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-   -   Number of points on elliptic curves over finite fields (https://www.mersenneforum.org/showthread.php?t=26453)

 RedGolpe 2021-01-28 11:34

Number of points on elliptic curves over finite fields

Hi all, I noticed that for elliptic curves of the form

[I]y[/I][SUP]2[/SUP] ≡ [I]x[/I][SUP]3[/SUP] + [I]a[/I] (mod [I]p[/I])

sometimes the number of points is always [I]p[/I]+1 for any choice of [I]a[/I]. This seems to be the case for all [I]p[/I] ≡ 5 (mod 6).

Moreover, when this does not happen, i.e., for [I]p[/I] ≡ 1 (mod 6), it looks like there are exactly zero curves of such form where the number of points is [I]p[/I]+1.

Can someone point me towards the right direction as to why this happens?

 Nick 2021-01-28 11:58

[QUOTE=RedGolpe;570326]Can someone point me towards the right direction as to why this happens?[/QUOTE]
[URL]https://en.wikipedia.org/wiki/Hasse%27s_theorem_on_elliptic_curves[/URL]

 RedGolpe 2021-01-28 12:06

I am familiar with Hasse's theorem, but it says nothing about the specific number of points: it only gives a bound.

 RedGolpe 2021-01-28 13:04

[URL="https://math.stackexchange.com/questions/875983/solutions-to-the-mordell-equation-modulo-p"]This[/URL] looks like a good answer.

 Dr Sardonicus 2021-01-28 13:33

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.

 RedGolpe 2021-01-28 14:03

You are just missing the identity point, or the point at infinity.

 Robert Holmes 2021-01-29 22:29

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.

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