Analytic rank and conductor of a newform

For a positive-weight newform, its coefficient Dirichlet series has the entire continuation ModularForm.L. This file defines the analytic rank to be the order of vanishing of that continuation at the central point k / 2. The continuation is not identically zero, because its Dirichlet coefficients have first coefficient one. Consequently the order is finite and is faithfully represented by a natural number.

The analytic conductor is stated in the arithmetic s-coordinate. If s_an = s - (k - 1) / 2 is the analytic normalization, its two archimedean parameters are s_an + (k - 1) / 2 and s_an + (k + 1) / 2. Thus

q(f, s) = N (|s_an + (k - 1) / 2| + 3) (|s_an + (k + 1) / 2| + 3).

The constant 3 and this normalization follow Iwaniec--Kowalski, §5.1 and (5.7).

Main definitions

• HeckeRing.GL2.Newform.analyticRank: the order of vanishing at k / 2.
• HeckeRing.GL2.Newform.analyticConductorAt: the analytic conductor at s.
• HeckeRing.GL2.Newform.analyticConductor: the conductor at the central point.

Main results

• HeckeRing.GL2.Newform.L_ne_zero: the entire continuation is not identically zero.
• HeckeRing.GL2.Newform.analyticOrderAt_L_ne_top: its order is finite at every point.
• HeckeRing.GL2.Newform.analyticRank_eq_analyticOrderNatAt: any entire continuation agreeing with the coefficient series on its convergence half-plane gives the same analytic rank.
• HeckeRing.GL2.Newform.analyticRank_eq_zero_iff: rank zero is equivalent to nonvanishing at the central point.
• HeckeRing.GL2.Newform.analyticConductorAt_eq: the conductor in the arithmetic coordinate.

References

• H. Iwaniec and E. Kowalski, Analytic Number Theory, §5.1, especially (5.7).

@[simp]

theorem HeckeRing.GL2.Newform.L_ne_zero {N : ℕ} [NeZero N] {k : ℤ} (f : Newform N k) (hk : 0 < k) : ModularForm.L hk f.toCuspForm ≠ 0

The normalized first coefficient of a newform ensures that its entire L-function is nonzero, and hence that its analytic order is finite.

@[simp]

theorem HeckeRing.GL2.Newform.analyticOrderAt_L_ne_top {N : ℕ} [NeZero N] {k : ℤ} (f : Newform N k) (hk : 0 < k) (s : ℂ) : analyticOrderAt (ModularForm.L hk f.toCuspForm) s ≠ ⊤

The analytic order of the entire L-function of a newform is finite at every point.

noncomputable def HeckeRing.GL2.Newform.analyticRank {N : ℕ} [NeZero N] {k : ℤ} (f : Newform N k) (hk : 0 < k) : ℕ

The analytic rank of a positive-weight newform is the order of vanishing of its entire L-function at the central point s = k / 2.

Equations

• f.analyticRank hk = analyticOrderNatAt (ModularForm.L hk f.toCuspForm) (↑k / 2)

theorem HeckeRing.GL2.Newform.analyticRank_def {N : ℕ} [NeZero N] {k : ℤ} (f : Newform N k) (hk : 0 < k) : f.analyticRank hk = analyticOrderNatAt (ModularForm.L hk f.toCuspForm) (↑k / 2)

The defining equation for the analytic rank.

theorem HeckeRing.GL2.Newform.analyticRank_eq_analyticOrderNatAt {N : ℕ} [NeZero N] {k : ℤ} (f : Newform N k) (hk : 0 < k) {F : ℂ → ℂ} (hF : Differentiable ℂ F) (hFL : ∀ {s : ℂ}, ↑k / 2 + 1 < s.re → F s = LSeries (fun (n : ℕ) => (PowerSeries.coeff n) (UpperHalfPlane.qExpansion 1 ⇑f.toCuspForm)) s) : f.analyticRank hk = analyticOrderNatAt F (↑k / 2)

Any entire continuation agreeing with the coefficient Dirichlet series on the known convergence half-plane computes the analytic rank. In particular, the definition is independent of the chosen construction of analytic continuation.

@[simp]

theorem HeckeRing.GL2.Newform.analyticRank_eq_zero_iff {N : ℕ} [NeZero N] {k : ℤ} (f : Newform N k) (hk : 0 < k) : f.analyticRank hk = 0 ↔ ModularForm.L hk f.toCuspForm (↑k / 2) ≠ 0

A newform has analytic rank zero exactly when its entire L-function does not vanish at the central point.

@[simp]

theorem HeckeRing.GL2.Newform.analyticRank_pos_iff {N : ℕ} [NeZero N] {k : ℤ} (f : Newform N k) (hk : 0 < k) : 0 < f.analyticRank hk ↔ ModularForm.L hk f.toCuspForm (↑k / 2) = 0

A newform has positive analytic rank exactly when its entire L-function vanishes at the central point.

noncomputable def HeckeRing.GL2.Newform.analyticConductorAt {N : ℕ} [NeZero N] {k : ℤ} (_f : Newform N k) (s : ℂ) : ℝ

The analytic conductor of a newform at an arithmetic parameter s.

Writing s_an = s - (k - 1) / 2, the two norm factors are the archimedean parameters s_an + (k - 1) / 2 and s_an + (k + 1) / 2.

Equations

• _f.analyticConductorAt s = ↑N * (‖s - (↑k - 1) / 2 + (↑k - 1) / 2‖ + 3) * (‖s - (↑k - 1) / 2 + (↑k + 1) / 2‖ + 3)

@[simp]

theorem HeckeRing.GL2.Newform.analyticConductorAt_eq {N : ℕ} [NeZero N] {k : ℤ} (f : Newform N k) (s : ℂ) : f.analyticConductorAt s = ↑N * (‖s‖ + 3) * (‖s + 1‖ + 3)

In the arithmetic s-coordinate, the two archimedean parameters simplify to s and s + 1.

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

The analytic conductor of a newform is its analytic conductor at the central point s = k / 2.

Equations

• f.analyticConductor = f.analyticConductorAt (↑k / 2)

theorem HeckeRing.GL2.Newform.analyticConductor_def {N : ℕ} [NeZero N] {k : ℤ} (f : Newform N k) : f.analyticConductor = f.analyticConductorAt (↑k / 2)

The defining equation for the central analytic conductor.

@[simp]

theorem HeckeRing.GL2.Newform.analyticConductor_eq {N : ℕ} [NeZero N] {k : ℤ} (f : Newform N k) : f.analyticConductor = ↑N * (‖↑k / 2‖ + 3) * (‖↑k / 2 + 1‖ + 3)

The central analytic conductor in the arithmetic coordinate.

theorem HeckeRing.GL2.Newform.analyticConductorAt_pos {N : ℕ} [NeZero N] {k : ℤ} (f : Newform N k) (s : ℂ) : 0 < f.analyticConductorAt s

The analytic conductor at every parameter is strictly positive.

theorem HeckeRing.GL2.Newform.analyticConductor_pos {N : ℕ} [NeZero N] {k : ℤ} (f : Newform N k) : 0 < f.analyticConductor

The central analytic conductor is strictly positive.