What should be textSpecmathbbZ[sqrtD]timesmathbbF1textSpecoverlinemathbbF1?
Sure, there's more than one definition.
I'm looking for any answer that uses at least one definition of scheme over mathbbF1.
This really is more a question of opinion.
What do you think this should be?
Some monoid that has something to do with textSpecmathbbZ[sqrtD][mathbbQ/mathbbZ] 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 overlinemathbbFp[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, textPic(SpecmathbbZ[sqrtD]timesmathbbF1textSpecoverlinemathbbF1text)?
(for whatever definition of Pic that should be natural here)
I've searched for information on mathbbF1, but most just talk about making textSpecmathbbZ 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.
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