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.

我们计算 Rankin-Selberg 中心值的某些“宽矩”$L$-功能$L（\pi \otimes \mathrm{\Omega },\frac{1}{2}）$在哪里$\pi$是一个尖部自同构表示${\mathrm{GL}<!--FUNCTION\; APPLICATION-->}_2$超过$\mathbb{Q}$和$\mathrm{\Omega }$是赫克特征（指挥家 $1$) 的虚二次场。该矩计算用于获得“弱同时”非零结果，这是不同Rankin-Selberg的非零结果$L$- 扭曲的乘积微不足道的函数。

证明依赖于关联广泛的时刻$L$- 使用 Waldspurger 公式在 Heegner 点评估的自守形式的通常矩的函数。为了实现这一目标，推导了沃尔德斯普格一般权自同构形式公式的经典版本，这可能具有独立意义。关键输入是 Heegner 点的均匀分布（具有明确的误差项），以及某些周期积分的非零结果。特别是，我们开发了一种软技术来获得三重卷积的非零性$L$-功能。