nt.number theory - What should Spec Z[sqrt{D}] x_{F_1} Spec bar{F_1} be?

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.