🚀 go-pugleaf

Date: 17 Aug 1987 14:47:06-EDT (Monday)
From: "Victor S. Miller" <VICTOR@yktvmz.bitnet>
Subject: Distribution of exponents of elliptic curves

Fix a prime p.  It is well known (by results of Deuring) what the distribution
of the set $\{N_p(E) | E$ an elliptic curve over $F_p\}$ looks like (where
$N_p(E)$ denotes the number of points in $E(F_p)$).  What, if anything is
known about the distribution of the following set:
   $\{exp_p(E) | E$ an elliptic curve over $F_p\}$
where $exp_p(E)$ denotes the exponent of the group of points over $F_p$?
Note, that this amounts to asking about the distribution of the group
structures.
  If instead, we now fix an elliptic curve E over F_p and look at the set:
   $\{exp_q(E) | q=p^n, n>0\}$ what can we say about that.  This amounts
to asking about the distribution of the group structures in various
extensions.
   For the first set it is customary to look at $\theta_p =
\arccos((p+1-N_p)/(2\sqrt{p})$ but what could be a corresponding normalization
for the second problem.
                        Victor S. Miller
                        IBM Research