The coefficients of a period polynomial are critical L-values

Let f be a cusp form of weight k = w + 2 on SL(2, ℤ). Expanding (X - τY)ʷ in its period polynomial r_f(X, Y) = ∫₀^{i∞} f(τ) (X - τY)ʷ dτ writes r_f in terms of the periods rₙ(f) = ∫₀^{i∞} f(τ) τⁿ dτ = iⁿ⁺¹ Λ(f, n + 1) (TauCeti.cuspIntegral_mul_pow_zero_infty):

r_f(X, Y) = ∑_{n=0}^{w} (-1)ⁿ (w choose n) iⁿ⁺¹ Λ(f, n + 1) Xʷ⁻ⁿ Yⁿ.

So the coefficients of r_f are, up to nonzero constants, the values of L(f, s) at the critical integers s = 1, …, w + 1. The parity involution P(X, Y) ↦ P(-X, Y) multiplies the monomial Xʲ Yʷ⁻ʲ by (-1)ʲ, so the even period polynomial r_f^+ collects the critical values L(f, n + 1) with w - n even and the odd period polynomial r_f^- those with w - n odd. In particular r_f^+ vanishes exactly when all the critical values L(f, n + 1) with w - n even vanish, and similarly for r_f^-. So a single nonzero value L(f, w + 1) makes r_f^+ nonzero, and, for w ≠ 0, a single nonzero value L(f, w) makes r_f^- nonzero. This is the route to the nonvanishing of the even and odd period polynomials of a Hecke eigenform: once the weight is large enough, both points w and w + 1 lie in the half-plane where the L-function of an eigenform is an absolutely convergent Euler product.

Main results

• TauCeti.periodPolynomial_eq_sum: r_f in the monomial basis, with the completed L-values Λ(f, n + 1) as coefficients.
• TauCeti.binaryFormMonomialBasis_repr_periodPolynomial: the coordinates of r_f in the monomial basis.
• TauCeti.evenPeriodPolynomial_eq_zero_iff, TauCeti.oddPeriodPolynomial_eq_zero_iff: r_f^+ (respectively r_f^-) vanishes if and only if L(f, n + 1) = 0 for every n ≤ w with w - n even (respectively odd).
• TauCeti.evenPeriodPolynomial_ne_zero, TauCeti.oddPeriodPolynomial_ne_zero: r_f^+ ≠ 0 if L(f, w + 1) ≠ 0, and r_f^- ≠ 0 if w ≠ 0 and L(f, w) ≠ 0.

References

• Y. I. Manin, Periods of parabolic forms and p-adic Hecke series, Mat. Sb. 92 (1973), 378–401, §1.
• W. Kohnen and D. Zagier, Modular forms with rational periods, in Modular Forms (R. A. Rankin, ed.), Ellis Horwood, 1984, 197–249, §1.1.

theorem TauCeti.periodPolynomial_eq_sum {k : ℤ} {w : ℕ} (hk : k = ↑w + 2) (f : CuspForm (Matrix.SpecialLinearGroup.mapGL ℝ).range k) : ↑((periodPolynomial hk) f) = ∑ j : Fin (w + 1), ((-1) ^ (w - ↑j) * ↑(w.choose ↑j) * Complex.I ^ (w - ↑j + 1) * ModularForm.Λ ⋯ f (↑(w - ↑j) + 1)) • (binaryFormMonomialBasis ℂ w) j

The period polynomial in the monomial basis. The coefficient of Xʲ Yʷ⁻ʲ in the period polynomial of a cusp form f of weight k = w + 2 on SL(2, ℤ) is (-1)ʷ⁻ʲ (w choose j) iʷ⁻ʲ⁺¹ Λ(f, w - j + 1).

theorem TauCeti.binaryFormMonomialBasis_repr_periodPolynomial {k : ℤ} {w : ℕ} (hk : k = ↑w + 2) (f : CuspForm (Matrix.SpecialLinearGroup.mapGL ℝ).range k) (j : Fin (w + 1)) : ((binaryFormMonomialBasis ℂ w).repr ↑((periodPolynomial hk) f)) j = (-1) ^ (w - ↑j) * ↑(w.choose ↑j) * Complex.I ^ (w - ↑j + 1) * ModularForm.Λ ⋯ f (↑(w - ↑j) + 1)

The coordinates of the period polynomial r_f in the monomial basis Xʲ Yʷ⁻ʲ.

theorem TauCeti.evenPeriodPolynomial_eq_zero_iff {k : ℤ} {w : ℕ} (hk : k = ↑w + 2) (f : CuspForm (Matrix.SpecialLinearGroup.mapGL ℝ).range k) : (evenPeriodPolynomial hk) f = 0 ↔ ∀ n ≤ w, Even (w - n) → ModularForm.L ⋯ f (↑n + 1) = 0

The even period polynomial vanishes exactly when the critical L-values L(f, n + 1) with w - n even vanish.

theorem TauCeti.oddPeriodPolynomial_eq_zero_iff {k : ℤ} {w : ℕ} (hk : k = ↑w + 2) (f : CuspForm (Matrix.SpecialLinearGroup.mapGL ℝ).range k) : (oddPeriodPolynomial hk) f = 0 ↔ ∀ n ≤ w, Odd (w - n) → ModularForm.L ⋯ f (↑n + 1) = 0

The odd period polynomial vanishes exactly when the critical L-values L(f, n + 1) with w - n odd vanish.

theorem TauCeti.evenPeriodPolynomial_ne_zero {k : ℤ} {w : ℕ} (hk : k = ↑w + 2) (f : CuspForm (Matrix.SpecialLinearGroup.mapGL ℝ).range k) (hL : ModularForm.L ⋯ f (↑w + 1) ≠ 0) : (evenPeriodPolynomial hk) f ≠ 0

A nonzero critical value makes the even period polynomial nonzero: if L(f, w + 1) ≠ 0, then r_f^+ ≠ 0.

theorem TauCeti.oddPeriodPolynomial_ne_zero {k : ℤ} {w : ℕ} (hk : k = ↑w + 2) (f : CuspForm (Matrix.SpecialLinearGroup.mapGL ℝ).range k) (hw : w ≠ 0) (hL : ModularForm.L ⋯ f ↑w ≠ 0) : (oddPeriodPolynomial hk) f ≠ 0

A nonzero critical value makes the odd period polynomial nonzero: if w ≠ 0 and L(f, w) ≠ 0, then r_f^- ≠ 0.