Abelian varieties with good reduction everywhere over function fields

Question

There is a famous theorem due to J.-M. Fontaine, Il n'y a pas de variété abélienne sur Z (and independently by V.A. Abrashkin) that there are no abelian varieties over Z. I was wondering whether there is a function field analog of this result. More precisely: Let $k$ be a finite field. Is it true that there are no non-isotrivial abelian varieties over $\mathbb{P}^1_{k}$ with good reduction everywhere?

The case of elliptic curves is elementary. There are related results for families of curves but my question really focuses on the case of higher dimensional abelian varieties over $k(T)$.

Answer 1

There are non-isotrivial families of supersingular abelian varieties of dimension $g$ over $\mathbb P^1_{\overline {\mathbb F_p}}$ if $g\geq 2$; see

Goren, E. Z.(3-MGL); Oort, F. Stratifications of Hilbert modular varieties. (English summary) J. Algebraic Geom. 9 (2000), no. 1, 111–154.

There are many other papers of Goren and Oort with explicit constructions.

One could now ask whether there exist non-isotrivial families of non-supersingular abelian varieties over the projective line. I can make one remark about this.

A result of Moret-Bailly shows that the locus of ordinary abelian varieties is quasi-affine. In particular, any non-isotrivial family of abelian varieties over $\mathbb P^1$ contains a non-ordinary fibre. You can find this result in

Moret-Bailly, Asterisque 129, Pinceaux de varieties abeliennes, p. 237 , Theoreme XI.5.2