerGamma1CharCuspRestrict hQ hQN h k χ = (TauCeti.atkinLehnerOperatorGamma1Cusp hQ hQN h k).restrict ⋯
Instances For
On underlying cusp forms, atkinLehnerGamma1CharCuspRestrict is
atkinLehnerOperatorGamma1Cusp. The cusp-form counterpart of
coe_atkinLehnerGamma1CharRestrict_apply, stated for the same reason.
The dependence on the Atkin–Lehner matrix: replacing W by γ W, for γ ∈ Γ₀(N),
precomposes the operator on M_k(Γ₁(N)) with the diamond operator of γ. Every Atkin–Lehner
matrix for Q is of the form γ W (IsAtkinLehnerMatrix.exists_mem_Gamma0_eq_mul_left).
On M_k(N, χ) the operator is determined by Q up to a scalar: replacing W by γ W
multiplies W_Q f by χ(d_γ), for d_γ the diamond label of γ ∈ Γ₀(N).
The dependence on the Atkin–Lehner matrix, on cusp forms: the S_k(Γ₁(N)) counterpart of
atkinLehnerOperatorGamma1_mul_left.
On S_k(N, χ) the operator is determined by Q up to a scalar: the cusp-form counterpart
of atkinLehnerOperatorGamma1_mul_left_of_mem_modFormCharSpace.
At the Fricke matrix the operator is the Fricke operator frickeOperator on
M_k(Γ₁(N)).
At the Fricke matrix the cusp-form operator is the Fricke operator frickeOperatorCusp
on S_k(Γ₁(N)).
The square of W_Q #
W * W is Q times an element γ of Γ₀(N) (IsAtkinLehnerMatrix.exists_mem_Gamma0_mul_self),
and the scalar Q slashes as Q ^ (k - 2), so on M_k(Γ₁(N)) the square of W_Q is
Q ^ (k - 2) times the diamond operator of γ. Its label is -1 modulo Q, and modulo N / Q
it is fixed by Q * u ≡ W₁₁ ^ 2. Under Atkin and Li's normalization W₁₁ ≡ 1 modulo N / Q
(atkinLiMatrix) the label is Q⁻¹ modulo N / Q, so on M_k(N, χ) with χ = χ_Q · χ_{N/Q}
the square is the constant Q ^ (k - 2) χ_Q(-1) χ_{N/Q}(Q)⁻¹ of Atkin and Li.
The square of W_Q on M_k(Γ₁(N)) is a diamond operator: W_Q ∘ W_Q = Q ^ (k - 2) ⟨u⟩,
where u is the unit that is -1 modulo Q and satisfies Q * u = W₁₁ ^ 2 modulo N, which
determines it modulo N / Q (IsAtkinLehnerMatrix.toHomUnits_gamma0Map_of_mul_self_eq).
The square of W_Q on S_k(Γ₁(N)) is a diamond operator: the cusp-form counterpart of
atkinLehnerOperatorGamma1_atkinLehnerOperatorGamma1.
Atkin and Li's square of W_Q on a nebentypus space: if the lower-right entry of W is
1 modulo N / Q (as for atkinLiMatrix) and f ∈ M_k(N, χ) with χ = χ_Q · χ_{N/Q} split
along N = Q · (N / Q), then W_Q (W_Q f) = Q ^ (k - 2) χ_Q(-1) χ_{N/Q}(Q)⁻¹ f.
Atkin and Li's square of W_Q on a nebentypus space of cusp forms: the cusp-form
counterpart of atkinLehnerOperatorGamma1_atkinLehnerOperatorGamma1_of_mem_modFormCharSpace.