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It is a well-known fact, that the stack of shtukas $\mathop{\mathcal{S}\!\mathit{ht}_{\pm 1}^1}$ with two legs and elementary modifications (i.e. those of type $(1,0,\dotsc,0)$ or $(0,\dotsc,0,-1)$) is etale (or smooth of relative dimension $2r-2$ in the rank $r$ case) over $X^2$. This can be derived from the fact that $\mathop{\mathcal{H}\!\mathit{ecke}}_{\pm 1}^1\to X^2\times \mathop{\mathcal{B}\!\mathit{un}}^1$
is etale (or smooth of relative dimension $2r-2$ in the higher rank case), where $\mathop{\mathcal{H}\!\mathit{ecke}}_{\pm 1}^1$ denotes the elementary Hecke stack of rank $1$. Compare Proposition 1.3 and Lemma 2.3 in Introduction to the stack of Shtukas, Ngo for details.

The above result describes actually a rather special situation, as modifications can be defined much more generally. For example, the stack of Shtuka with type of modification bounded by or equal to a a dominant coweight for $\mathrm{GL}_r$ (in the rank $r$ case) is also a geometric object of interest, compare Definition 3.1.6. in On generalized D Shtuka.

My particular interest is about the case where one does not specify any type of modification at all and I am happy if I understand the rank $1$ case.

More precisely, let $\mathop{\mathcal{H}\!}\mathit{ecke}^1$ be the stack over $\mathbb{F}_q$ which associates to $S\in(\mathbf{Sch}/{\mathbb{F}_q})_{fppf}$ the groupoid consisting of objects $(\mathcal{L},\mathcal{L}',(x_1,x_2),\alpha)$ where
$\bullet$ $\mathcal{L},\mathcal{L}'$ are line bundles on $X_S=X\times_{\mathrm{Spec}(\mathbb{F})_q}S$
$\bullet$ $(x_1,x_2)\in X(S)^2$
$\bullet$ $\alpha\colon \mathcal{L}\mid_{X_S-\Gamma_{x_1}\cup\Gamma_{x_2}}\overset{\sim}{\to}\mathcal{L}'\mid_{X_S-\Gamma_{x_1}\cup\Gamma_{x_2}}$is an isomorphism and isomorphisms of such data. The stack of Shtukas without specified modification can then be defined by the usual pullback diagram i.e. as the pullback of $\mathop{\mathcal{H}\!\mathit{ecke}}^1\to \mathop{\mathcal{B}\!\mathit{un}}^1\times \mathop{\mathcal{B}\!\mathit{un}}^1$ along $\mathop{\mathcal{B}\!\mathit{un}}^1\overset{(\mathrm{id},\mathrm{Frob}^*)}{\to} \mathop{\mathcal{B}\!\mathit{un}}^1\times \mathop{\mathcal{B}\!\mathit{un}}^1$.

My question is, whether in this situation one can still prove the etaleness of $\mathop{\mathcal{H}\!\mathit{ecke}}^1\to X^2\times \mathop{\mathcal{B}\!\mathit{un}}^1$ and $\mathop{\mathcal{S}\!\mathit{ht}}^1\to X^2$, or what could go wrong in this case.

I could not find this case in literature and I am not sure, whether etaleness (smoothness) is still given or not.

Thank You for Your efforts

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    $\begingroup$ +1 for the first sentence. $\endgroup$
    – Olivier
    Dec 20, 2019 at 13:18

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