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

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Research

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.

Keywords

Supersingular elliptic curves Equidistribution Gross points Heegner points Mean values of L-functions  \(L^4\) norm 

Mathematics Subject Classification

11M41 

Notes

Acknowledgements

R. M. and M. Y. were supported by the National Science Foundation under agreement Nos. DMS-1162535 (R. M.) and DMS-1101261 (M. Y.). Any opinions, findings and conclusions or recommendations expressed in this material are those of the authors and do not necessarily reflect the views of the National Science Foundation.

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© Liu et al. 2015

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Authors and Affiliations

  1. 1.Department of MathematicsWashington State UniversityPullmanUSA
  2. 2.Department of MathematicsTexas A&M UniversityCollege StationUSA

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