Is there a theorem which guarentees the existance of an elliptic curve with given number of points over $\mathbf{F}_p$ for a given $p$.
Thanks
Is there a theorem which guarentees the existance of an elliptic curve with given number of points over $\mathbf{F}_p$ for a given $p$.
Thanks
Deuring proved that for every $a, |a| < 2\sqrt{p}$, there exists an elliptic curve with $p+1-a$ points over $\mathbb{F}_p$.
M Deuring, Die Typen der Multiplikatorenringe elliptischer Funktionenkörper, Abh. Math. Sem. Univ. Hamburg 14 (1941), 197-272.
Perhaps you are looking for Honda-Tate theory, see http://projecteuclid.org/download/pdf_1/euclid.jmsj/1260463295 Honda, Taira (1968), "Isogeny classes of abelian varieties over finite fields", Journal of the Mathematical Society of Japan, 20: 83–95. (The eigenvalues of the Frobenius determine the number of $\mathbf{F}_{q^n}$-rational points by the Lefschetz trace formula.)