2 correction

### 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.

1

# Why no abelian varieties over Z?

### 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

### Question

Why is the set of 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.