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 $n=2$.
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$\begingroup$ What assumptions are you making on the weight of your Siegel form? The Galois representations are typically constructed ``directly'' when the weight is cohomological, and by p-adic approximation for small weights, and it's much harder to keep track of p-adic Hodge theoretic properties when there's a p-adic approximation process involved. Note that 2 is a small weight for $\mathrm{GSp}_4$! $\endgroup$– David LoefflerCommented Sep 22, 2016 at 8:17
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1$\begingroup$ Hold on: these Galois representations don't actually exist. The Galois representation associated to an automorphic rep of $\mathrm{GSp}_{2n}$ should go into the L-group, which is $\mathrm{GSpin}_{2n+1}$. There are exceptional isomorphisms $\mathrm{GSpin}_3 \cong \mathrm{GL}_2$ and $\mathrm{GSpin}_5 \cong \mathrm{GSp}_4$; but there is no such isomorphism for $n \ge 3$. $\endgroup$– David LoefflerCommented Sep 22, 2016 at 9:17
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$\begingroup$ Thank you, David! A silly mistake on my part. Question edited. Cohomological weight is good enough for me $\endgroup$– Eins NullCommented Sep 22, 2016 at 15:13
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1 Answer
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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:
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 $\ell$.
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, or this paper of Jorza. 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. 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.)
answered Sep 22, 2016 at 14:04