arvind@utcsri.UUCP (08/18/87)
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