Is there a theorem which guarentees the existance of an elliptic curve with given number of points over $\mathbf{F}_p$ for a given $p$.
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$\begingroup$ See also mathoverflow.net/questions/317531/…, mathoverflow.net/questions/87916 $\endgroup$– WatsonJun 18, 2021 at 16:44
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2 Answers
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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.
answered Dec 12, 2016 at 8:25
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$\begingroup$ Do we have any way to construct such an elliptic curve in polynomial in $logp$ steps or can anyone tell about algorithms on constructing them. $\endgroup$ Dec 12, 2016 at 9:02
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1$\begingroup$ That's a different question, and people are unlikely to spot it here as a comment on an answer. You should ask a different question. $\endgroup$– zntDec 12, 2016 at 11:15
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1$\begingroup$ @HimanshuShukla Deuring's proof developed into what's known as the CM method of constructing elliptic curves. I believe the construction is polynomial in $D{\log p}$ steps where $D = 4p - a^2$. $\endgroup$ Dec 12, 2016 at 13:13
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$\begingroup$ @FelipeVoloch can you please give me a link of the paper by Deuring $\endgroup$ Dec 16, 2016 at 5:36
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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.)