TITLE: Why no abelian varieties over Z?
QUESTION [31 upvotes]: Motivation
I learned about this question from a wonderful article Rational points on curves by Henri Darmon. He gives a list of statements (some are theorems, some conjectures) of the form 

the set $\{$ objects $\dots$ over field $K$ with good reduction everywhere except set $S$ $\}$ is finite/empty

One interesting thing he mentions is about abelian schemes in the most natural case $K = \mathbb Q$, $S$ empty. I think according to the definition we have a trivial example of relative dimension 0.
Question

Why is the set of non-trivial abelian schemes over $\mathop{\text{Spec}}\mathbb Z$ empty?

Reference
This is proven in Il n'y a pas de variété abélienne sur Z by Fontaine, but I'm asking because: (1) Springer requires subscription, (2) there could be new ideas after 25 years, (3) the text is French and could be hard to read (4) this knowledge is worth disseminating.

REPLY [12 votes]: Here's another motivation for believing that there is no abelian variety over Z: the existence of such a beast would imply that the "motivic" expectations concerning L-functions are false. More precisely, it is not very difficult to show (Mestre does this in a paper in Compositio in 1986 using explicit formulas) that if an abelian variety over the rationals of dimension g at least 1 has the property that its L-function is entire with the expected functional equation, then its conductor is at least 10^{g}, and in particular is >1.  
(There's a very short proof of a weaker fact, sufficient to "imply" the theorem of Fontaine and Abrashkin, in Th. 5.51 of my book with Iwaniec, though the printed version has an unfortunate mistake, and -- I must apologize here for not checking the history at the time -- we did not mention Abrashkin...)