Today, while I was lecturing on the Legendre symbol, I realized that the phenomenon: "the product of two non-squares is a square" isn't so foreign. For example, for $\mathbb R^\times$, the squares are just $\mathbb R^{>0}$. Additionally, two non-squares (negative numbers) multiply to a square (positive number). I was feeling particularly happy with this example, and decided to define a real analogue of the Legendre symbol. $$ \left( \frac{a}{\infty} \right) : = \frac{a}{| \ a \ |} $$ Of course I immediately identified this as the sgn character. Now, you can lift $\left( \frac{a}{p} \right)$ to a function on $\mathbb Q_p^\times$ through the projection $\mathbb Q_p^\times \rightarrow \mathbb F_p^\times$. These are now all examples of Hecke characters. My questions, which aren't extremely well-defined are as follows:
How serious should I take this analogy? Are there other examples that will help me put this in a broader context?
Can we make an analogy at complex places? The obvious thing doesn't seem to yield anything interesting because the squaring map is surjective on $\mathbb C^\times$.
-Spooky
