Wide moments of L-functions I : Twists by class group characters of imaginary quadratic fields

We calculate certain “wide moments” of central values of Rankin–Selberg $L$-functions $L\left(\pi \otimes \mathrm{\Omega },\frac{1}{2}\right)$ where $\pi$ is a cuspidal automorphic representation of ${\mathrm{GL}<!--FUNCTION\; APPLICATION-->}_2$ over $\mathbb{Q}$ and $\mathrm{\Omega }$ is a Hecke character (of conductor $1$) of an imaginary quadratic field. This moment calculation is applied to obtain “weak simultaneous” nonvanishing results, which are nonvanishing results for different Rankin–Selberg $L$-functions where the product of the twists is trivial.

The proof relies on relating the wide moments of $L$-functions to the usual moments of automorphic forms evaluated at Heegner points using Waldspurger’s formula. To achieve this, a classical version of Waldspurger’s formula for general weight automorphic forms is derived, which might be of independent interest. A key input is equidistribution of Heegner points (with explicit error terms), together with nonvanishing results for certain period integrals. In particular, we develop a soft technique for obtaining the nonvanishing of triple convolution $L$-functions.