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 "$a\equiv 0 \pmod p$".

By this we can easily deduce (using Hensel's lemma): $f$ is irreducible over $F$ iff $E$ has supersingular reduction. We then disguss 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, we know $D_{cris}(V)$ 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?

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?

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 "$E$ is supersingular" is equivalent to "$a\equiv 0 \pmod p$".

By this we can easily deduce (using Hensel's lemma): $f$ is irreducible over $F$ iff $E$ is supersingular. We then disguss 2 cases:

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

2. If $E$ is ordinary 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, we know $D_{cris}(V)$ is simple iff $E$ is supersingular.

I have 2 questions:

1. Is the deduction right?

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

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 "$a\equiv 0 \pmod p$".

By this we can easily deduce (using Hensel's lemma): $f$ is irreducible over $F$ iff $E$ has supersingular reduction. We then disguss 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, we know $D_{cris}(V)$ 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?