= Point-count criterion for y squared equals x cubed plus k x
{title2=$\#E(\mathbb F_p)=p+1\iff p\equiv3\pmod4$}
For odd $p$ and $k\not\equiv0\pmod p$, take $E:y^2=x^3+kx$. If $p\equiv3\pmod4$, pairing $x$ with $-x$ cancels the <Legendre symbols>. If $p\equiv1\pmod4$, the <coefficient formula for trace of Frobenius modulo p> gives
$$
a_p\equiv\binom{(p-1)/2}{(p-1)/4}k^{(p-1)/4}\not\equiv0\pmod p.
$$
The <binomial coefficient> is nonzero because every factorial involved is shorter than $p$. Thus the <Trace of Frobenius> is not zero in the second case.
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