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I want to determine $\mathrm D_{\mathrm{cris}}$ of certain twists of the Galois representations attached to modular forms. For one particular twist it is not clear to me how $\mathrm D_{\mathrm{cris}}$ looks like.

Let $f\in\mathrm S_k(\Gamma_1(N),\psi)$ be a newform of weight $k\ge2$, level $N$, nebentype $\psi$, and assume $p$ is a prime with $p\mid N$ such that $f$ is ordinary at $p$ and the $p$-part of $\psi$ is nontrivial. Let $V_f$ be the representation attached to $f$. Then $$V_f|_{\mathrm G_{\mathbb Q_p}}\cong\begin{pmatrix}\delta&*\\&\varepsilon\end{pmatrix}$$ with characters $\delta$ and $\varepsilon$, and $\delta$ is unramified. To be precise, I want the representation $V_f$ to be characterized by characteristic polynomials of geometric Frobenii, while I normalize class field theory arithmetically. In particular, $V_f$ has determinant $\psi^{-1}\kappa^{1-k}$, where $\kappa$ is the cyclotomic character.

If we twist with $\psi\kappa^n$ ($1\le n\le k-1$), tensor with $\mathrm B_{\mathrm{cris}}$ and take cohomology we get a long exact sequence \begin{align*} 0&\rightarrow\mathrm D_{\mathrm{cris}}(\delta\psi\kappa^n)\rightarrow\mathrm D_{\mathrm{cris}}(V_f(\psi)(n))\rightarrow\mathrm D_{\mathrm{cris}}(\varepsilon\psi\kappa^n)\\ &\rightarrow\mathrm H^1(\mathbb Q_p,\mathrm B_{\mathrm{cris}}\otimes\delta\psi\kappa^n)\rightarrow\mathrm H^1(\mathbb Q_p,\mathrm B_{\mathrm{cris}}\otimes V_f(\psi)(n))\rightarrow\dotsm \end{align*} and we know:

  • $\mathrm D_{\mathrm{cris}}(\delta\psi\kappa^n)=0$ since $\delta$ is unramified and $\psi$ is ramified,
  • $\dim\mathrm D_{\mathrm{cris}}(\varepsilon\psi\kappa^n)=1$ since $\varepsilon|_{I_p}=(\delta\varepsilon)|_{I_p}=(\psi^{-1}\kappa^{1-k})|_{I_p}$, so $\varepsilon\psi$ is unramified.

I want to determine $\mathrm D_{\mathrm{cris}}(V_f(\psi)(n))$. Clearly, if $f$ is a CM form, then $V_f$ decomposes, and it is easy to see that $\dim\mathrm D_{\mathrm{cris}}(V_f(\psi)(n))=1$ in this case. But I guess there are also cases where the space vanishes. Are there any general results about this? Maybe these cases can be characterized?

More specifically, assume that we know in addition that $\mathrm H^i_{\mathrm f}(\mathbb Q,V)=\mathrm H^i_{\mathrm f}(\mathbb Q,V^*(1))=0$ for $V=V_f(\psi)(n)$ and $i=0,1$. Can we say something more under this assumption? My hope would be that $\mathrm D_{\mathrm{cris}}(V_f(\psi)(n))$ vanishes in these cases.

Here $\mathrm H^i_{\mathrm f}$ should be as defined by Bloch and Kato, or as in Fukaya-Kato's paper "A formulation of conjectures on $p$-adic zeta functions in non-commutative Iwasawa theory", §2.4.2. In fact, my question arises from calculating certain characteristic polynomials as in 4.2.21 (iii) in this article and the assumption on $\mathrm H^i_{\mathrm f}$ comes from (i) there; the situation described above is the only one in which it is not clear to me how to calculate these.

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  • $\begingroup$ What do you mean by ordinary in this context? $\endgroup$ – Will Sawin May 31 '17 at 17:57
  • $\begingroup$ I should have fixed a place $\mathfrak p$ lying over $p$ of the number field generated by the Fourier coefficients of $f$; $V_f$ should then be the $\mathfrak p$-adic representation attached to $f$ and that $f$ is ordinary should mean that its $p$-th Hecke eigenvalue is a $\mathfrak p$-adic unit. Then $V_f$ is of the upper triangular form with $\delta$ and $\varepsilon$ as described above. $\endgroup$ – Michael Fütterer May 31 '17 at 19:51
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The isomorphism class of the $G_{\mathbf{Q}_p}$-representation $V_f$ determines (up to scaling) a class in $H^1(\mathbf{Q}_p, \delta \epsilon^{-1})$. The condition that $\mathbf{D}_{\mathrm{cris}}(V_f(\psi)(n))$ is 1-dimensional is exactly requiring that this extension is in Bloch--Kato's $H^1_{\mathrm{f}}$.

Now, you know that $V_f$ is de Rham (because it comes from a modular form) so the extension class automatically lies in $H^1_{\mathrm{g}}$. However, there are formulae for the dimensions of $H^1_{\mathrm{f}}$ and $H^1_{\mathrm{g}}$, and in particular the two are almost always equal. For instance, I'm pretty sure that $H^1_{\mathrm{f}}$ and $H^1_{\mathrm{g}}$ will always coincide for 1-dimensional representations that are not crystalline, since the error terms are all dimensions of subquotients of $D_{\mathrm{cris}}$. So this shows that $\mathbf{D}_{\mathrm{cris}}(V_f(\psi)(n))$ is 1-dimensional in your setting.

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