Rankin-Selberg L-functions and the reduction of CM elliptic curves Academic Article uri icon

abstract

  • 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.

published proceedings

  • RESEARCH IN THE MATHEMATICAL SCIENCES

author list (cited authors)

  • Liu, S., Masri, R., & Young, M. P.

citation count

  • 7

complete list of authors

  • Liu, Sheng-Chi||Masri, Riad||Young, Matthew P

publication date

  • December 2015