The goal is to prove the following theorem.
Theorem 1 (Fontaine). Let $A$ be an abelian variety over $\mathcal{O}_ K$, where $K$ is a $p$-adic field. Let $L$ be the field generated by the coordinates of points of $A[p^n]$. Let $e = v_K(p)$. Then we have $$ u_{L/K} \le e (n + 1/(p-1)). $$
For $E/K$ finite, let us write $$ \mathfrak{m}_ {E/K}^t = \lbrace x \in \mathcal{O}_ E : v_K(x) \ge t \rbrace $$ so that the actual maximal ideal is $\mathfrak{m}_ {E/K}^{1/e_{E/K}}$.
Definition 2. We say that $P_m(L/K)$ is satisfied when for all $E/K$ finite, if there exists a $\mathcal{O}_ K$-algebra homomorphism $\mathcal{O}_ L \to \mathcal{O}_ E / \mathfrak{m}_ {E/K}^m$ then there exists an inclusion $\mathcal{O}_ L \hookrightarrow \mathcal{O}_ E$.
Note that if $L/K$ is unramified, then $P_m(L/K)$ is satisfied for all $m$ by Hensel’s lemma.
Theorem 3. If $m \gt u_{L/K}$ then $P_m(L/K)$ is true. On the other hand, $P_{u_{L/K} - 1/e_{L/K}}(L/K)$ is false.
Remark 4. Yoshida further shows that if $m \lt u_{L/K}$ then $P_m(L/K)$ is false.
To prove this, recall Krasner’s lemma. This says that if $\alpha, \beta \in \bar{K}$ are such that $\lvert \alpha - \beta \rvert \lt \lvert \alpha - \sigma(\alpha) \rvert$ for all $\sigma \in \operatorname{Gal}(\bar{K}/K)$, then $\alpha \in K(\beta)$.
Using this, let us show that if $m \gt u_{L/K}$ then $P_m(L/K)$ is true. Let $P(x) \in \mathcal{O}_ K[x]$ be the minimal polynomial of $\pi_L$ over $K$. Let $\eta \colon \mathcal{O}_ L \to \mathfrak{m}_ {E/K}^t$. Let $\beta \in \mathcal{O}_ E$ be a lift of $\eta(\pi_L)$ so that $$ v_K(P(\beta)) \ge m \gt u_{L/K}. $$ What we need to do is show that $v_K(\beta - \sigma \pi_L) \ge m$ for some $\sigma \in \operatorname{Gal}(L/K)$. Let $\sigma_0 \in \operatorname{Gal}(L/K)$ be such that $\sigma_0 \pi_L$ is closest to $\beta$. Then $$ \beta - \sigma \pi_L = \beta - \sigma_0 \pi_L + \sigma_0(\pi_L - \sigma_0^{-1} \sigma \pi_L) $$ and so $$ v_K(\beta - \sigma \pi_L) = \min(v_K(\beta - \sigma_0 \pi_L), v_K(\pi_L - \sigma_0^{-1} \sigma \pi_L)). $$ Taking the product over all $\sigma$, we get $$ v_K(P(\beta)) = \sum_{\sigma \in \operatorname{Gal}(L/K)} \min(v_K(\beta - \sigma_0 \pi_L), i_{L/K}(\sigma_0^{-1} \sigma)). $$ That is, we have $$ v_K(\beta - \sigma_0 \pi_L) \gt \varphi_{L/K}^{-1}(u_{L/K}) = i_{L/K}. $$ Now can apply Krasner’s lemma to conclude that $$ L \subseteq K(\beta) \subseteq E. $$
We now show that $u_{L/K} - 1/e_{L/K}$ is false. We may as well assume that $L/K$ is totally ramified, because we can replace $K$ with an unramified extension in $L$. If $L/K$ is tamely ramified, we have $u_{L/K}$ and even $P_1(L/K)$ is false because we have $\mathcal{O}_ L \to \mathcal{O}_ K/\mathfrak{m}_ {K/K}^1$ but there is no map $L \to K$.
If $L/K$ is wildly ramified, write $t = u_{L/K} - 1/e_{L/K}$ and write $t = r + s/e_{L/K}$ for $0 \le s \lt e_{L/K}$. Let $P(x) = \mathcal{O}_ K[x]$ be the minimal polynomial of $\pi_L$ over $K$, and now we consider the polynomial $Q(x) = P(x) - \pi_K^r x^s$ which is still Eisenstein. We let $\beta$ be a root of $Q$ and $E = K(\beta)$ which is totally ramified over $K$. Then we have $$ \mathcal{O}_ L \to \mathcal{O}_ E / \mathfrak{m}_ {E/K}^t; \quad \pi_L \mapsto \beta $$ because $P(\beta) = \pi_K \beta^s$ and hence $v_K(P(\beta)) = r + s v_K(\beta) = r + s/e_{L/K} = t$. We now claim that there is no map $L \to E$. If there is, then we would have $L = E$ because they have the same degree, and so $v_K(\sigma \pi_L - \beta) \in e_{L/K}^{-1} \mathbb{Z}$. Now if we consider $$ e_{L/K} \max_{\sigma} v_K(\sigma \pi_L - \beta) = e_{L/K} \varphi_{L/K}^{-1}(v_K(P(\beta))) = e_{L/K} \varphi_{L/K}^{-1}(t) \in \mathbb{Z}. $$ Then we have $$ \varphi_{L/K}^{-1}(u_{L/K} - 1/e_{L/K}) = i_{L/K} - \frac{1}{e_{L/K} \lvert G_{i_{L/K}-\epsilon} \rvert} \in e_{L/K}^{-1} \mathbb{Z}. $$ This is a contradiction.
Proof of Fontaine’s theorem
Last time we saw the following.
Proposition 5. Let $A$ be a finite flat $\mathcal{O}_ K$-algebra of the form $A = \mathcal{O}_ K[[x_1, \dotsc, x_m]] / (f_1, \dotsc, f_m)$. Let $X = \operatorname{Spec} A$ and suppose that there exists an element $0 \neq a \in \mathcal{O}_ K$ such that $a \Omega_{A/\mathcal{O}_ K}^1 = 0$ and $\Omega_{A/\mathcal{O}_ K}^1$ is a free $A/aA$-module. Suppose that $S$ is a finite flat $\mathcal{O}_ K$-algebra and $I \subseteq S$ be a topologically nilpotent pd-ideal. Then a map $A \to S/aI$ lifts to $A \to S$.
Recall that if $A$ is an abelian scheme over $\mathcal{O}_ K$ and $L = K(A(\bar{K})[p^n])$, we want to prove that $u_{L/K} \le e(n+(p-1)^{-1})$. Writing $t = e(n+1/(p-1))$, by Yoshida it suffices to prove the property $P_t(L/K)$.
So what we need to show is that if $E/K$ is finite then $\mathcal{O}_ L \to \mathcal{O}_ E / \mathfrak{m}_ {E/K}^t$ lifts to $\mathcal{O}_ L \to \mathcal{O}_ E$. We will apply the proposition above to $A[p^n]$.
Theorem 6 (Schoof). If $k$ is a perfect field of characteristic $p$ and $G = \operatorname{Spec} A$ is a connected finite flat group scheme, then $A \cong k[x_1, \dotsc, x_n] / (x_1^{p^{e_1}}, \dotsc, x_n^{p^{e_n}})$ as a $k$-algebra.
Corollary 7. If $R$ is a complete discrete valuation ring with perfect residue field and $G = \operatorname{Spec} A$ is a connected finite flat group scheme, then $A \cong R[[x_1, \dotsc, x_n]] / (f_1, \dotsc, f_n)$.
For the torsion points of an abelian variety, we can check that $$ \Omega_{A / \mathcal{O}_ K}^1 = \Omega_{\mathscr{A}/\mathcal{O}_ K}^1 / p^n \Omega_{\mathscr{A}/\mathcal{O}_ K}^1. $$
Lemma 8. The ideal $\mathfrak{m}_ {E/K}^s \subseteq \mathcal{O}_ E$ is a topologically nilpotent pd-ideal if and only if $s \gt e_K / (p-1)$.
For each $L$-point of the generic fiber of $\mathscr{A}[p^n]$, we can now lift $$ A \to \mathcal{O}_ L \to \mathcal{O}_ E / \mathfrak{m}_ {E/K}^t $$ to $A \to \mathcal{O}_ E$. This means that $E$ has to contain $L$.
No abelian schemes over ℤ
A consequence of Fontaine’s theorem is that $$ v_K(\mathcal{D}_ {L/K}) = u_{L/K} - i_{L/K} \le u_{L/K} \le e_K \Bigl( n + \frac{1}{p-1} \Bigr). $$ So now if $\mathscr{A}$ is an abelian scheme over $\mathbb{Z}$ then we have $$ L = Q(\mathscr{A}\lbrack p^n \rbrack(\bar{\mathbb{Q}})) $$ is unramified away from $p$ and moreover $$ \Delta_{L/Q} = N_{L/\mathbb{Q}}(\mathcal{D}_ {L/\mathbb{Q}}) \le (p^{n+1/(p-1)})^{[L:\mathbb{Q}]}. $$
Taking $n = 1$ and $p = 3$, we get a bound on the root discriminant $$ \Delta_{L/\mathbb{Q}}^{1/[L:\mathbb{Q}]} \le 3^{3/2}. $$ This will be the source of the contradiction.