The goal of this question is to find a "geometric" definition of Frobenius element in $\text{Gal}(\overline{\mathbb{Q}}/\mathbb{Q})$.


Here are two definitions that don't work, but that should help explain what I mean. Fix an algebraic closure of $\mathbb{Q}$ and, for a prime $p$, fix an algebraic closure of $\mathbb{F}\_p$. If there were a canonical morphism $f : \mathbb{Q} \to \mathbb{F}\\_p$, then the induced map $f\\_{\ast} : \pi\_1(\text{Spec } \mathbb{F}\_p) \to \pi\_1(\text{Spec } \mathbb{Q})$ of étale fundamental groups (basepoints the algebraic closures above, suppressed) would be an obvious way to define Frobenius elements: we would just take the image of a generator of $\pi\_1(\text{Spec } \mathbb{F}\_p) \simeq \hat{\mathbb{Z}}$. (The general motivation being: "gee, if I had a category $C$ and a functor, let's call it $\pi\_1 : C \to \text{Grp}$, wouldn't it be nice if I could find an object, let's call it $S^1$, with $\pi\_1(S^1) \simeq \mathbb{Z}$, so I could probe elements of $\pi\_1(X)$ using morphisms $S^1 \to X$," and it seems for schemes that $\pi\_1(S^1) \simeq \hat{\mathbb{Z}}$ is the next best thing.)


But of course this is nonsense since no such $f$ exists. The next obvious thing (from my perspective, being someone who knows nothing about all this étale stuff) is to look at the canonical morphism $f : \mathbb{Z} \to \mathbb{F}\\_p$, which induces a map $f\\_{\ast} : \pi\_1(\text{Spec } \mathbb{F}\_p) \to \pi\_1(\text{Spec } \mathbb{Z})$, and then to look at the canonical morphism $g : \mathbb{Z} \to \mathbb{Q}$ and the induced map $g\\_{\ast} : \pi\_1(\text{Spec } \mathbb{Q}) \to \pi\_1(\text{Spec } \mathbb{Z})$. Perhaps the Frobenius elements at $p$ are nothing more than the inverse image under $g\\_{\ast}$ of the image of a generator of $\pi\_1(\text{Spec } \mathbb{F}\_p)$ under $f\\_{\ast}$ (up to conjugacy to account for changes in basepoint).


But this is also nonsense since $\pi\_1(\text{Spec } \mathbb{Z})$ is trivial. 



> 
> So what is the correct version of this construction?
> 
> 
> 


Here's my guess: instead of using $\mathbb{Z}$, we have to use the localization $R = \mathbb{Z}\_{(p)}$. As before there are morphisms $f : R \to \mathbb{F}\_p, g : R \to \mathbb{Q}$ inducing maps $f\\_{\ast} : \pi\_1(\text{Spec } \mathbb{F}\_p) \to \pi\_1(\text{Spec } R)$ and $g\\_{\ast} : \pi\_1(\text{Spec } \mathbb{Q}) \to \pi\_1(\text{Spec } R)$, and maybe now something like the statement "the Frobenius elements in $\pi\_1(\text{Spec } \mathbb{Q})$ at $p$ are the inverse image under $g\\_{\ast}$ of the image of a generator of $\hat{\mathbb{Z}}$ under $f\\_{\ast}$ (up to conjugacy)" is finally true. Is it? And what does $g\\_{\ast}$ look like?



Frobenius elements from the point of view of étale fundamental groups
Your guess is right. $\pi\_1(Spec(R))$ is the automorphism group of the maximal extension $\mathbb{Q}^{p\textrm{-}ur} \subset \overline{\mathbb{Q}}$ unramified at $p$. (This is a special case of SGA 1 Exp. V Prop 8.2 about the fundamental group of normal schemes). The morphism $g\_\*$ is simply the restriction map $Gal(\overline{\mathbb{Q}}/\mathbb{Q}) \to Gal(\mathbb{Q}^{p\textrm{-}ur}/\mathbb{Q})$. 


