Fricke transport of the good Hecke operators

At an index n coprime to N, the Fricke operator intertwines Tₙ with ⟨n⟩⁻¹ Tₙ on forms for Γ₁(N). On a nebentypus space this gives the scalar χ(n) when Tₙ is moved past Fricke from the source to the target character space. The same identities hold for the Petersson-normalized Fricke operator and for cusp forms.

This relation transports good Hecke eigensystems to the inverse-nebentypus space; it is the Hecke-theoretic input to the Fricke pseudo-eigenvalue theorem for primitive forms.

References

• T. Miyake, Modular forms, §4.6, Theorem 4.6.15.
• F. Diamond and J. Shurman, A first course in modular forms, §5.5.

theorem TauCeti.frickeOperator_heckeTNat {N n : ℕ} [NeZero N] [NeZero n] {k : ℤ} (hn : n.Coprime N) (f : ModularForm (Subgroup.map (Matrix.SpecialLinearGroup.mapGL ℝ) (CongruenceSubgroup.Gamma1 N)) k) : (frickeOperator k) ((HeckeRing.GL2.heckeTNat k n) f) = (diamondOp k (ZMod.unitOfCoprime n hn)⁻¹) ((HeckeRing.GL2.heckeTNat k n) ((frickeOperator k) f))

At a good index, Fricke carries Tₙ to its inverse-diamond multiple on modular forms.

theorem TauCeti.frickeOperatorCusp_heckeTCuspNat {N n : ℕ} [NeZero N] [NeZero n] {k : ℤ} (hn : n.Coprime N) (f : CuspForm (Subgroup.map (Matrix.SpecialLinearGroup.mapGL ℝ) (CongruenceSubgroup.Gamma1 N)) k) : (frickeOperatorCusp k) ((HeckeRing.GL2.heckeTCuspNat k n) f) = (diamondOpCusp k (ZMod.unitOfCoprime n hn)⁻¹) ((HeckeRing.GL2.heckeTCuspNat k n) ((frickeOperatorCusp k) f))

At a good index, Fricke carries Tₙ to its inverse-diamond multiple on cusp forms.

theorem TauCeti.normalizedFrickeOperator_heckeTNat {N n : ℕ} [NeZero N] [NeZero n] {k : ℤ} (hn : n.Coprime N) (f : ModularForm (Subgroup.map (Matrix.SpecialLinearGroup.mapGL ℝ) (CongruenceSubgroup.Gamma1 N)) k) : (normalizedFrickeOperator k) ((HeckeRing.GL2.heckeTNat k n) f) = (diamondOp k (ZMod.unitOfCoprime n hn)⁻¹) ((HeckeRing.GL2.heckeTNat k n) ((normalizedFrickeOperator k) f))

The Petersson normalization leaves the Fricke–Hecke intertwining relation unchanged.

theorem TauCeti.normalizedFrickeOperatorCusp_heckeTCuspNat {N n : ℕ} [NeZero N] [NeZero n] {k : ℤ} (hn : n.Coprime N) (f : CuspForm (Subgroup.map (Matrix.SpecialLinearGroup.mapGL ℝ) (CongruenceSubgroup.Gamma1 N)) k) : (normalizedFrickeOperatorCusp k) ((HeckeRing.GL2.heckeTCuspNat k n) f) = (diamondOpCusp k (ZMod.unitOfCoprime n hn)⁻¹) ((HeckeRing.GL2.heckeTCuspNat k n) ((normalizedFrickeOperatorCusp k) f))

The normalized Fricke–Hecke intertwining relation on cusp forms.

theorem TauCeti.frickeOperator_heckeTNat_of_mem_modFormCharSpace {N n : ℕ} [NeZero N] [NeZero n] {k : ℤ} (hn : n.Coprime N) {χ : (ZMod N)ˣ →* ℂˣ} {f : ModularForm (Subgroup.map (Matrix.SpecialLinearGroup.mapGL ℝ) (CongruenceSubgroup.Gamma1 N)) k} (hf : f ∈ modFormCharSpace k χ) : (frickeOperator k) ((HeckeRing.GL2.heckeTNat k n) f) = ↑(χ (ZMod.unitOfCoprime n hn)) • (HeckeRing.GL2.heckeTNat k n) ((frickeOperator k) f)

On M_k(N, χ), moving a good Tₙ past Fricke introduces χ(n). The output of Fricke has nebentypus χ⁻¹.

theorem TauCeti.frickeOperatorCusp_heckeTCuspNat_of_mem_cuspFormCharSpace {N n : ℕ} [NeZero N] [NeZero n] {k : ℤ} (hn : n.Coprime N) {χ : (ZMod N)ˣ →* ℂˣ} {f : CuspForm (Subgroup.map (Matrix.SpecialLinearGroup.mapGL ℝ) (CongruenceSubgroup.Gamma1 N)) k} (hf : f ∈ cuspFormCharSpace k χ) : (frickeOperatorCusp k) ((HeckeRing.GL2.heckeTCuspNat k n) f) = ↑(χ (ZMod.unitOfCoprime n hn)) • (HeckeRing.GL2.heckeTCuspNat k n) ((frickeOperatorCusp k) f)

On S_k(N, χ), moving a good Tₙ past Fricke introduces χ(n).

theorem TauCeti.normalizedFrickeOperator_heckeTNat_of_mem_modFormCharSpace {N n : ℕ} [NeZero N] [NeZero n] {k : ℤ} (hn : n.Coprime N) {χ : (ZMod N)ˣ →* ℂˣ} {f : ModularForm (Subgroup.map (Matrix.SpecialLinearGroup.mapGL ℝ) (CongruenceSubgroup.Gamma1 N)) k} (hf : f ∈ modFormCharSpace k χ) : (normalizedFrickeOperator k) ((HeckeRing.GL2.heckeTNat k n) f) = ↑(χ (ZMod.unitOfCoprime n hn)) • (HeckeRing.GL2.heckeTNat k n) ((normalizedFrickeOperator k) f)

On a nebentypus space, the normalized Fricke operator intertwines Tₙ with χ(n)Tₙ. No parity or reality hypothesis on χ is needed.

theorem TauCeti.normalizedFrickeOperatorCusp_heckeTCuspNat_of_mem_cuspFormCharSpace {N n : ℕ} [NeZero N] [NeZero n] {k : ℤ} (hn : n.Coprime N) {χ : (ZMod N)ˣ →* ℂˣ} {f : CuspForm (Subgroup.map (Matrix.SpecialLinearGroup.mapGL ℝ) (CongruenceSubgroup.Gamma1 N)) k} (hf : f ∈ cuspFormCharSpace k χ) : (normalizedFrickeOperatorCusp k) ((HeckeRing.GL2.heckeTCuspNat k n) f) = ↑(χ (ZMod.unitOfCoprime n hn)) • (HeckeRing.GL2.heckeTCuspNat k n) ((normalizedFrickeOperatorCusp k) f)

The normalized Fricke–Hecke relation on cusp forms with general nebentypus.

theorem TauCeti.heckeTCuspNat_normalizedFrickeOperatorCusp_eq_smul_iff_heckeTCuspNat_eq_smul {N n : ℕ} [NeZero N] [NeZero n] {k : ℤ} (hn : n.Coprime N) {χ : (ZMod N)ˣ →* ℂˣ} {f : CuspForm (Subgroup.map (Matrix.SpecialLinearGroup.mapGL ℝ) (CongruenceSubgroup.Gamma1 N)) k} (hf : f ∈ cuspFormCharSpace k χ) (c : ℂ) : (HeckeRing.GL2.heckeTCuspNat k n) ((normalizedFrickeOperatorCusp k) f) = ((↑(χ (ZMod.unitOfCoprime n hn)))⁻¹ * c) • (normalizedFrickeOperatorCusp k) f ↔ (HeckeRing.GL2.heckeTCuspNat k n) f = c • f

Fricke transports a good Hecke eigenrelation by multiplying its eigenvalue by χ(n)⁻¹. The equivalence includes the zero form and does not require normalization of its coefficients.