 mersenneforum.org > Math Number of points on elliptic curves over finite fields
 Register FAQ Search Today's Posts Mark Forums Read 2021-01-28, 11:34 #1 RedGolpe   Aug 2006 Monza, Italy 73 Posts Number of points on elliptic curves over finite fields Hi all, I noticed that for elliptic curves of the form y2 ≡ x3 + a (mod p) sometimes the number of points is always p+1 for any choice of a. This seems to be the case for all p ≡ 5 (mod 6). Moreover, when this does not happen, i.e., for p ≡ 1 (mod 6), it looks like there are exactly zero curves of such form where the number of points is p+1. Can someone point me towards the right direction as to why this happens? Last fiddled with by RedGolpe on 2021-01-28 at 11:46 Reason: more info   2021-01-28, 11:58   #2
Nick

Dec 2012
The Netherlands

2×3×293 Posts Quote:
 Originally Posted by RedGolpe Can someone point me towards the right direction as to why this happens?
https://en.wikipedia.org/wiki/Hasse%...lliptic_curves   2021-01-28, 12:06 #3 RedGolpe   Aug 2006 Monza, Italy 73 Posts I am familiar with Hasse's theorem, but it says nothing about the specific number of points: it only gives a bound.   2021-01-28, 13:04 #4 RedGolpe   Aug 2006 Monza, Italy 1118 Posts This looks like a good answer.   2021-01-28, 13:33 #5 Dr Sardonicus   Feb 2017 Nowhere 512910 Posts If p == 5 (mod 6) then gcd(3, p-1) = 1, so x -> x^3 (mod p) is invertible. In fact, x -> x^((2p-1)/3) (mod p) is the inverse map. Thus if p == 5 (mod 6), x^3 is "any residue mod p" and x^3 + a is "any residue mod p." If x^3 + a is one of the (p-1)/2 quadratic non-residues (mod p) there are no points (x, y) on the curve y^2 = x^3 + a. If x^3 + a is one of the (p-1)/2 nonzero squares (mod p) there are two points (x,y) and (x, -y) on the curve. If x^3 + a = 0 (mod p) there is one point (x, 0) on the curve. That makes p points in all. I seem to be missing one point. Last fiddled with by Dr Sardonicus on 2021-01-28 at 13:44 Reason: xifnig posty   2021-01-28, 14:03 #6 RedGolpe   Aug 2006 Monza, Italy 1118 Posts You are just missing the identity point, or the point at infinity.   2021-01-29, 22:29 #7 Robert Holmes   Oct 2007 2·53 Posts The key term here is supersingular elliptic curves. For p = 2 mod 3, any curve of the form y^2 = x^3 + B is supersingular, p > 3. Another common case is p = 3 mod 4, in which case y^2 = x^3 + x is also known to be supersingular.  Thread Tools Show Printable Version Email this Page Similar Threads Thread Thread Starter Forum Replies Last Post toktarev Math 1 2019-12-05 14:43 WraithX Math 12 2010-09-29 09:34 meng_luckywolf Math 6 2007-12-13 04:21 bongomongo Factoring 5 2006-12-21 18:19 otkachalka Factoring 5 2005-11-20 12:22

All times are UTC. The time now is 22:07.

Wed Dec 1 22:07:09 UTC 2021 up 131 days, 16:36, 1 user, load averages: 1.22, 1.15, 1.21