Are there nonobvious cases where equations have finitely many algebraic integer solutions?

2011-10-10T02:07:28Z

Let $X$ be a scheme of finite type over $\mathbb{Z}$. Let $R$ be the ring of algebraic integers. My intuition is that $X(R)$ is practically always infinite.

More specifically, suppose that $X$ is faithfully flat over $\mathbb{Z}$, of relative dimension $\geq 1$, and the generic fiber is geometrically irreducible. Is that enough to guarantee infinitely many algebraic integer points?

This question is inspired by this one; I have no application in mind.

Answer by ACL for Are there nonobvious cases where equations have finitely many algebraic integer solutions?

2011-10-10T07:28:01Z

This is basically true, in view of a density theorem due to Robert Rumely (Arithmetic over the ring of all algebraic integers, J. reine u. angew. Math. 368, 1986, p. 127-133). It relies on Rumely's capacity theory, and his extension of the theorem of Fekete-Szegö.

For a generalization, and an algebraic proof, see also Laurent Moret-Bailly, Groupes de Picard et problèmes de Skolem. II. Annales scientifiques de l'École Normale Supérieure, Sér. 4, 22 no. 2 (1989), p. 181-194. (Numdam, http://www.numdam.org/item?id=ASENS_1989_4_22_2_181_0)

The hypothesis of Moret-Bailly's Theorem is that $X$ be irreducible, surjective and of positive relative dimension over ${\rm Spec}\mathbf Z$, and that its generic fiber be geometrically irreducible. Then, he proves that $X$ has $\overline{\mathbf Z}$-points which can be chosen arbitrarily close to a given $p$-adic point (end even more...).

Are there nonobvious cases where equations have finitely many algebraic integer solutions? - MathOverflow

Are there nonobvious cases where equations have finitely many algebraic integer solutions?

Answer by ACL for Are there nonobvious cases where equations have finitely many algebraic integer solutions?