Average of the First Invariant Factor of the Reductions of Abelian Varieties of Cm Type
نویسنده
چکیده
For a field of definition k of an abelian variety A and prime ideal p of k which is of a good reduction for A, the structure of A(Fp) as abelian group is: (1) A(Fp) ' Z/d1(p)Z⊕ · · · ⊕ Z/dg(p)Z⊕ Z/e1(p)Z⊕ · · · ⊕ Z/eg(p)Z, where di(p)|di+1(p), dg(p)|e1(p), and ei(p)|ei+1(p) for 1 ≤ i < g. We are interested in finding an asymptotic formula for the number of prime ideals p with Np < x, A has a good reduction at p, d1(p) = 1. We succeed in proving this under the assumption of the Generalized Riemann Hypothesis (GRH). Unconditionally, we achieve a short range asymptotic for abelian varieties of CM type, and the full cyclicity theorem for elliptic curves over a number field containing the CM field.
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