I have been stuck on this question for a while now to no success. Help would be appreciated.
Consider $x$ and $y$ such that $(x, p) =(y, p) = 1$. For what $p$ does their exist $x$ and $y$ such that $x^2 + y^2 \equiv 0 \pmod p$?
(Also, if you could help answer in terms of the Jacobian that would be appreciated, as that is the relevant section in the book)
In case anyone comes across a question similar and wants to know an answer, I eventually figured it out:
$x^2 + y^2 \equiv 0$ mod p = $x^2 \equiv -y^2$ mod p = $(x/y)^2 \equiv -1$ mod p.