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This is a followup to this question.

Let $p \ge 3$ be prime, and let $V$ be a crystalline 2-dimensional representation of $G_{\mathbb{Q}_p}$ and $T$ a lattice in $V$. I'm going to assume just about every niceness condition on $V$ that I can think of:

  • $V$ is irreducible;

  • $\operatorname{Fil}^0 \mathbb{D}_{\mathrm{cris}}(V)$ is 1-dimensional (so one Hodge-Tate weight of $V$ is $\le 0$ and the other is $> 0$)

  • none of the eigenvalues of Frobenius on $\mathbb{D}_\mathrm{cris}(V)$ are integral powers of $p$;

  • the Hodge filtration of $V$ has length $< (p-1)$, so $T$ corresponds to a strongly divisible $\mathbb{Z}_p$-lattice $\mathbb{D}(T)$ in $\mathbb{D}_{\mathrm{cris}}(V)$ via Fontaine-Laffaille theory.

Let $T$ be a lattice in $V$, and let $\omega$ be a $\mathbb{Z}_p$-basis of the "tangent space" $t_T = \mathbb{D}(T) / \operatorname{Fil}^0 \mathbb{D}(T)$. The Tamagawa number of $T$ over $K_n = \mathbb{Q}_p(\mu_{p^n})$ is given by $$ \operatorname{Tam}^0_{K_n, \omega}(T) = \frac{[H^1_f(K_n, T) : \exp(\mathcal{O}_{K_n} \omega)]}{[\mathbb{D}(T) : (1- \varphi) \mathbb{D}(T)]}$$

where $[ A : B ]$ is a generalised index (so $[ \mathbb{Z}_p : \tfrac{1}{p} \mathbb{Z}_p] = \tfrac{1}{p}$ etc).

Question: Is it true that under the above hypotheses this Tamagawa number is always 1?

I know this is true for all $n$ if $V$ corresponds to an elliptic curve (because the Tamagawa number has an alternative definition in terms of the index of the nonsingular points in the special fibre of the Neron model) and, if I've understood correctly, for $n = 0$ it is true for any $V$ satisfying the hypotheses above (by a theorem of Bloch and Kato).

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I guess $K_n$ should be $\mathbb{Q}_p(\mu_{p^n}$. – Chris Wuthrich May 24 '11 at 11:49
Yes, it should -- thanks Chris. – David Loeffler May 24 '11 at 12:15
I think the answer to your question is yes. Philosophically, it is certainly yes because under your hypotheses, the natural definition of the local algebraic p-adic L-function will indeed interpolate the values of the algebraic p-adic L-function. In order to prove it, the best is simply to reproduce the proof of Bloch and Kato with coefficients, the point being that the trivialization of the local complex producing the local algebraic p-adic L-function comes from the map 1-phi on D and from the short exact sequence defining the exponential map. – Olivier May 24 '11 at 18:38
Because these arguments are a priori equally valid over any extension of Qp, I don't expect any special difficulties in generalizing the proof of Bloch-Kato. So it should be just an exercise. I'll do it, if you want me to. – Olivier May 24 '11 at 18:40
Dear Olivier: I'd be very glad if you would explain how to do this "exercise". The proof of Bloch and Kato relies critically on Fontaine--Laffaille theory, and as such it seems very much non-obvious how to extend this to unramified bases. – David Loeffler May 24 '11 at 18:56
up vote 8 down vote accepted

I think that for $K_n=\mathbf{Q}_p$ what you're looking for is in my (unpublished) paper

see proposition II.2 for instance.

This paper is unpublished because it was rewritten and massively expanded with/by Denis Benois. There is some stuff in the paper with Benois about going up the cyclotomic tower which may help. I'm sorry I don't have time to look in Benois-Berger to see if there's an answer to your full question.

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It seems to me that Berger-Benois corollary 4.4.3 and the paragraph page 66 starting with D'autre part should do the job. In particular, this shows that I was indeed way too optimistic: the question of David turned out to be more or less as hard as the local epsilon conjecture. – Olivier May 26 '11 at 11:40
Thank you Olivier! In the published version, this would be corollary 4.21 and page 674. – Laurent Berger May 26 '11 at 14:00
@Laurent: Not a direct comment on your answer, but just a reaffirmation of the consensus here that you were the best-informed TA ever hired by our department ;-) – Jim Humphreys May 26 '11 at 17:36
Dear Laurent (or Olivier): could you elaborate how these statements prove what we need? It seems that the paper proves conjecture $C_{Iw}$ (which is invariant under twisting). This implies $C_{EP}$ for all twists, but $C_{EP}$ amounts to a statement about the quotient of Tamagawa numbers for $V$ and $V^*(1)$, so this doesn't give a formula for the Tamagawa numbers of $V$ themselves. – David Loeffler May 27 '11 at 10:10
(I can see how knowing $C_{Iw}$ might allow one to read off the Tamagawa number in cases where $H^1_f(V)$ is either zero, or the whole of $H^1$; but when $H^1_f$ is 1-dimensional, how does one proceed?) – David Loeffler May 27 '11 at 10:12

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