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Let $p \equiv 3 \pmod 4$ be a prime, and $E/F_{p^r}$ be the elliptic curve given by $y^2 = x^3 − x$. Find the zeta-function of $E/F_p$ and use it to determine $|E(F_{p^r} )|$ for all r>0.

Jyrki Lahtonen
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mathnerd007
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    Hint: Look at the polynomial $m(x)=x^3-x.$ It is odd, right? Because $p\equiv3\pmod4$ exactly one of $m(a)$ and $m(-a)$ is a quadratic residue (obviously $a\neq0,\pm1$). – Jyrki Lahtonen Nov 20 '15 at 21:56
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    Partially duplicate of this older question. – Jyrki Lahtonen Nov 20 '15 at 22:01
  • Is this an assignment*homework question? What have you tried so far? – Klangen Dec 17 '18 at 10:09
  • @Klangen See my edit for a simpler way of typesetting a congruence. Also, the question is 3 years old, and the asker hasn't been heard from since then. It is safe to assume that the eventual homework is overdue. If you are inclined to answer the question, feel free to do so! The question is a bit lacking in context though. – Jyrki Lahtonen Dec 18 '18 at 13:04

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