The conjugate form f_ρ(τ) = conj (f (-conj τ)) #
The conjugate of a modular form f is f_ρ(τ) = conj (f (-conj τ)). It is holomorphic, and if
f = ∑ aₙ qⁿ then f_ρ = ∑ conj (aₙ) qⁿ. Mathlib's slash action of GL(2, ℝ) builds complex
conjugation into the action of a matrix of negative determinant, so f_ρ = f ∣[k] J for the
reflection J = !![-1, 0; 0, 1] (UpperHalfPlane.J), and Mathlib's ModularForm.translate makes
f_ρ a modular form for J⁻¹ 𝒢 J. Conjugation by J changes the signs of the off-diagonal
entries, so it preserves Γ₀(N) and Γ₁(N).
On Γ₁(N) the conjugate form carries the nebentypus χ to its complex conjugate χ⁻¹, since
the diamond eigenvalues are roots of unity, and it intertwines the Hecke operators Tₙ: their
coefficient formula has real coefficients apart from the values of χ. So the conjugate of a Hecke
eigenform of nebentypus χ is a Hecke eigenform of nebentypus χ⁻¹ with the complex-conjugate
eigenvalues. This is the form that the Fricke involution relates a newform to: for a newform f
of level N, f ∣ W_N is a multiple of f_ρ (Miyake, Theorem 4.6.15), the multiple being the
Atkin–Li pseudo-eigenvalue.
Main definitions #
ModularForm.conj,CuspForm.conj: the conjugate formf_ρ, with the antilinear mapsModularForm.conjₗandCuspForm.conjₗ.CuspForm.conjCharSpace: the conjugate form as an antilinear equivalenceS_k(N, χ) ≃ S_k(N, χ⁻¹).
Main results #
TauCeti.qExpansion_slash_J: slashing a modular form byJconjugates itsq-expansion coefficients;ModularForm.qExpansion_conjandCuspForm.qExpansion_conjare its forms forf_ρ.ModularForm.conj_conj:f ↦ f_ρis an involution.CuspForm.conj_levelRaise:f ↦ f_ρcommutes with the level-raising operatorsV_d.TauCeti.Gamma0_map_le_conjAct_inv_J,TauCeti.Gamma1_map_le_conjAct_inv_J: conjugation byJpreservesΓ₀(N)andΓ₁(N).ModularForm.conj_mem_modFormCharSpace,CuspForm.conj_mem_cuspFormCharSpace:f ∈ M_k(N, χ)givesf_ρ ∈ M_k(N, χ⁻¹), and likewise for cusp forms.CuspForm.heckeRingHomCuspCharSpace_conjCharSpace:Tₙ (f_ρ) = (Tₙ f)_ρonS_k(N, χ), andCuspForm.heckeRingHomCuspCharSpace_conjCharSpace_eq_smul: the conjugate of aTₙ-eigenform is aTₙ-eigenform with the conjugate eigenvalue.
References #
- T. Miyake, Modular forms, §4.6, where the conjugate form is written
f_ρ. - A. O. L. Atkin and W.-C. W. Li, Twists of newforms and pseudo-eigenvalues of
W-operators, Invent. Math. 48 (1978), 221–243.
The matrix J = !![-1, 0; 0, 1] is its own inverse.
Slashing by J conjugates the value at the reflected point J • τ = -conj τ, in every
weight.
The q-parameter at the reflected point J • τ = -conj τ is the conjugate of the
q-parameter at τ.
A subgroup contained in its conjugate J⁻¹ 𝒢 J by the involution J is equal to it.
A strict period of 𝒢 is a strict period of J⁻¹ 𝒢 J: conjugating !![1, h; 0, 1] by J
gives its inverse !![1, -h; 0, 1].
Slashing by J conjugates the q-expansion coefficients. For a modular form f of
weight k on 𝒢 and a strict period h of 𝒢, the q-expansion of
f ∣[k] J = conj ∘ f ∘ (τ ↦ -conj τ) is that of f with every coefficient conjugated.
The congruence subgroups #
The J-conjugate !![a, -b; -c, d] of !![a, b; c, d] ∈ SL(2, ℤ).
Instances For
Conjugation by J realizes conjJ.
conjJ leaves the lower-right entry, hence the diamond label, alone.
Conjugation by J preserves Γ₀(N), so the conjugate of a form on Γ₀(N) is again a
form on Γ₀(N).
Conjugation by J preserves Γ₁(N), so the conjugate of a form on Γ₁(N) is again a
form on Γ₁(N).
Nebentypus #
The conjugate form f_ρ(τ) = conj (f (-conj τ)) of a modular form f for 𝒢, as a
modular form for any 𝒢' conjugated into 𝒢 by J = !![-1, 0; 0, 1]. It is the slash of f
by the determinant -1 matrix J, so its q-expansion has the complex-conjugate
coefficients (ModularForm.qExpansion_conj).
Equations
Instances For
The conjugate form is an involution.
The conjugate form, as a ℂ-antilinear map.
Equations
- ModularForm.conjₗ hJ = { toFun := ModularForm.conj hJ, map_add' := ⋯, map_smul' := ⋯ }
Instances For
The q-expansion of the conjugate form has the conjugate coefficients:
a_n(f_ρ) = conj (a_n(f)).
The conjugate of a form of nebentypus χ has nebentypus χ⁻¹: f_ρ ∈ M_k(N, χ⁻¹).
The conjugate cusp form f_ρ(τ) = conj (f (-conj τ)), the cusp-form counterpart of
ModularForm.conj.
Equations
- CuspForm.conj hJ f = CuspForm.ofLe hJ (CuspForm.translate f UpperHalfPlane.J)
Instances For
The conjugate cusp form is the conjugate of the underlying modular form.
The conjugate cusp form, as a ℂ-antilinear map.
Equations
- CuspForm.conjₗ hJ = { toFun := CuspForm.conj hJ, map_add' := ⋯, map_smul' := ⋯ }
Instances For
The q-expansion of the conjugate cusp form has the conjugate coefficients.
Conjugation commutes with the level-raising operators: (V_d f)_ρ = V_d (f_ρ), since
the reflection τ ↦ -conj τ commutes with τ ↦ d τ.
Nebentypus and Hecke operators #
The conjugate of a cusp form of nebentypus χ has nebentypus χ⁻¹:
f_ρ ∈ S_k(N, χ⁻¹).
Conjugating twice, first on S_k(N, χ) and then on S_k(N, χ⁻¹), gives back the original
form.
The conjugate form intertwines the Hecke operators: for f ∈ S_k(N, χ) and n ≠ 0,
T_n (f_ρ) = (T_n f)_ρ, where on the left T_n acts on S_k(N, χ⁻¹). Since the conjugation is
antilinear, the conjugate of a T_n-eigenform with eigenvalue λ is a T_n-eigenform with
eigenvalue conj λ.
The conjugate of a Hecke eigenform is an eigenform with the conjugate eigenvalue: if
Tₙ f = λ f on S_k(N, χ), then Tₙ f_ρ = conj λ • f_ρ on S_k(N, χ⁻¹).