Connection between sgn character and the Legendre symbol

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:

1. How serious should I take this analogy? Are there other examples that will help me put this in a broader context?

2. 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

• Hi Spooky, I think you already realized that the Legendre symbol is for the finite place and the sign character is for the infinity place. This is really well explained in Kato's number theory book. – 包殿斌 Apr 26 '18 at 18:38