What should be $\text{Spec } \mathbb{Z}[\sqrt{D}] \times_{\mathbb{F}_1} \text{Spec } \overline{\mathbb{F}}_{1}$?

Sure, there's more than one definition. I'm looking for any answer that uses at least one definition of scheme over $\mathbb{F}_1$.

This really is more a question of opinion. What do you think this should be? Some monoid that has something to do with $\text{Spec }\mathbb{Z}[\sqrt{D}][\mathbb{Q}/\mathbb{Z}]$ would be my guess (where the second brackets mean group ring).

This interests me from the point of view that, say, hyperelliptic curves over a finite field come (geometrically) from the group scheme of a quadratic extension of $\overline{\mathbb{F}}_p [t]$. In this case the frobenius acts on ideal classes, and satisfies a quadratic equation. But, from what I understand, the natural analogue of frobenius in the arithmetic case, is like taking any positive power, and taking limits to 0 (or something of the sort). Would this satisfy some kind of equation on, say, $\text{Pic(Spec } \mathbb{Z}[\sqrt{D}] \times_{\mathbb{F}_1} \text{Spec } \overline{\mathbb{F}}_{1}\text{)}$? (for whatever definition of Pic that should be natural here)

I've searched for information on $\mathbb{F}_1$, but most just talk about making $\text{Spec }\mathbb{Z}$ into a curve, getting zeta functions to be Riemann's, etc. Instead, I want to ask questions that are not just about proving the Riemann hypothesis, like the one above.