Petersson inner product

For weight-

k

cusp forms on

Γ

, the Petersson inner product is

⟨ f, g ⟩ = \int_{Γ\ h} f (τ) \overline{g (τ)} y^{k} \frac{d x d y}{y ^{2}} .

(1)

Cusp decay makes the integral convergent.

Table of contents
- Hecke operators are self-adjoint for the Petersson inner product Petersson inner product
- Rankin–Selberg method Petersson inner product
  - Rankin–Selberg integral for holomorphic cusp forms Rankin–Selberg method
    - Positive unfolding of a cusp-form square Rankin–Selberg integral for holomorphic cusp forms
      - Absolute convergence of cusp-form L-series from square coefficients Positive unfolding of a cusp-form square
  - Rankin–Selberg convolution Rankin–Selberg method

Hecke operators are self-adjoint for the Petersson inner product ( $⟨ T_{n} f, g ⟩ = ⟨ f, T_{n} g ⟩$ )

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On level-one cusp forms, determinant-normalized Hecke operators satisfy

⟨ T_{n} f, g ⟩ = ⟨ f, T_{n} g ⟩

. Unfolding their finite correspondences transfers a matrix to its inverse; its adjugate has the same determinant and reverses the correspondence. Since these operators commute, the cusp space has a common orthonormal basis of Hecke eigenforms, with real eigenvalues.

Rankin–Selberg method

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The Rankin–Selberg method represents Dirichlet series built from automorphic forms as integrals against Eisenstein series and studies them by unfolding those integrals.

Rankin–Selberg integral for holomorphic cusp forms ( $I (f, g, w) = \int_{Γ\ H} f \overline{g} E (τ, w) y^{k} d μ$ )

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For two weight-

k

level-one cusp forms, use the primitive real-analytic Eisenstein series and

d μ = d x d y / y^{2}

. The Rankin–Selberg unfolding identity is

I (f, g, w) = \frac{Γ ( w + k - 1 )}{( 4 π ) ^{w + k - 1}} \sum_{n \geq 1} \frac{a _{n} b _{n}}{n ^{w + k - 1}} .

(1)

Initially this follows by absolute convergence, unfolding to

0 \leq x < 1

, and integrating the Fourier series in

x

. Exponential cusp decay and polynomial growth of the primitive real-analytic Eisenstein series make the integral holomorphic for

Re w > 1

, continuing the Dirichlet series to

Re s > k

Positive unfolding of a cusp-form square ( $\sum_{n \geq 1} ∣ a_{n} ∣^{2} n^{- s} < \infty (s > k)$ )

 0  0

For real

w > 1

, the finite Rankin–Selberg integral

I (f, f, w)

has a nonnegative integrand. Tonelli theorem unfolds it without an a priori bound on the Dirichlet series. Integration over

x

gives

\int_{0}^{1} ∣ f (x + i y) ∣^{2} d x = \sum_{n \geq 1} ∣ a_{n} ∣^{2} e^{- 4 πn y}

. A second application of Tonelli theorem gives the finite sum in the displayed identity with

s = w + k - 1

. Positivity, rather than analytic continuation alone, proves absolute convergence.

Absolute convergence of cusp-form L-series from square coefficients ( $\sum_{n \geq 1} ∣ a_{n} ∣ n^{- σ} < \infty (σ > (k + 1) /2)$ )

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By positive unfolding of a cusp-form square,

\sum ∣ a_{n} ∣^{2} n^{- β} < \infty

for

β > k

. Choose

k < β < 2 σ - 1

. The Cauchy-Schwarz inequality bounds the absolute sum for the L-function of a cusp form by

(\sum_{n \geq 1} ∣ a_{n} ∣^{2} n^{- β})^{1/2} (\sum_{n \geq 1} n^{- (2 σ - β)})^{1/2} < \infty.

(1)

Rankin–Selberg convolution ( $L (f, g, s)$ )

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For cusp forms

f = \sum a_{n} q^{n}

and

g = \sum b_{n} q^{n}

, their Rankin–Selberg convolution in the elementary normalization is

L (f, g, s) = \sum_{n \geq 1} a_{n} \overline{b_{n}} n^{- s}

Euler factor of a coefficientwise product of Hecke eigenforms ( $\sum_{r \geq 0} a_{p^{r}} \overline{b}_{p^{r}} X^{r} = \frac{1 - p ^{2 k - 2} X ^{2}}{\prod _{u \in {α_{p}, β_{p}}, v \in {\overline{γ}_{p}, \overline{δ}_{p}}} ( 1 - uv X )}$ )

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For normalized level-one Hecke eigenforms, let

α_{p} + β_{p} = a_{p}

α_{p} β_{p} = p^{k - 1}

, and similarly

γ_{p} + δ_{p} = b_{p}

γ_{p} δ_{p} = p^{k - 1}

. The Hecke multiplication relations give

a_{p^{r}} = (α_{p}^{r + 1} - β_{p}^{r + 1}) / (α_{p} - β_{p})

, with the repeated-root case obtained by continuity. Multiplying the two expressions and summing four geometric series proves the displayed local factor. Multiplicativity gives an Euler product; multiplying by

ζ (2 s - 2 k + 2)

cancels its local numerator and produces the degree-four convolution factors.

Zeta-completed Rankin-Selberg coefficient series ( $R (s) = 2 π^{- w} Γ (w) ζ (2 w) (4 π)^{- s} Γ (s) F (s), w = s - k + 1$ )

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For two level-one weight-

k

cusp forms,

F (s) = \sum a_{n} \overline{b}_{n} n^{- s}

. Unfolding against the nonholomorphic lattice sum normalized by

π^{- w} Γ (w)

gives the displayed completion as the integral of

f \overline{g} y^{k}

against that Eisenstein series. Its functional equation is

R (s) = R (2 k - 1 - s)

, and its only possible simple poles are at

k - 1, k

. The raw series is obtained by dividing out the gamma and zeta factors, so it does not inherit the same unqualified pole statement; zeros of the denominator zeta factor can yield additional possible poles. The raw series is regular at