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


Ontology type: schema:ScholarlyArticle      Open Access: True


Article Info

DATE

2015-12

AUTHORS

Sheng-Chi Liu, Riad Masri, Matthew P. Young

ABSTRACT

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≫εq18+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 ∑χL(f×Θχ,1/2)of central values of the Rankin–Selberg L-functions L(f×Θχ,s) where f is a fixed weight 2, level q arithmetically normalized Hecke cusp form and Θχ 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. More... »

PAGES

22

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URI

http://scigraph.springernature.com/pub.10.1186/s40687-015-0040-y

DOI

http://dx.doi.org/10.1186/s40687-015-0040-y

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