Hasse bound for elliptic curves (glitches?)

2018-03-12 20:21:33

Hasse theorem says (in the simplest case), for any elliptic curve $E$ (with integer coefficients), the number of solutions mod $p$ is within $2\sqrt p$ of the number of points of a projective line $\mathbb P^1(\mathbb F_p)$ (i.e. $p+1$)

$$ |p+1-N_p| \leq 2\sqrt p $$

for all $p$ (not just asymptotically). I was playing around with it (by the simplest brute-force calculation) but I encountered a few cases, such as $y^2=x^3-1$, where it just goes over the Hasse bounds by a little bit, for a few primes.

https://beta.observablehq.com/@liuyao12/elliptic-curves-counting-solutions#Hasse

It must be a silly mistake somewhere. Then it is a shameless promotion on my part.