Research

Open Access

Research

Open Access

- Sheng-Chi Liu
^{1}, - Riad Masri
^{2}Email author and - Matthew P. Young
^{2}

Research in the Mathematical Sciences2015**2**:22

https://doi.org/10.1186/s40687-015-0040-y

© Liu et al. 2015

**Received: **16 March 2015

**Accepted: **26 August 2015

**Published: **29 September 2015

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 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 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.

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

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