Hasse's bound for elliptic curves over finite fields
(Theorem)
Let be an elliptic curve defined over a finite field with elements (
is a prime). The following theorem gives a bound of the size of
, , i.e. the number points of defined over
. This was first conjectured by Emil Artin (in his thesis!) and proved by Helmut Hasse in the 1930's.
Theorem 1 (Hasse)
Remark: Let
as in the definition of the L-series of an ellitpic curve. Then Hasse's bound reads:
This fact is key for the convergence of the L-series of .
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