The sign of the functional equation

Hecke's functional equation relates the completed L-function of a cusp form f of level Γ₁(N) to that of its Petersson-normalized Fricke companion: Λ_N(k - s, f) = i^k Λ_N(s, 𝒲_N f), a relation between two forms. When f is an eigenvector of the normalized Fricke operator, 𝒲_N f = ε • f, the companion is f again up to the scalar ε, and the relation becomes a functional equation for Λ_N(·, f) alone, with sign i^k ε.

On a newform of trivial nebentypus the eigenvalue is the Fricke sign ε_N(f) ∈ {1, -1} and the weight is even, so the sign of the functional equation is i^k ε_N(f), again 1 or -1. A sign different from 1 forces the completed L-function to vanish at the central point s = k / 2, and hence makes the analytic rank of the newform positive.

Main results

• CuspForm.frickeCompletedL_sub_eq_of_normalizedFrickeOperatorCusp_eq_smul: the one-form functional equation Λ_N(k - s, f) = i^k ε Λ_N(s, f) on a Fricke eigenvector.
• CuspForm.Λ_sub_eq_of_normalizedFrickeOperatorCusp_eq_smul: the same equation for Mathlib's ModularForm.Λ, where the change of normalization contributes a factor N ^ (s - k / 2).
• CuspForm.Λ_eq_zero_of_normalizedFrickeOperatorCusp_eq_smul, CuspForm.L_eq_zero_of_normalizedFrickeOperatorCusp_eq_smul: a sign different from 1 kills the central value of the completed and of the ordinary L-function.
• HeckeRing.GL2.Newform.frickeCompletedL_sub_eq: the functional equation of a newform of trivial nebentypus, with sign i^k ε_N(f).
• HeckeRing.GL2.Newform.I_zpow_mul_frickeSign_eq_one_or_neg_one: that sign is 1 or -1.
• HeckeRing.GL2.Newform.analyticRank_pos_of_I_zpow_mul_frickeSign_ne_one: a sign different from 1 makes the analytic rank positive.

References

• F. Diamond and J. Shurman, A first course in modular forms, Theorem 5.10.2.
• T. Miyake, Modular forms, Theorem 4.3.5.

The functional equation of a Fricke eigenvector

theorem CuspForm.frickeCompletedL_sub_eq_of_normalizedFrickeOperatorCusp_eq_smul {N : ℕ} [NeZero N] {k : ℤ} (f : CuspForm (Subgroup.map (Matrix.SpecialLinearGroup.mapGL ℝ) (CongruenceSubgroup.Gamma1 N)) k) {ε : ℂ} (hk : 0 < k) (hε : (TauCeti.normalizedFrickeOperatorCusp k) f = ε • f) (s : ℂ) : f.frickeCompletedL (N.toPNat ⋯) (↑k - s) = Complex.I ^ k * ε * f.frickeCompletedL (N.toPNat ⋯) s

The one-form functional equation. If a cusp form of level Γ₁(N) and positive weight k is an eigenvector of the normalized Fricke operator with eigenvalue ε, then its level-N completed L-function satisfies

Λ_N(k - s, f) = i^k ε Λ_N(s, f).

The sign of the functional equation is therefore i^k ε; the factor i^k comes from the two-form equation frickeCompletedL_functional_equation_gamma1.

theorem CuspForm.Λ_sub_eq_of_normalizedFrickeOperatorCusp_eq_smul {N : ℕ} [NeZero N] {k : ℤ} (f : CuspForm (Subgroup.map (Matrix.SpecialLinearGroup.mapGL ℝ) (CongruenceSubgroup.Gamma1 N)) k) {ε : ℂ} (hk : 0 < k) (hε : (TauCeti.normalizedFrickeOperatorCusp k) f = ε • f) (s : ℂ) : ModularForm.Λ hk f (↑k - s) = Complex.I ^ k * ε * ↑N ^ (s - ↑k / 2) * ModularForm.Λ hk f s

The functional equation in Mathlib's normalization. For a Fricke eigenvector with eigenvalue ε, Mathlib's completed L-function satisfies

Λ(k - s, f) = i^k ε N^(s - k/2) Λ(s, f),

the level-dependent factor N^(s - k/2) being what distinguishes ModularForm.Λ from the level-N completion Λ_N = N^(s/2) Λ; it is 1 at the central point s = k / 2.

Vanishing at the central point

theorem CuspForm.frickeCompletedL_eq_zero_of_normalizedFrickeOperatorCusp_eq_smul {N : ℕ} [NeZero N] {k : ℤ} (f : CuspForm (Subgroup.map (Matrix.SpecialLinearGroup.mapGL ℝ) (CongruenceSubgroup.Gamma1 N)) k) {ε : ℂ} (hk : 0 < k) (hε : (TauCeti.normalizedFrickeOperatorCusp k) f = ε • f) (hsign : Complex.I ^ k * ε ≠ 1) : f.frickeCompletedL (N.toPNat ⋯) (↑k / 2) = 0

The central point of a functional equation with sign different from 1. The central point s = k / 2 is the fixed point of s ↦ k - s, so the functional equation reads Λ_N(k/2, f) = i^k ε Λ_N(k/2, f) there; unless the sign i^k ε is 1, the value vanishes.

theorem CuspForm.Λ_eq_zero_of_normalizedFrickeOperatorCusp_eq_smul {N : ℕ} [NeZero N] {k : ℤ} (f : CuspForm (Subgroup.map (Matrix.SpecialLinearGroup.mapGL ℝ) (CongruenceSubgroup.Gamma1 N)) k) {ε : ℂ} (hk : 0 < k) (hε : (TauCeti.normalizedFrickeOperatorCusp k) f = ε • f) (hsign : Complex.I ^ k * ε ≠ 1) : ModularForm.Λ hk f (↑k / 2) = 0

Mathlib's completed L-function vanishes at the central point when the sign of the functional equation is different from 1: the central point is the fixed point of s ↦ k - s, where the normalization factor N ^ (s - k / 2) is 1.

theorem CuspForm.L_eq_zero_of_normalizedFrickeOperatorCusp_eq_smul {N : ℕ} [NeZero N] {k : ℤ} (f : CuspForm (Subgroup.map (Matrix.SpecialLinearGroup.mapGL ℝ) (CongruenceSubgroup.Gamma1 N)) k) {ε : ℂ} (hk : 0 < k) (hε : (TauCeti.normalizedFrickeOperatorCusp k) f = ε • f) (hsign : Complex.I ^ k * ε ≠ 1) : ModularForm.L hk f (↑k / 2) = 0

The central value of the L-function vanishes when the sign of the functional equation is different from 1.

The functional equation of a newform of trivial nebentypus

theorem HeckeRing.GL2.Newform.frickeCompletedL_sub_eq {N : ℕ} [NeZero N] {k : ℤ} (f : Newform N k) (hχ : f.χ = 1) (hk : 0 < k) (s : ℂ) : f.frickeCompletedL (N.toPNat ⋯) (↑k - s) = Complex.I ^ k * f.frickeSign hχ * f.frickeCompletedL (N.toPNat ⋯) s

The functional equation of a newform of trivial nebentypus, with sign i^k ε_N(f) for ε_N(f) the Fricke sign:

Λ_N(k - s, f) = i^k ε_N(f) Λ_N(s, f).

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

The sign of the functional equation of a newform of trivial nebentypus is 1 or -1. A nonzero form with trivial nebentypus has even weight, so i^k is (-1) ^ (k / 2), and the Fricke sign is itself ± 1.

Positive analytic rank

theorem HeckeRing.GL2.Newform.analyticRank_pos_of_I_zpow_mul_frickeSign_ne_one {N : ℕ} [NeZero N] {k : ℤ} (f : Newform N k) (hχ : f.χ = 1) (hk : 0 < k) (hsign : Complex.I ^ k * f.frickeSign hχ ≠ 1) : 0 < f.analyticRank hk

A newform of trivial nebentypus whose functional equation has sign different from 1 has positive analytic rank: its L-function vanishes at the central point s = k / 2.