When are the $l$-local $p$-adic Galois representations of Siegel modular forms semistable? By this I mean $\rho\_{f}: G\_{\mathbb{Q}}\to \operatorname{GSpin}\_{2n+1}(\overline{\mathbb{Q}}\_p)$ restricted to the decomposition group at $l$. Is this controlled by the level? I am primarily interested in this when $l=p$ and $n=2$.



Semistability of local Siegel Galois rep:
As I commented above, the question needs some adjustment for $n \ge 3$, since the Galois representation doesn't go into $\operatorname{GSp}\_{2n}$ but rather into its $L$-group, which is $\operatorname{GSpin}\_{2n + 1}$; this is isomorphic to $\operatorname{GSp}\_{2n}$ if $n = 1$ or $n = 2$, but totally different if $n \ge 3$.


What one **expects** is the following:



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> If the automorphic representation $\Pi$ generated by an eigenform $f$ has a non-zero vector invariant under the Iwahori subgroup at $\ell$ (that is, the subgroup of $\mathrm{GSp}\_{2n}(\mathbf{Z}\_\ell)$ that is the preimage of the Borel subgroup mod $\ell$), then $\rho\_f$ (or rather $\rho\_{\Pi}$) is semistable at $\ell$.
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Here we understand "semistable" to mean "image of inertia is unipotent" for $\ell \ne p$, and in the sense of Fontaine theory for $\ell = p$. (Note that the Iwahori-stable vector needn't be $f$ itself, because we didn't assume anything about $f$ being a newform; indeed I don't think anyone knows what "newform" should mean once $n \ge 3$.)


For $n = 2$ there is an extensive literature on the local properties of $\rho\_\Pi$, see e.g. [this paper of Sorensen](https://web.math.princeton.edu/~csorense/gal_final.pdf), or [this paper of Jorza](https://www3.nd.edu/~ajorza/papers/gsp4gl2.pdf). These deal with almost all cases of my boxed statement for $n = 2$, but not quite all; it may well now be possible to clean up the remaining cases using Arthur's classification of automorphic representations of classical groups, but I'm not sure.


For general $n$, the existence of the $\mathrm{GSpin}$-valued representation was only proved about a week ago (!) in [this preprint of Kret and Shin](https://arxiv.org/pdf/1609.04223.pdf). Among their results is the statement that if $\Pi$ has nonzero Iwahori-invariants at $p$, then either $\rho\_\Pi$ or its quadratic twist must be semistable at $p$. (They don't directly address the case of $\ell \ne p$ in their paper, but it can probably be dealt with along the same lines.)


