Rankin–Selberg L-functions and the reduction of CM elliptic curves
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© 2015 Liu et al. Let q be a prime and K =Q(−D) be an imaginary quadratic field such that q is inert in K. If q is a prime above q in the Hilbert class field of K, there is a reduction map rq: Eℓℓ(OK ) −→ Eℓℓss (Fq2 ) from the set of elliptic curves over Q with complex multiplication by the ring of integers OK to the set of supersingular elliptic curves over Fq2 . 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 ≫ε q18+ε . 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 ∑ L(f ×-χ, 1/2) χ of central values of the Rankin–Selberg L-functions L(f ×-χ, s) where f is a fixed weight 2, level q arithmetically normalized Hecke cusp form and-χ varies over the weight 1, level D theta series associated to an ideal class group character χ of K. We apply this result to study the arithmetic of Abelian varieties, subconvexity, and L4 norms of autormorphic forms.
author list (cited authors)
Liu, S., Masri, R., & Young, M. P.