Rankin–Selberg L-functions and the reduction of CM elliptic curves

Abstract

Let q be a prime and $K={\mathbb Q}(\sqrt{-D})$ be an imaginary quadratic field such that q is inert in K. If $\mathfrak {q}$ is a prime above q in the Hilbert class field of K, there is a reduction map

$$\begin{aligned} r_{\mathfrak q}:\;{\mathcal {E\ell \ell }}({\mathcal {O}}_K) \longrightarrow {\mathcal {E\ell \ell }}^{ss}({\mathbb F}_{q^2}) \end{aligned}$$

from the set of elliptic curves over $\overline{{\mathbb Q}}$ with complex multiplication by the ring of integers ${\mathcal {O}}_K$ to the set of supersingular elliptic curves over ${\mathbb {F}}_{q^2}.$ We prove a uniform asymptotic formula for the number of CM elliptic curves which reduce to a given supersingular elliptic curve and use this result to deduce that the reduction map is surjective for $D \gg _{\varepsilon } q^{18+\varepsilon }.$ This can be viewed as an analog of Linnik’s theorem on the least prime in an arithmetic progression. We also use related ideas to prove a uniform asymptotic formula for the average

$$\begin{aligned} \sum _{\chi }L(f \times \Theta _\chi ,1/2) \end{aligned}$$

of central values of the Rankin–Selberg L-functions ${L(f \times {\Theta _{\chi}},s)}$ where f is a fixed weight 2, level q arithmetically normalized Hecke cusp form and $\Theta _\chi$ varies over the weight 1, level D theta series associated to an ideal class group character $\chi$ of K. We apply this result to study the arithmetic of Abelian varieties, subconvexity, and $L^4$ norms of autormorphic forms.