# Re: Number of elliptic curves over finite fields

• From: Jyrki Lahtonen <lahtonen@xxxxxx>
• Date: Tue, 16 Jan 2007 14:49:51 +0200

Pubkeybreaker wrote:
Rolf Bardeli wrote:
In v.z. Gathen and Gerhard: "Modern Computer Algebra", the number of
elliptic curves of orders 3 to 13 over the finite field F7 is given as
1, 4, 3, 6, 4, 6, 4, 6, 3, 4, 1

What is known about the sequence of orders for finite fields Fp in
general? Does it always have such remarkable symmetry?

There is a theorem by Deuring which says that the distribution follows
a sine squared
distribution over the range [p - 2sqrt(p) + 1, p + 2sqrt(p) + 1]

This is true (in a suitable asymptotic sense that you are probably
more knowledgeable about than I am).

I think that the OP was asking for the exact "mirror symmetry" of the
distribution about the midpoint q+1 (q= size of the field). This can
be seen by replacing the term y^2 with g*y^2, where g is a non-square
element of GF(q). If so desired one can then go back to the Weierstrass
form, but this change reflects the number of points as described.

In characteristic 2 (i.e. q=2^n) this trick is not available. There
one can use the trace description of the number of solutions of
(y/x)^2+(y/x)=x+a+b/x. Replacing 'a' with 'a+e', where 'e' is an element
with trace 1 accounts for the "mirror curves" in that case.

IOW while Deuring's result is much deeper, the EXACT mirror symmetry
of the distrubution is easy to account for.

Cheers,

Jyrki
.