The Fricke sign and the Fricke pseudo-eigenvalue of a newform

For a newform f of level N, weight k and trivial nebentypus, the normalized Fricke operator 𝒲_N f = (√N) ^ (2 - k) • (f ∣[k] !![0, -1; N, 0]) is ε_N · f for a sign ε_N ∈ {1, -1}, the Fricke sign (or Fricke eigenvalue) of f. The sign of the functional equation of L(s, f) is i ^ k · ε_N.

The argument has three inputs.

• 𝒲_N preserves the new subspace. It preserves the old subspace (TauCeti.normalizedFrickeOperatorCusp_mem_cuspFormsOld) and its Petersson adjoint is (-1) ^ k • 𝒲_N, so it preserves the orthogonal complement.
• On S_k(N, 1) the operator 𝒲_N commutes with the Hecke operators Tₙ for n prime to N. At the trivial character the nebentypus-twisted Hecke sum is the plain Γ₀(N) slash sum (HeckeRing.GL2.twistedHeckeSlashSum_eq_heckeSlashSum). That sum commutes with the Fricke slash coset by coset (TauCeti.commute_atkinLehnerOperatorCusp_heckeSlashGamma0CuspFormEnd), and the ring elements Tₙ with (n, N) = 1 are generated by double cosets of determinant prime to N (HeckeRing.GL2.heckeTCompositeGamma0_mem_closure_coprimeDetCoset).
• Multiplicity one, in the form HeckeRing.GL2.exists_eq_smul_of_commute_heckeRingHomCusp_of_mem_cuspFormsNew for a commuting involution. 𝒲_N f is then a good Hecke eigenvector in the new part with the same eigenvalues as f, hence a multiple ε • f, and 𝒲_N ∘ 𝒲_N = (-1) ^ k forces ε ^ 2 = 1, since (-1) ^ k = 1 by the parity lemma when f ≠ 0 has trivial nebentypus.

For a nontrivial nebentypus χ the operator 𝒲_N carries S_k(N, χ) to S_k(N, χ⁻¹), and a newform goes to a multiple of its conjugate newform f_ρ, not of itself (Miyake, Theorem 4.6.15(2)): 𝒲_N f = λ_N(f) • f_ρ, the scalar λ_N(f) being the Fricke pseudo-eigenvalue of Atkin and Li. The argument is the same multiplicity-one step, now in S_k(N, χ⁻¹): Fricke multiplies the good eigenvalue λ_p by χ(p)⁻¹ (TauCeti.heckeTCuspNat_normalizedFrickeOperatorCusp_eq_smul_iff_heckeTCuspNat_eq_smul), and χ(p)⁻¹ λ_p = conj λ_p is the eigenvalue of f_ρ at p. Unitarity of 𝒲_N and of f ↦ f_ρ for the Petersson product gives |λ_N(f)| = 1. For trivial nebentypus f_ρ = f, and the pseudo-eigenvalue is the Fricke sign.

Main results

• TauCeti.normalizedFrickeOperatorCusp_mem_cuspFormsNew, TauCeti.cuspFormsNew_map_normalizedFrickeOperatorCusp: 𝒲_N maps the new subspace onto itself.
• TauCeti.commute_normalizedFrickeCharCuspOneEnd_heckeRingHomCuspCharSpace: on S_k(N, 1), 𝒲_N commutes with Tₙ for every n prime to N.
• TauCeti.exists_normalizedFrickeOperatorCusp_eq_smul_of_mem_cuspFormsNew: a nonzero form in the new part of S_k(N, 1) that is an eigenvector of every good Tₚ satisfies 𝒲_N f = ε • f with ε = 1 or ε = -1.
• HeckeRing.GL2.Newform.frickeSign: the canonical Fricke sign of a newform of trivial nebentypus, together with its eigenvalue equation and sign law.
• HeckeRing.GL2.Newform.exists_normalizedFrickeOperatorCusp_eq_smul_conj: 𝒲_N f is a multiple of the conjugate newform f_ρ, for every nebentypus.
• HeckeRing.GL2.Newform.frickePseudoEigenvalue: the Fricke pseudo-eigenvalue λ_N(f), with its equation 𝒲_N f = λ_N(f) • f_ρ, its uniqueness, and HeckeRing.GL2.Newform.norm_frickePseudoEigenvalue: |λ_N(f)| = 1.
• HeckeRing.GL2.Newform.frickePseudoEigenvalue_eq_frickeSign: for trivial nebentypus the pseudo-eigenvalue is the Fricke sign.

References

• A. O. L. Atkin and J. Lehner, Hecke operators on Γ₀(m), Math. Ann. 185 (1970), 134–160.
• A. O. L. Atkin and W.-C. W. Li, Twists of newforms and pseudo-eigenvalues of W-operators, Invent. Math. 48 (1978), 221–243.
• T. Miyake, Modular forms, Theorem 4.6.15.
• F. Diamond and J. Shurman, A first course in modular forms, §5.10.

𝒲_N preserves the new subspace

theorem TauCeti.normalizedFrickeOperatorCusp_mem_cuspFormsNew {N : ℕ} [NeZero N] {k : ℤ} {f : CuspForm (Subgroup.map (Matrix.SpecialLinearGroup.mapGL ℝ) (CongruenceSubgroup.Gamma1 N)) k} (hf : f ∈ cuspFormsNew N k) : (normalizedFrickeOperatorCusp k) f ∈ cuspFormsNew N k

The normalized Fricke operator preserves the new subspace: 𝒲_N maps S_k(Γ₁(N))ⁿᵉʷ into itself.

theorem TauCeti.cuspFormsNew_map_normalizedFrickeOperatorCusp (N : ℕ) [NeZero N] (k : ℤ) : Submodule.map (normalizedFrickeOperatorCusp k) (cuspFormsNew N k) = cuspFormsNew N k

The normalized Fricke operator carries the new subspace onto itself.

𝒲_N commutes with the good Hecke operators on S_k(N, 1)

noncomputable def TauCeti.normalizedFrickeCharCuspOneEnd {N : ℕ} [NeZero N] (k : ℤ) : Module.End ℂ ↥(cuspFormCharSpace k 1)

The normalized Fricke operator on the trivial-nebentypus space S_k(N, 1), as the endomorphism obtained by specializing normalizedFrickeCharCuspRestrict to the trivial character.

Equations

• TauCeti.normalizedFrickeCharCuspOneEnd k = (TauCeti.normalizedFrickeOperatorCusp k).restrict ⋯

@[simp]

theorem TauCeti.coe_normalizedFrickeCharCuspOneEnd_apply {N : ℕ} [NeZero N] {k : ℤ} (f : ↥(cuspFormCharSpace k 1)) : ↑((normalizedFrickeCharCuspOneEnd k) f) = (normalizedFrickeOperatorCusp k) ↑f

On underlying cusp forms, normalizedFrickeCharCuspOneEnd is normalizedFrickeOperatorCusp.

theorem TauCeti.commute_normalizedFrickeCharCuspOneEnd_heckeRingHomCuspCharSpace {N : ℕ} [NeZero N] {k : ℤ} {n : ℕ} (hn : n.Coprime N) : Commute (normalizedFrickeCharCuspOneEnd k) ((HeckeRing.GL2.heckeRingHomCuspCharSpace k 1) (HeckeRing.GL2.heckeTCompositeGamma0 N n))

The normalized Fricke operator commutes with the good Hecke operators on S_k(N, 1): for n prime to N, 𝒲_N Tₙ = Tₙ 𝒲_N.

The Fricke sign

theorem TauCeti.exists_normalizedFrickeOperatorCusp_eq_smul_of_mem_cuspFormsNew {N : ℕ} [NeZero N] {k : ℤ} {f : ↥(cuspFormCharSpace k 1)} (ha : ∀ (p : ℕ), Nat.Prime p → p.Coprime N → ∃ (c : ℂ), ((HeckeRing.GL2.heckeRingHomCuspCharSpace k 1) (HeckeRing.GL2.heckeTCompositeGamma0 N p)) f = c • f) (hf : ↑f ∈ cuspFormsNew N k) (hf0 : ↑f ≠ 0) : ∃ (ε : ℂ), (ε = 1 ∨ ε = -1) ∧ (normalizedFrickeOperatorCusp k) ↑f = ε • ↑f

The Fricke sign: a nonzero cusp form in the new part of S_k(N, 1) that is an eigenvector of Tₚ at every prime p ∤ N is an eigenvector of the normalized Fricke operator, with eigenvalue 1 or -1.

theorem HeckeRing.GL2.Newform.exists_normalizedFrickeOperatorCusp_eq_smul {N : ℕ} [NeZero N] {k : ℤ} (f : Newform N k) (hχ : f.χ = 1) : ∃ (ε : ℂ), (ε = 1 ∨ ε = -1) ∧ (TauCeti.normalizedFrickeOperatorCusp k) f.toCuspForm = ε • f.toCuspForm

The Fricke sign of a newform of trivial nebentypus: 𝒲_N f = ε • f with ε = 1 or ε = -1.

noncomputable def HeckeRing.GL2.Newform.frickeSign {N : ℕ} [NeZero N] {k : ℤ} (f : Newform N k) (hχ : f.χ = 1) : ℂ

The Fricke sign ε_N(f) of a newform with trivial nebentypus.

Equations

• f.frickeSign hχ = ⋯.choose

theorem HeckeRing.GL2.Newform.normalizedFrickeOperatorCusp_eq_frickeSign_smul {N : ℕ} [NeZero N] {k : ℤ} (f : Newform N k) (hχ : f.χ = 1) : (TauCeti.normalizedFrickeOperatorCusp k) f.toCuspForm = f.frickeSign hχ • f.toCuspForm

The normalized Fricke operator acts on a trivial-nebentypus newform by its Fricke sign.

theorem HeckeRing.GL2.Newform.frickeSign_eq_one_or_neg_one {N : ℕ} [NeZero N] {k : ℤ} (f : Newform N k) (hχ : f.χ = 1) : f.frickeSign hχ = 1 ∨ f.frickeSign hχ = -1

The Fricke sign of a trivial-nebentypus newform is 1 or -1.

theorem HeckeRing.GL2.Newform.frickeSign_eq_of_normalizedFrickeOperatorCusp_eq_smul {N : ℕ} [NeZero N] {k : ℤ} (f : Newform N k) (hχ : f.χ = 1) {ε : ℂ} (hε : (TauCeti.normalizedFrickeOperatorCusp k) f.toCuspForm = ε • f.toCuspForm) : f.frickeSign hχ = ε

A scalar satisfying the normalized Fricke eigenvalue equation is the Fricke sign.

The Fricke pseudo-eigenvalue

theorem HeckeRing.GL2.Newform.exists_normalizedFrickeOperatorCusp_eq_smul_conj {N : ℕ} [NeZero N] {k : ℤ} (f : Newform N k) : ∃ (c : ℂ), (TauCeti.normalizedFrickeOperatorCusp k) f.toCuspForm = c • f.conj.toCuspForm

The Fricke involution sends a newform to a multiple of its conjugate newform (Miyake, Theorem 4.6.15(2)): 𝒲_N f = c • f_ρ for a newform f of any nebentypus χ, where f_ρ is the conjugate newform HeckeRing.GL2.Newform.conj. The scalar is HeckeRing.GL2.Newform.frickePseudoEigenvalue.

noncomputable def HeckeRing.GL2.Newform.frickePseudoEigenvalue {N : ℕ} [NeZero N] {k : ℤ} (f : Newform N k) : ℂ

The Fricke pseudo-eigenvalue λ_N(f) of a newform f (Atkin–Li): the scalar with 𝒲_N f = λ_N(f) • f_ρ, where f_ρ is the conjugate newform. It has absolute value 1 (HeckeRing.GL2.Newform.norm_frickePseudoEigenvalue), and for trivial nebentypus it is the Fricke sign (HeckeRing.GL2.Newform.frickePseudoEigenvalue_eq_frickeSign).

Equations

• f.frickePseudoEigenvalue = ⋯.choose

theorem HeckeRing.GL2.Newform.normalizedFrickeOperatorCusp_eq_frickePseudoEigenvalue_smul {N : ℕ} [NeZero N] {k : ℤ} (f : Newform N k) : (TauCeti.normalizedFrickeOperatorCusp k) f.toCuspForm = f.frickePseudoEigenvalue • f.conj.toCuspForm

The normalized Fricke operator sends a newform to its pseudo-eigenvalue times its conjugate newform.

theorem HeckeRing.GL2.Newform.frickePseudoEigenvalue_eq_of_normalizedFrickeOperatorCusp_eq_smul {N : ℕ} [NeZero N] {k : ℤ} (f : Newform N k) {c : ℂ} (hc : (TauCeti.normalizedFrickeOperatorCusp k) f.toCuspForm = c • f.conj.toCuspForm) : f.frickePseudoEigenvalue = c

A scalar satisfying the equation 𝒲_N f = c • f_ρ is the Fricke pseudo-eigenvalue.

theorem HeckeRing.GL2.Newform.norm_frickePseudoEigenvalue {N : ℕ} [NeZero N] {k : ℤ} (f : Newform N k) : ‖f.frickePseudoEigenvalue‖ = 1

The Fricke pseudo-eigenvalue has absolute value 1.

theorem HeckeRing.GL2.Newform.frickePseudoEigenvalue_eq_frickeSign {N : ℕ} [NeZero N] {k : ℤ} (f : Newform N k) (hχ : f.χ = 1) : f.frickePseudoEigenvalue = f.frickeSign hχ

For trivial nebentypus the pseudo-eigenvalue is the Fricke sign: such a newform is its own conjugate (HeckeRing.GL2.Newform.conj_eq_self_of_χ_eq_one), so 𝒲_N f = λ_N(f) • f.