Petersson inner product

For weight-$k$ cusp forms on $\Gamma$, the Petersson inner product isCusp decay makes the integral convergent.

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.

The Rankin–Selberg method relates arithmetic Dirichlet series to integrals of automorphic forms against Eisenstein series. Expanding the Eisenstein series as a sum over a coset space lets one replace an integral over a fundamental domain by an integral over a simpler region. This step is called unfolding.

For a weight-$k$ level-one cusp form $f(\tau )={\sum }_{n\ge 1}{a}_n{e}^{2\pi in\tau }$, the primitive real-analytic Eisenstein series $E(\tau ,w)$ gives, initially for large $\mathrm{Re}w$,Unfolding produces the strip $0\le x<1$, integration in $x$ removes unequal Fourier modes, and integration in $y$ supplies the gamma function. The Rankin–Selberg integral for holomorphic cusp forms makes this link precise for two forms. Analytic properties of the Eisenstein series then control the coefficient Dirichlet series. For real $w>1$, positivity also proves convergence through positive unfolding of a cusp-form square; analytic continuation alone would not justify convergence of the original sum.

For two weight-$k$ level-one cusp forms, use the primitive real-analytic Eisenstein series and $d\mu =dxdy/y^2$. The Rankin–Selberg unfolding identity isInitially this follows by absolute convergence, unfolding to $0\le x<1$, 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 $\mathrm{Re}w>1$, continuing the Dirichlet series to $\mathrm{Re}s>k$.

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 ${\int }_0^1|f(x+iy){|}^{2}dx={\sum }_{n\ge 1}|a_n{|}^2{e}^{-4\pi ny}$. 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.

By positive unfolding of a cusp-form square, $\sum |a_n{|}^2{n}^{-\beta }<\infty$ for $\beta >k$. Choose $k<\beta <2\sigma -1$. The Cauchy-Schwarz inequality bounds the absolute sum for the L-function of a cusp form by

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 $L(f,g,s)={\sum }_{n\ge 1}{a}_n\overline{b_n}{n}^{-s}$.

For normalized level-one Hecke eigenforms, let ${\alpha }_p+{\beta }_p=a_p$, ${\alpha }_p{\beta }_p=p^{k-1}$, and similarly ${\gamma }_p+{\delta }_p=b_p$, ${\gamma }_p{\delta }_p=p^{k-1}$. The Hecke multiplication relations give $a_{p^r}=({\alpha }_p^{r+1}-{\beta }_p^{r+1})/({\alpha }_p-{\beta }_p)$, 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 $\zeta (2s-2k+2)$ cancels its local numerator and produces the degree-four convolution factors.

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 ${\pi }^{-w}\Gamma (w)$ gives the displayed completion as the integral of $f\overline{g}{y}^k$ against that Eisenstein series. Its functional equation is $\mathcal{R}(s)=\mathcal{R}(2k-1-s)$, 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 $k-1$ and can have a simple pole at $k$.

Let $f$ and $g$ be level-one cusp forms of respective weights $k$ and $l$, with $r=k-l$ an even integer greater than two. Use the unnormalized lattice Eisenstein sumwhose constant Fourier coefficient is $2\zeta (r)$. With the Petersson inner product linear in its first argument and the elementary Rankin–Selberg convolution $L(f,g,s)={\sum }_{n\ge 1}{a}_n\overline{b_n}{n}^{-s}$, unfolding givesThe sum over primitive bottom rows unfolds the fundamental domain to the translation strip. Integration in the real direction keeps equal Fourier modes, and ${\int }_0^{\infty }{e}^{-4\pi ny}{y}^{k-2}dy=\Gamma (k-1)/(4\pi n{)}^{k-1}$ supplies the factor. If instead one uses the normalized Eisenstein series $E_r=G_r/(2\zeta (r))$, the factor $2\zeta (r)$ is absent. The stated level and normalization are essential to the displayed formula.