Hasse’s theorem

Helmut Hasse’s theorem is in German .

The number of points on an elliptic curve over a finite field GF(q) is within the range

$q+1\pm 2{\sqrt {q}}$,

that is to say, of all q² points in GF(q)⨉GF(q), the number of them that satisfy any given elliptic curve equation always falls in this range.

For “hyperelliptic” curves or other Abelian varieties of genus g>1, Hasse’s theorem is still applicable when the permissible range is broadened by a factor of g:

$q+1\pm 2g{\sqrt {q}}$.

This result was proved by André Weil, and is known as the Hasse–Weil theorem.