= Vanishing trace criterion for a congruent number curve
{title2=$a_p=0\iff p\equiv3\pmod4$}
At an odd good prime, use a model $y^2=x^3-d^2x$ with $d$ a unit. If $p\equiv3\pmod4$, pairing $x$ with $-x$ cancels their <Legendre symbols>, so the trace is zero. If $p\equiv1\pmod4$, the four distinct rational <2-torsion> points force the group order to be divisible by four, whereas $p+1$ is two modulo four. The trace cannot vanish. This proves the converse without an explicit complex-multiplication character-sum evaluation.
Back to article page
