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Let $E$ be an elliptic curve defined over a finite field $\mathbb{F}_q$ with $q=p^r$ elements ($p\in\Ints$ is a prime). The following theorem gives a bound of the size of $E(\mathbb{F}_q)$ , $N_q$ , i.e. the number points of $E$ defined over $\mathbb{F}_q$ . This was first conjectured by Emil Artin (in his thesis!) and proved by Helmut Hasse in the 1930's.
Theorem 1 (Hasse) $$\mid N_q -q -1 \mid \leq 2\sqrt{q} $$
Remark: Let $a_p=p+1-N_p$ as in the definition of the L-series of an ellitpic curve. Then Hasse's bound reads:
$$\mid a_p \mid \leq 2\sqrt{p}$$
This fact is key for the convergence of the L-series of $E$ .
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