On the average exponent of CM Elliptic Curves Modulo p

Sungjin Kim

doi:10.1016/j.ffa.2014.07.003

On the average exponent of CM Elliptic Curves Modulo p

Number Theory 2015-09-08 v4

Authors: Sungjin Kim

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Abstract

Let $E$ be an elliptic curve defined over $\Q$ and with complex multiplication by $\mO_K$ , the ring of integers in an imaginary quadratic field $K$ . It is known that $E(\F_p)$ has a structure E(\F_p)\simeq \Z/d_p\Z \oplus \Z/e_p\Z. with $d_p|e_p$ . We give an asymptotic formula for the average order of $e_p$ , with improved error term, and upper bound estimate for the average of $d_p$ .

Keywords

CM elliptic curves complex multiplication average exponent group structure asymptotic formula error term upper bound

Cite

@article{arxiv.1207.6652,
  title  = {On the average exponent of CM Elliptic Curves Modulo p},
  author = {Sungjin Kim},
  journal= {arXiv preprint arXiv:1207.6652},
  year   = {2015}
}

Comments

7 pages, improvement in Theorem 2, conductor dependence

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