Poincaré series and Petersson inner products

For an integer m≥1, start with exp(2πimτ), invariant under Γ∞ in the weight-k slash action. Summing its slash translates over left cosets of Γ∞ in Γ defines Pₖ,ₘ. Right multiplication by another group element permutes these cosets and gives the modular law once convergence is established. At even k>2, comparison with a convergent lattice sum gives absolute local uniform convergence, and the resulting function is a cusp form. Weight two is excluded from this elementary convergence statement.

A nonzero seed need not produce a nonzero averaged form. Averaging can cancel, and a cusp-form space of dimension zero forces every member Pₖ,ₘ to vanish. The case m=0 gives an Eisenstein-type average rather than the cuspidal construction used here. It is important to distinguish an individual orbit sum from a basis theorem: infinitely many Poincaré series may be linearly dependent even when their span is the whole finite-dimensional cusp space. Neither convergence nor symmetrization alone asserts that they are independent.

Define the Petersson pairing as the integral of f(τ) times the conjugate of g(τ), multiplied by y^k, against hyperbolic measure dx dy/y² over one modular fundamental region. We do not divide by the domain volume or group index. This convention is linear in f and conjugate-linear in g. The factor y^k cancels the absolute-square automorphy factors, so the density descends to the quotient. Replacing it with ordinary Euclidean area without the corresponding power adjustment breaks that invariance.

For cusp forms, exponential decay at the cusp dominates the polynomial y factor, giving a finite integral. The pairing is positive definite on the cusp space because the integral of |f|² times a positive density is positive for a nonzero holomorphic function. A noncusp Eisenstein series generally has divergent norm with this unregularized integral; E₄ tends to one, leaving an integral comparable to the divergent integral of y². Pairing a cusp form with a holomorphic noncusp form can still converge, but that does not extend a finite norm to all modular forms.

Insert the Poincaré sum into the second slot and use absolute convergence to interchange summation with integration. The translated fundamental regions tile a strip for Γ∞, so the integral unfolds to x between zero and one and y>0. Conjugating the Fourier seed produces exp(−2πimx)exp(−2πmy). The x integral annihilates every Fourier mode of f except m, and the remaining y integral is a gamma integral with exponential rate 4πm. Both factors of two in that rate have a visible origin.

If a cusp form is orthogonal to every Pₖ,ₘ, the extraction formula makes all its Fourier coefficients zero; the identity theorem then makes the form zero. This proves that the Poincaré series span the cusp space. It also tests nonvanishing in particular cases: a nonzero extracted coefficient forces the corresponding Poincaré series to be nonzero. The logic does not require positivity of every individual summand. The inner product combines the symmetry and the Fourier expansion into a useful duality between a function space and coefficient functionals.