It's a result related in spirit to Minkowski's theorem that $\mathbb Q$ admits no non-trivial unramified extensions. If $A$ is an abelian variety over $\mathbb Q$ with everywhere good reduction, then for any integer $n$ the $n$-torsion scheme $A[n]$ is a finite flat group scheme over $\mathbb Z$. Although this group scheme will be ramified at primes $p$ dividing $n$, Fontaine's theory shows that the ramification is of a rather mild type: so mild, that
a non-trivial such family of $A[n]$ can't exist.
In the last 25 years, there has been much research on related questions, including by
Brumer--Kramer, Schoof, and F. Calegari, among others. (One particularly interesting recent variation is a joint paper of F. Calegari and Dunfield in which they use related ideas to construct a tower of closed hyperbolic 3-manifolds that are rational homology spheres, but whose injectivity radii grow without bound.)
EDIT: I should add that the case of elliptic curves is older, due to Tate I believe,
and uses a different argument: he considers the equation computing the discriminant of
a cubic polynomial $f(x)$ (corresponding to the elliptic curve $y^2 = f(x)$) and shows
that this solution equation has no integral solutions giving a discriminant of $\pm 1$.
This direction of argument generalizes in different ways, but is related to a result of Shafarevic (I think) proving that there are only finitely many elliptic curves with good reduction outside a finite set of primes. (A result which was generalized by Faltings to abelian varieties as part of his proof of Mordell's conjecture.)
Finally, one could add that in Faltings's argument, he also relied crucially on ramification results for $p$-divisible groups, due also to Tate, I think, results which Fontaine's theory generalizes. So one sees that the study of ramification of finite flat groups schemes and $p$-divisible groups (and more generally Fontaine's $p$-adic Hodge theory) plays a crucial role in these sorts of Diophantine questions. A colleague describes it as the ``black magic'' that makes all Diophantine arguments (including Wiles' proof of FLT as well) work.
P.S. It might be useful to give a toy illustrative example of how finite flat group schemes give rise to mildly ramified extensions: consider all the quadratic extensions of $\mathbb Q$ ramified only at $2$: they are ${\mathbb Q}(\sqrt{-1}),$ ${\mathbb Q}(\sqrt{2})$, and ${\mathbb Q}(\sqrt{-2})$, with discriminants $-4$, $8$, and $-8$ respectively. Thus ${\mathbb Q}(\sqrt{-1})$ is the least ramified, and not coincidentally, it is the splitting field of the finite flat group scheme $\mu_4$ of 4th roots of unity.
In fact Shimura handled the case of an abelian variety $A$ with complex multiplication by an order $O$ inside the maximal order $O_K$ of the CM field $K$.
A very good modern reference is the following article by Marco Streng:
An explicit version of Shimura's reciprocity law for Siegel modular functions.
I also recommend his PhD thesis.
Both are available here:
http://pub.math.leidenuniv.nl/~strengtc/research.html
Now for the statement (see Theorem 2.2 of the paper above):
let $\tau$ be the element of the Siegel space that corresponds to the abelian variety $A$ above. The Siegel modular functions of level $N$ with $q$-expansion in $\mathbb{Q}(\zeta_N)$ evaluated at $\tau$ generate an abelian extension $H(N)$ of the reflex field $K^r$ (for the type norm $\Phi$ associated to $A$).
This abelian extension is associated by class field theory to the class group
$I_K(NF)/H_{\Phi,O}(N)$ where
- $F$ is the conductor of $O$
- $I_K(NF)$ are the fractional ideals of $K$ prime to $NF$
- $H_{\Phi,O}(N)$ are the ideals $\mathfrak{a} \in I_K(NF)$ such that $\exists \mu \in K$ with $N_{\Phi^r,O}(\mathfrak{a})=\mu O$, $\mu \overline{\mu}=N(\mathfrak{a})\in \mathbb{Q}$ and $\mu \equiv 1 {\bmod^\times} NO$.
(Here $N_{\Phi^r,O}$ is the type norm from the reflex field $K^r$ to $K$.)
Now specializing to elliptic curves, we get that
- If $E$ has CM by $O_K$, then $K(j(E),h(E[𝔪]))$ is the ray class field associated to the class group $I_K(m)/\{ \mu O_K \mid \mu \equiv 1 {\bmod^\times} m\}$
- If $E$ has CM by $O$ where the conductor of $O$ is $F \in \mathbb{Z}$, then $K(j(E))$ is the ring class field of $O$, meaning the extension
associated to the class group $I_K(F)/\{ \text{principal ideals of $O$ primes to $F$}\}$. And finally, $K(j(E),h(E[𝔪]))$ will correspond to
$I_K(mF)/\{ \mu O \mid \text{$\mu$ is prime to $F$ and}\ \mu \equiv 1 {\bmod^\times} mO\}$.
(At least we have the inclusion, but I am pretty sure that in the elliptic curve case, the evaluation of Weber functions at the points of $m$-torsion over the $j$-invariant give the field generated by level-$m$ modular forms evaluated at $\tau$.) So in this case we have something intermediate between a ring class field and a ray class field. [I have not checked carefully but it should be the compositium of the ring class field of conductor $F$ and the ray class field of modulus $m$ when $m$ is prime to $F$].
Best Answer
Here is an example.
Let $K = \mathbf{Q}(\sqrt{-21})$. Then the class group of $K$ is $C_2 \times C_2$ and its Hilbert class field is $H_K = \mathbf{Q}(\sqrt{-1}, \sqrt{3}, \sqrt{7})$. In particular, $H_K$ is a CM-field and its maximal totally real subfield $H_K^+$ is $\mathbf{Q}(\sqrt{3}, \sqrt{7})$.
The LMFDB database has an elliptic curve 4.4.7056.1-1.1.a1 over $H_K^+$ which has everywhere good reduction and has CM by $\mathcal{O}_K$. Base-extending this from $H_K^+$ to $H_K$ gives the example you seek.
Similar examples should exist whenever $H_K$ has class number 1 and all units of $H_K$ are in the kernel of the norm map to $K$. (The class number condition may well not be needed, but the condition on the units certainly is). I have no idea if there are infinitely many such fields $K$, but there definitely some! This happens for $\mathbf{Q}(\sqrt{-d})$ for $d = 21, 33, 42, 57, 66, 77, 93$ (and no others for $d < 100$).