Return to Answer

Added: I think the isogeny theorem can also fail in the case of good ordinary reduction, using the same argument. Consider again elliptic curves over $\mathbb Q_p$, now of good ordinary reduction. Let's restrict to those elliptic curves of a fixed reduction $\overline E$ over $\mathbb F_p$. We have $a_p \ne 0$, and let's assume for simplicity that the two roots of $x^2-a_p x + p$ are in $\mathbb Z_p$, so they are $u$, $pv$ with $u$, $v \in \mathbb Z_p^\times$. Then there's a basis of $M$ (the Fontaine-Laffaille object associated to $E[p^\infty]$) such that $\phi = \begin{bmatrix}u & \\ & pv\end{bmatrix}$. The possible choices of $M^1$ are precisely the lines generated by the vector $\begin{bmatrix} px\\ 1\end{bmatrix}$, where $x \in \mathbb Z_p$. The isomorphism class of $M$ (and thus of the Galois representation $T_p E$) only depends on the valuation of $x$, because a diagonal change of basis leaves $\phi$ invariant. But there are uncountably many $x$ of any fixed positive valuation (and thus deformations of $\overline E$ to $E$). By the above argument, these cannot all be isogenous.

Added: I think the isogeny theorem can also fail in the case of good ordinary reduction, using the same argument. Consider again elliptic curves over $\mathbb Q_p$, now of good ordinary reduction. Let's restrict to those elliptic curves of a fixed reduction $\overline E$ over $\mathbb F_p$. We have $a_p \ne 0$, and let's assume for simplicity that the two roots of $x^2-a_p x + p$ are in $\mathbb Z_p$, so they are $u$, $pv$ with $u$, $v \in \mathbb Z_p^\times$. Then there's a basis of $M$ (the Fontaine-Laffaille object associated to $E[p^\infty]$) such that $\phi = \begin{bmatrix}u & \\\ & pv\end{bmatrix}$. The possible choices of $M^1$ are precisely the lines generated by the vector $\begin{bmatrix} px\\\ 1\end{bmatrix}$, where $x \in \mathbb Z_p$. The isomorphism class of $M$ (and thus of the Galois representation $T_p E$) only depends on the valuation of $x$, because a diagonal change of basis leaves $\phi$ invariant. But there are uncountably many $x$ of any fixed positive valuation (and thus deformations of $\overline E$ to $E$). By the above argument, these cannot all be isogenous.

Suppose we have two elliptic curves $E_1$, $E_2$ over $\mathbb Z_p$ as above (i.e., supersingular reduction and $p > 3$). Then we know that $E_1[p^\infty] \cong E_2[p^\infty]$ and that $E_1 \times \mathbb F_p$ is isogenous to $E_2 \times \mathbb F_p$. But in order to apply the theorem of Serre-Tate and deduce the existence of an isogeny $E_1 \to E_2$ we need to know that we can choose a map $\alpha : E_1[p^\infty] \to E_2[p^\infty]$ of $p$-divisible groups over $\mathbb Z_p$ and a map $\beta : E_1 \times \mathbb F_p \cong E_2 \times \mathbb F_p$ of elliptic curves such that $\alpha \times \mathbb F_p = \beta[p^\infty]$ as map of $p$-divisible groups $E_1[p^\infty] \to E_2[p^\infty]$ over $\mathbb F_p$.

[Finally, a small comment/question: I think Tate's result over finite fields assumes that $l$ is prime to the characteristic. Milne-Waterhouse (1971) have a version for $l = p$ where $T_l$ is replaced by the Dieudonné module. In this case $T_p E$ is just the first crystalline cohomology group. Is there a general crystalline analogue of the Tate conjecture?]

Suppose we have two elliptic curves $E_1$, $E_2$ over $\mathbb Z_p$ as above (i.e., supersingular reduction and $p > 3$). Then we know that $E_1[p^\infty] \cong E_2[p^\infty]$ and that $E_1 \times \mathbb F_p$ is isogenous to $E_2 \times \mathbb F_p$. But in order to apply the theorem of Serre-Tate and deduce the existence of an isogeny $E_1 \to E_2$ we need to know that we can choose a map $\alpha : E_1[p^\infty] \to E_2[p^\infty]$ of $p$-divisible groups over $\mathbb Z_p$ and a map $\beta : E_1 \times \mathbb F_p \to E_2 \times \mathbb F_p$ of elliptic curves such that $\alpha \times \mathbb F_p = \beta[p^\infty]$ as map of $p$-divisible groups $E_1[p^\infty] \to E_2[p^\infty]$ over $\mathbb F_p$.

[Finally, a small comment/question: I think Tate's result over finite fields assumes that $l$ is prime to the characteristic. Milne-Waterhouse (1971) have a version for $l = p$ where $T_l$ is replaced by the Dieudonné module. In this case $T_p E$ is just the first crystalline cohomology group. Is there a general crystalline analogue of the Tate conjecture?]

Added: I think the isogeny theorem can also fail in the case of good ordinary reduction, using the same argument. Consider again elliptic curves over $\mathbb Q_p$, now of good ordinary reduction. Let's restrict to those elliptic curves of a fixed reduction $\overline E$ over $\mathbb F_p$. We have $a_p \ne 0$, and let's assume for simplicity that the two roots of $x^2-a_p x + p$ are in $\mathbb Z_p$, so they are $u$, $pv$ with $u$, $v \in \mathbb Z_p^\times$. Then there's a basis of $M$ (the Fontaine-Laffaille object associated to $E[p^\infty]$) such that $\phi = \begin{bmatrix}u & \\ & pv\end{bmatrix}$. The possible choices of $M^1$ are precisely the lines generated by the vector $\begin{bmatrix} px\\ 1\end{bmatrix}$, where $x \in \mathbb Z_p$. The isomorphism class of $M$ (and thus of the Galois representation $T_p E$) only depends on the valuation of $x$, because a diagonal change of basis leaves $\phi$ invariant. But there are uncountably many $x$ of any fixed positive valuation (and thus deformations of $\overline E$ to $E$). By the above argument, these cannot all be isogenous.

The category of $p$-divisible groups over $\mathbb Z_p$ is equivalent, by results of Fontaine-Laffaille, to the category of quadruples $(M,M^1,\phi,\phi^1)$, where $M$ is a finite free $\mathbb Z_p$-module, $M^1$ a $\mathbb Z_p$-direct summand ("Hodge filtration"), and $\phi : M \to M$, $\phi^1 : M^1 \to M$ are $\mathbb Z_p$-linear maps such that $p\phi^1 = \phi|_{M^1}$ and $\phi(M) + \phi^1(M^1) = M$. [In our case $M$ is of rank 2 and $M^1$ of rank 1.] The Dieudonne module of the special fibre of the given $p$-divisible group is recovered as follows (reference?): we need to give a $\mathbb Z_p[F,V]/(FV-p)$-module $D$ that is finite free as $\mathbb Z_p$-module. We take $D := M$, $F := \phi$, and $V := pF^{-1}$ (this is defined on $M[1/p]$ initially, but the final condition on $M$ forces it to preserve $M$). In other words, one forgets the Hodge filtration. So lifting $\beta[p^\infty]$ to $\mathbb Z_p$ is difficult because a given map of Dieudonne modules doesn't need to preserve the Hodge filtration. (The countable flexibility we have with $\beta$ is not enough to move a given line to any other.)

The category of $p$-divisible groups over $\mathbb Z_p$ is equivalent, by results of Fontaine-Laffaille, to the category of quadruples $(M,M^1,\phi,\phi^1)$, where $M$ is a finite free $\mathbb Z_p$-module, $M^1$ a $\mathbb Z_p$-direct summand ("Hodge filtration"), and $\phi : M \to M$, $\phi^1 : M^1 \to M$ are $\mathbb Z_p$-linear maps such that $p\phi^1 = \phi|_{M^1}$ and $\phi(M) + \phi^1(M^1) = M$. [In our case $M$ is of rank 2 and $M^1$ of rank 1.] The Dieudonne module of the special fibre of the given $p$-divisible group is recovered as follows (reference?): we need to give a $\mathbb Z_p[F,V]/(FV-p)$-module $D$ that is finite free as $\mathbb Z_p$-module. We take $D := M$, $F := \phi$, and $V := pF^{-1}$ (this is defined on $M[1/p]$ initially, but the final condition on $M$ forces it to preserve $M$). In other words, one forgets the Hodge filtration. So lifting $\beta[p^\infty]$ to $\mathbb Z_p$ is difficult because a given map of Dieudonne modules doesn't need to preserve the Hodge filtration. (The countable flexibility we have with $\beta$ is not enough to move a given line to any other. Note that $M^1 \otimes \mathbb F_p$ is uniquely determined as the kernel of $\phi \otimes \mathbb F_p$; but apart from that $M^1$ is free to vary.)