The Atkin–Li pseudo-eigenvalue of a newform at an exact divisor

Let f be a newform of level N, weight k and nebentypus χ = χ_Q · χ_{N/Q}, split along an exact divisor Q ∥ N. Write 𝒲_Q = (√Q) ^ (2 - k) • (· ∣[k] W) for the normalized slash by the Atkin–Li matrix W = atkinLiMatrix N Q, whose lower-right entry is 1 modulo N / Q. Then 𝒲_Q f = λ_Q(f) • g, where g = f.atkinLehnerTransform h is the newform of nebentypus χ_Q⁻¹ · χ_{N/Q} on the line of 𝒲_Q f. The scalar λ_Q(f) is the pseudo-eigenvalue of Atkin and Li. The choice of W matters: another Atkin–Lehner matrix for Q changes λ_Q(f) by a value of χ, and under the Atkin–Li normalization 𝒲_Q ∘ 𝒲_Q is the constant χ_Q(-1) χ_{N/Q}(Q)⁻¹ on S_k(N, χ).

The results here are what the operator identities give without further analytic input.

• The product law λ_Q(f) · λ_Q(g) = χ_Q(-1) χ_{N/Q}(Q)⁻¹, from the square of 𝒲_Q and the fact that transporting twice returns f.
• The two endpoints: at Q = 1 the Atkin–Li matrix lies in Γ₁(N), so λ_1(f) = 1; at Q = N it is γ W_N for the Fricke matrix W_N and some γ ∈ Γ₀(N) of diamond label -1, so g = f_ρ and λ_N(f) is χ(-1) times the Fricke pseudo-eigenvalue.
• The norm identity |λ_Q(f)|² ⟪g, g⟫ = ⟪f, f⟫, from the Petersson unitarity of 𝒲_Q. Hence |λ_Q(f)| = 1 exactly when f and g have the same Petersson norm.
• That equality of norms, and so |λ_Q(f)| = 1, in the two cases where g is identified with a newform of known norm. If χ_Q is trivial then g = f, and f is a genuine 𝒲_Q-eigenform with λ_Q(f)² = χ_{N/Q}(Q)⁻¹. If χ_{N/Q} is trivial then g = f_ρ is the conjugate newform, whose norm is that of f.

For a general nebentypus, both factors nontrivial, the equality ⟪g, g⟫ = ⟪f, f⟫ is not proved here.

Main definitions

All declarations are in the namespace HeckeRing.GL2.Newform.

• atkinLehnerPseudoEigenvalue: the pseudo-eigenvalue λ_Q(f).

Main results

• smul_atkinLehnerOperatorGamma1Cusp_eq_atkinLehnerPseudoEigenvalue_smul: 𝒲_Q f = λ_Q(f) • g, with uniqueness of the scalar in atkinLehnerPseudoEigenvalue_eq_of_atkinLehnerOperatorGamma1Cusp_eq_smul.
• atkinLehnerPseudoEigenvalue_mul_atkinLehnerPseudoEigenvalue_atkinLehnerTransform: the product law, with the two factors of the nebentypus read off as values of χ at CRT units; and atkinLehnerPseudoEigenvalue_mul_atkinLehnerPseudoEigenvalue_atkinLehnerTransform_of_χ_eq_mul for a given splitting χ = χ_Q · χ_{N/Q}.
• atkinLehnerPseudoEigenvalue_one: λ_1(f) = 1.
• atkinLehnerTransform_self, atkinLehnerPseudoEigenvalue_self: at Q = N the transported newform is f_ρ and λ_N(f) = χ(-1) · frickePseudoEigenvalue f.
• norm_atkinLehnerPseudoEigenvalue_eq_one_iff: |λ_Q(f)| = 1 if and only if ⟪g, g⟫ = ⟪f, f⟫.
• atkinLehnerTransform_eq_self_of_χ_eq_comp_right: if χ is pulled back from a character modulo N / Q, then g = f, so |λ_Q(f)| = 1 (norm_atkinLehnerPseudoEigenvalue_of_χ_eq_comp_right) and λ_Q(f)² = χ_{N/Q}(Q)⁻¹ (atkinLehnerPseudoEigenvalue_sq_of_χ_eq_comp_right).
• atkinLehnerTransform_eq_conj_of_χ_eq_comp_left: if χ is pulled back from a character modulo Q, then g = f_ρ, so |λ_Q(f)| = 1 (norm_atkinLehnerPseudoEigenvalue_of_χ_eq_comp_left).

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.
• T. Miyake, Modular forms, Theorem 4.6.16.

noncomputable def HeckeRing.GL2.Newform.atkinLehnerPseudoEigenvalue {N Q : ℕ} [NeZero N] {k : ℤ} (f : Newform N k) (h : TauCeti.Nat.IsExactDivisor Q N) : ℂ

The Atkin–Li pseudo-eigenvalue λ_Q(f) of a newform f at an exact divisor Q of the level: the scalar with 𝒲_Q f = λ_Q(f) • g, where 𝒲_Q is the slash by the Atkin–Li matrix atkinLiMatrix N Q normalized by (√Q) ^ (2 - k) and g = f.atkinLehnerTransform h is the transported newform. Since g has first Fourier coefficient 1, it is the first Fourier coefficient of 𝒲_Q f.

theorem HeckeRing.GL2.Newform.atkinLehnerPseudoEigenvalue_def {N Q : ℕ} [NeZero N] {k : ℤ} (f : Newform N k) (h : TauCeti.Nat.IsExactDivisor Q N) : f.atkinLehnerPseudoEigenvalue h = TauCeti.atkinLehnerNormalizer Q k * (PowerSeries.coeff 1) (UpperHalfPlane.qExpansion 1 ⇑((TauCeti.atkinLehnerOperatorGamma1Cusp ⋯ ⋯ ⋯ k) f.toCuspForm))

Defining equation for atkinLehnerPseudoEigenvalue: it is (√Q) ^ (2 - k) times the first Fourier coefficient of the slash of f by the Atkin–Li matrix.

theorem HeckeRing.GL2.Newform.smul_atkinLehnerOperatorGamma1Cusp_eq_atkinLehnerPseudoEigenvalue_smul {N Q : ℕ} [NeZero N] {k : ℤ} (f : Newform N k) (h : TauCeti.Nat.IsExactDivisor Q N) : TauCeti.atkinLehnerNormalizer Q k • (TauCeti.atkinLehnerOperatorGamma1Cusp ⋯ ⋯ ⋯ k) f.toCuspForm = f.atkinLehnerPseudoEigenvalue h • (f.atkinLehnerTransform h).toCuspForm

𝒲_Q f = λ_Q(f) • g: the normalized slash of a newform by the Atkin–Li matrix is its pseudo-eigenvalue times the transported newform.

theorem HeckeRing.GL2.Newform.atkinLehnerPseudoEigenvalue_eq_of_atkinLehnerOperatorGamma1Cusp_eq_smul {N Q : ℕ} [NeZero N] {k : ℤ} (f : Newform N k) (h : TauCeti.Nat.IsExactDivisor Q N) {c : ℂ} (hc : TauCeti.atkinLehnerNormalizer Q k • (TauCeti.atkinLehnerOperatorGamma1Cusp ⋯ ⋯ ⋯ k) f.toCuspForm = c • (f.atkinLehnerTransform h).toCuspForm) : f.atkinLehnerPseudoEigenvalue h = c

A scalar c with 𝒲_Q f = c • g is the pseudo-eigenvalue.

theorem HeckeRing.GL2.Newform.atkinLehnerPseudoEigenvalue_ne_zero {N Q : ℕ} [NeZero N] {k : ℤ} (f : Newform N k) (h : TauCeti.Nat.IsExactDivisor Q N) : f.atkinLehnerPseudoEigenvalue h ≠ 0

The pseudo-eigenvalue is nonzero, 𝒲_Q being injective.

theorem HeckeRing.GL2.Newform.atkinLehnerPseudoEigenvalue_mul_atkinLehnerPseudoEigenvalue_atkinLehnerTransform_of_χ_eq_mul {N Q : ℕ} [NeZero N] {k : ℤ} (f : Newform N k) (h : TauCeti.Nat.IsExactDivisor Q N) (ψ : (ZMod Q)ˣ →* ℂˣ) (φ : (ZMod (N / Q))ˣ →* ℂˣ) (hχ : f.χ = ψ.comp (ZMod.unitsMap ⋯) * φ.comp (ZMod.unitsMap ⋯)) : f.atkinLehnerPseudoEigenvalue h * (f.atkinLehnerTransform h).atkinLehnerPseudoEigenvalue h = ↑(ψ (-1) * (φ (ZMod.unitOfCoprime Q ⋯))⁻¹)

The product law of Atkin and Li: if the nebentypus of f is χ_Q · χ_{N/Q} along N = Q · (N / Q), then λ_Q(f) · λ_Q(g) = χ_Q(-1) χ_{N/Q}(Q)⁻¹, where g is the transported newform. Indeed 𝒲_Q g = λ_Q(g) • f because transporting twice returns f, so both sides are the constant by which 𝒲_Q ∘ 𝒲_Q acts on f.

theorem HeckeRing.GL2.Newform.atkinLehnerPseudoEigenvalue_mul_atkinLehnerPseudoEigenvalue_atkinLehnerTransform {N Q : ℕ} [NeZero N] {k : ℤ} (f : Newform N k) (h : TauCeti.Nat.IsExactDivisor Q N) : f.atkinLehnerPseudoEigenvalue h * (f.atkinLehnerTransform h).atkinLehnerPseudoEigenvalue h = ↑(f.χ (h.unitsEquivProd.symm (-1, 1)) * (f.χ (h.unitsEquivProd.symm (1, ZMod.unitOfCoprime Q ⋯)))⁻¹)

The product law of Atkin and Li, for an arbitrary nebentypus: λ_Q(f) · λ_Q(g) is χ(u) χ(v)⁻¹, where the CRT units u and v have components (-1, 1) and (1, Q) modulo Q and modulo N / Q. These are the values χ_Q(-1) and χ_{N/Q}(Q) of the two factors of the canonical splitting of χ along N = Q · (N / Q) (TauCeti.Nat.IsExactDivisor.eq_mul_comp_unitsMap).

theorem HeckeRing.GL2.Newform.sq_norm_atkinLehnerPseudoEigenvalue_mul_peterssonInnerCosets {N Q : ℕ} [NeZero N] {k : ℤ} (f : Newform N k) (h : TauCeti.Nat.IsExactDivisor Q N) : ↑‖f.atkinLehnerPseudoEigenvalue h‖ ^ 2 * (f.atkinLehnerTransform h).peterssonInnerCosets (f.atkinLehnerTransform h).toCuspForm = f.peterssonInnerCosets f.toCuspForm

The norm identity |λ_Q(f)|² ⟪g, g⟫ = ⟪f, f⟫, for g the transported newform: the normalized operator 𝒲_Q preserves the Petersson product.

theorem HeckeRing.GL2.Newform.norm_atkinLehnerPseudoEigenvalue_eq_one_iff {N Q : ℕ} [NeZero N] {k : ℤ} (f : Newform N k) (h : TauCeti.Nat.IsExactDivisor Q N) : ‖f.atkinLehnerPseudoEigenvalue h‖ = 1 ↔ (f.atkinLehnerTransform h).peterssonInnerCosets (f.atkinLehnerTransform h).toCuspForm = f.peterssonInnerCosets f.toCuspForm

|λ_Q(f)| = 1 exactly when f and the transported newform g have the same Petersson norm.

theorem HeckeRing.GL2.Newform.atkinLehnerTransform_eq_self_of_χ_eq_comp_right {N Q : ℕ} [NeZero N] {k : ℤ} (f : Newform N k) (h : TauCeti.Nat.IsExactDivisor Q N) (φ : (ZMod (N / Q))ˣ →* ℂˣ) (hχ : f.χ = φ.comp (ZMod.unitsMap ⋯)) : f.atkinLehnerTransform h = f

If the Q-part of the nebentypus is trivial, transport fixes the newform: 𝒲_Q f is then a multiple of f itself. Transport changes neither the nebentypus nor any good eigenvalue, so strong multiplicity one applies.

theorem HeckeRing.GL2.Newform.atkinLehnerPseudoEigenvalue_sq_of_χ_eq_comp_right {N Q : ℕ} [NeZero N] {k : ℤ} (f : Newform N k) (h : TauCeti.Nat.IsExactDivisor Q N) (φ : (ZMod (N / Q))ˣ →* ℂˣ) (hχ : f.χ = φ.comp (ZMod.unitsMap ⋯)) : f.atkinLehnerPseudoEigenvalue h ^ 2 = ↑(φ (ZMod.unitOfCoprime Q ⋯))⁻¹

If the Q-part of the nebentypus is trivial, λ_Q(f) ^ 2 = χ_{N/Q}(Q)⁻¹. Here 𝒲_Q f = λ_Q(f) • f (atkinLehnerTransform_eq_self_of_χ_eq_comp_right), so λ_Q(f) is a genuine eigenvalue of 𝒲_Q, and the product law reads λ_Q(f) ^ 2 = χ_{N/Q}(Q)⁻¹.

theorem HeckeRing.GL2.Newform.norm_atkinLehnerPseudoEigenvalue_of_χ_eq_comp_right {N Q : ℕ} [NeZero N] {k : ℤ} (f : Newform N k) (h : TauCeti.Nat.IsExactDivisor Q N) (φ : (ZMod (N / Q))ˣ →* ℂˣ) (hχ : f.χ = φ.comp (ZMod.unitsMap ⋯)) : ‖f.atkinLehnerPseudoEigenvalue h‖ = 1

If the Q-part of the nebentypus is trivial, |λ_Q(f)| = 1.

theorem HeckeRing.GL2.Newform.atkinLehnerTransform_eq_conj_of_χ_eq_comp_left {N Q : ℕ} [NeZero N] {k : ℤ} (f : Newform N k) (h : TauCeti.Nat.IsExactDivisor Q N) (ψ : (ZMod Q)ˣ →* ℂˣ) (hχ : f.χ = ψ.comp (ZMod.unitsMap ⋯)) : f.atkinLehnerTransform h = f.conj

If the nebentypus is trivial away from Q, transport is conjugation: the transported newform is the conjugate newform f_ρ. Both have nebentypus χ⁻¹ and good eigenvalues χ(p)⁻¹ a_p = conj a_p, so strong multiplicity one applies.

theorem HeckeRing.GL2.Newform.norm_atkinLehnerPseudoEigenvalue_of_χ_eq_comp_left {N Q : ℕ} [NeZero N] {k : ℤ} (f : Newform N k) (h : TauCeti.Nat.IsExactDivisor Q N) (ψ : (ZMod Q)ˣ →* ℂˣ) (hχ : f.χ = ψ.comp (ZMod.unitsMap ⋯)) : ‖f.atkinLehnerPseudoEigenvalue h‖ = 1

If the nebentypus is trivial away from Q, |λ_Q(f)| = 1: the transported newform is the conjugate newform, whose Petersson norm is that of f.

The endpoints Q = 1 and Q = N

@[simp]

theorem HeckeRing.GL2.Newform.atkinLehnerPseudoEigenvalue_one {N : ℕ} [NeZero N] {k : ℤ} (f : Newform N k) (h : TauCeti.Nat.IsExactDivisor 1 N) : f.atkinLehnerPseudoEigenvalue h = 1

At Q = 1 the pseudo-eigenvalue is 1: the Atkin–Li matrix atkinLiMatrix N 1 lies in Γ₁(N), so 𝒲_1 is the identity on S_k(Γ₁(N)) and λ_1(f) is the first Fourier coefficient of f.

@[simp]

theorem HeckeRing.GL2.Newform.atkinLehnerTransform_self {N : ℕ} [NeZero N] {k : ℤ} (f : Newform N k) (h : TauCeti.Nat.IsExactDivisor N N) : f.atkinLehnerTransform h = f.conj

At Q = N transport is conjugation: the Fricke matrix is an Atkin–Lehner matrix for N, and the normalized Fricke operator sends f to a multiple of the conjugate newform f_ρ.

@[simp]

theorem HeckeRing.GL2.Newform.atkinLehnerPseudoEigenvalue_self {N : ℕ} [NeZero N] {k : ℤ} (f : Newform N k) (h : TauCeti.Nat.IsExactDivisor N N) : f.atkinLehnerPseudoEigenvalue h = ↑(f.χ (-1)) * f.frickePseudoEigenvalue

At Q = N the pseudo-eigenvalue is χ(-1) times the Fricke pseudo-eigenvalue: the Atkin–Li matrix atkinLiMatrix N N is γ W_N for the Fricke matrix W_N and a matrix γ ∈ Γ₀(N) with lower-right entry -1 modulo N, so 𝒲_N f computed with the Atkin–Li matrix is χ(-1) times the normalized Fricke operator applied to f.