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: E(OK ) Ess (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 q18+ . This can be viewed as an analog of Linniks 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 -, 1/2) of central values of the RankinSelberg L-functions L(f -, s) where f is a fixed weight 2, level q arithmetically normalized Hecke cusp form and- varies over the weight 1, level D theta series associated to an ideal class group character of K. We apply this result to study the arithmetic of Abelian varieties, subconvexity, and L4 norms of autormorphic forms.