Irreducibility of Tate module (as a Galois representation) of elliptic curves with good reduction

Question

This question is following the previous question.

Definitions:

Suppose $F$ is an unramified finite extension of $\mathbb Q_p$ and $E$ is an elliptic curve defined over $F$ with good reduction. Denote $V=\mathbb Q_p\otimes_{\mathbb Z_p}T_p(E)$, and consider $D_{cris}(V)$ (by comparison theorem it's equal to $H_{dR}^1(E)$) the Frobenius on $D_{cris}(V)$ is denoted by $\varphi$.

I have the following deduction about the irreducibility of $D_{cris}(V)$ and I wonder if it's right:

By David Loeffler's answer in that post we know the Frobenius on $D_{cris}(V)$ has characteristic polynomial $f(x)=x^2-ax+p$, and by basic theory of elliptic curves we know "$E$ has supersingular reduction" is equivalent to "$a\equiv 0 \pmod p$". Thus we can easily deduce (using Hensel): $f$ is irreducible over $F$ iff $E$ has supersingular reduction.

There are 2 cases:

1. If $E$ has supersingular reduction, then $D_{cris}(V)$ is clearly irreducible (or, simple) since $\varphi$ doesn't has an eigenvalue in $F$.

2. If $E$ has ordinary reduction then $f$ has 2 roots $\lambda_1,\lambda_2$ in $F$ with valuation $0,1$. By theorem 8.3.6 of CMI notes we know $D_{cris}(V)$ is reducible.

To sum up, $D_{cris}(V)$ is simple iff $E$ has supersingular reduction.

I have 2 questions:

1. Is the deduction right?

2. If it's right, is there a similar property for general Abelian varieties?

Answer 1

Let me upgrade my comment to an answer:

There exist (simple) abelian varieties over $\mathbb{Q}_{p}$ of every dimension $g\geq 2$ with supersingular good reduction and non-semisimple Tate module. Abelian surface examples are constructed in M. Volkov, Abelian surfaces with supersingular good reduction and non-semisimple Tate module, Int. J. Number Theory (2010), no.4, 811–818. Examples are constructed where the special fibre is a product of supersingular elliptic curves, and where the special fibre is a simple supersingular abelian surface.

The strategy is to write down an admissible filtered $\phi$-module $(D,\mathrm{Fil},\varphi)$ over $\mathbb{Q}_{p}$ which is not semisimple, has Hodge-Tate weights $(0,1)$ and totally isotropic $\mathrm{Fil}$ under a non-degenerate pairing, and such that the characteristic polynomial of $\varphi$ is a supersingular $p$-Weil polynomial. The conditions guarantee that there exists an abelian variety $A$ over $\mathbb{Q}_{p}$ with good reduction and $\mathbb{D}_{\mathrm{cris}}(V_{p}(A))=(D,\mathrm{Fil},\varphi)$ by M. Volkov, A class of $p$-adic Galois representations arising from abelian varieties over $\mathbb{Q}_{p}$, Math. Ann. 331 (2005), no.4, 889–923, and one knows that the special fibre is supersingular by Honda-Tate theory.