We have just finished proving, using Hasse-Minkowski, the following:
Let $a, b, c$ be pairwise coprime squarefree integers. Then
$$aX^2 + bY^2 + cZ^2 = 0$$
has nontrivial solutions in $\mathbb Q$ iff the following holds:
$a, b, c$ are not all positive or not all negative.
For each odd prime $p$ dividing $a$, there exists $r \in \mathbb Z$ with $b + r^2c \equiv 0 \ \mod \ p$, and similarly for the primes
dividing $b$ and $c$.If $a, b, c$ are all odd, then there are two of them whose sum is divisible by 4.
If $a$ is even, then either $b + c$ or $a + b + c$ is divisible by 8 (similarly if $b$ or $c$ is even).
The author then notes that Cassels' Local Fields book has a proof that does not use condition 1, which suggests that a solution in $\mathbb Q_p$ for each $p$ guarantees a solution in $\mathbb R$, which makes us wonder if the behavior of this equation locally at $p$ for all but one prime $p \leq \infty$ determines the behavior at the final prime. Gouvea says this is true, vaguely mentions that it follows from "Reciprocity," and says nothing more. What, in a general sense, is going on here, and what does Reciprocity have to do with all this?