Average of the first invariant factor of the reductions of abelian varieties of CM type

作者：Kim Sungjin^*

来源：International Journal of Number Theory, 2016, 12(2): 445-463.

DOI：10.1142/S1793042116500263

• Abelian variety
• complex multiplication
• reduction
• first invariant

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(F-p) as abelian group is:
A(F-p) similar or equal to Z/d(1)(p) Z circle plus ... circle plus Z/d(g)(p) Z circle plus Z/e(1)(p)Z circle plus ... circle plus Z/e(g)(p)Z,
where d(i)(p) vertical bar d(i+1)(p), d(g)(p) vertical bar e(1)(p), and e(i)(p) vertical bar e(i+1)(p) for 1 <= i < g. We are interested in finding an asymptotic formula for the number of prime ideals p with N-p < x, A has a good reduction at p, d(1)(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.

• 出版日期2016-3