Let $X$ be a smooth projective curve over the finite field $\mathbb{F}$. Fix $n\in \mathbb{N}$. Presumably we can naturally define the moduli functor of rank $n$, convergent isocrystals (for some fixed coefficient field) on $X$. Is it known if this functor is reasonable? For example is it representable? If not does it form a reasonable stack?
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$\begingroup$ I would be pleasantly surprised if the fibered category were representable. I seem to recall that in the complex setting, the groupoid of rank $n$ connections is not a reasonable stack. $\endgroup$– S. Carnahan ♦Feb 26, 2011 at 17:04
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