The holomorphic kernel of the Rankin-Selberg convolution (Q1840691)

The standard way to deduce the analytic continuation and the functional equation of the convolution \(L\)-function \(L(s, f\otimes g)\) for a cusp form \(f\) and a modular form \(g\) is to express it as an inner product of \(fg\) with a non-holomorphic Eisenstein series. In this very interesting paper, a new approach to the convolution \(L\)-function is given without invoking the Eisenstein series. The main idea is to interpret the linear map \(f \to L(s, f\otimes g)\) as the inner product of \(f\) with a certain kernel function (cusp form) depending on \(g\) and \(s\). As the main result, the Fourier series of the kernel function is determined in terms of generalized Ramanujan sums. As an application of the method, a new simplified proof is given for a celebrated formula of \textit{B. Gross} and \textit{D. B. Zagier} [Invent. Math. 84, 225-320 (1986; Zbl 0608.14019)] concerning heights of Heegner points.