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There is a folklore conjecture that $\operatorname{Ext}^2$ vanishes in the category of geometric $p$-adic Galois representations (i.e. representations that are unramified almost everywhere and de Rham at places dividing $p$) over a number field. See for example Conjecture II.3.2.2 of Fontaine--Perin-Riou's Autour des conjectures de Bloch et Kato. This is an avatar in Galois representations of Beilinson's conjecture that the category of mixed motives over a finitely-generated field $k$ has cohomological dimension at most the Kronecker dimension of $k$ (c.f. Remarks 4.12(c) in Uwe Jannsen, Motivic Sheaves and Filtrations on Chow Groups, Proceedings of Symposia in Pure Mathematics 55 (1994), Part 1, pp. 245–302, doi:10.1090/pspum/055.1, pdf).

What is the current state of the art on this conjecture? I was hoping that the experts here could piece together a complete picture of what's currently known.

Part 2 of Theorems A and B of

  • Patrick B. Allen, Deformations of polarized automorphic Galois representations and adjoint Selmer groups, Duke Math. J. 165, no. 13 (2016), 2407-2460, doi:10.1215/00127094-3477342, arXiv:1411.7661,

building on earlier work by Flach and Wiles on modularity lifting, proves specific cases of $\operatorname{Ext}^2$ in the category of $p$-adic Galois representations unramified outside a finite set of places, but with no $p$-adic Hodge theory condition. However, as emphasized to me by David Loeffler, this does NOT prove the corresponding statement in the category of geometric Galois representations.

I would surmise that if there are results in this directions, they would be proven by Galois deformations or Iwasawa theory. In the latter case, I wonder if it might require computationally checking that a certain $p$-adic $L$-value does not vanish in order to prove the conjecture for a specific Galois representation.

David Roberts
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David Corwin
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  • Fontaine--Perrin-Riou is kind of hard to find without a subscription (and my institution doesn't have one), do you know an open-access reference for this conjecture? – David Loeffler Feb 12 '21 at 18:52
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    I'll look for one. For now, check your email. – David Corwin Feb 12 '21 at 19:25
  • Allen proves the vanishing of $\mathrm{Ext}^2$ only for a very particular class of representations. Also, although Flach had proved a special case earlier, the application of modularity lifting to this question is due to Wiles, not Flach. – naf Feb 13 '21 at 03:14
  • @naf: You are correct about the attribution. However, as is now clarified in the question, all these results of Flach, Wiles, Allen, etc are answering a different question from the one David is currently asking. I am not aware of any case at all where the "intrinsic" Ext group in the category of geometric Galois reps has been computed. – David Loeffler Feb 13 '21 at 16:17
  • @DavidLoeffler: The fact that what was proved by Flach, Wiles, Allen, etc is different from the conjecture of Fontaine and Perrin-Riou was clear in the first version of the question, but the way the question is phrased is still misleading: it seems to suggest that vanishing of $\mathrm{Ext}^2$ is known in full generality in the category of all $p$-adic representations, whereas what is proved is the vanishing of $H^2$ for a very special class of represntations. – naf Feb 14 '21 at 03:49
  • @naf Good point. I tried to make it clearer. – David Corwin Feb 15 '21 at 04:04
  • Maybe it's worth pointing out that $\mathrm{Ext}^2$ does not in fact always vanish in the category of all Galois representations (unramified outside a fixed finite set of places). When $H^2$ should vanish is predicted by a conjecture of Jannsen (Conjecture 1 in "On the l-adic cohomology of varieties over number fields and its Galois cohomology"); he only considers representations arising from smooth projective varieties, but his conjecture can be reformulated to apply to all (pure) geometric Galois representations. – naf Feb 15 '21 at 10:40

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