Are there nonobvious cases where equations have finitely many algebraic integer solutions? - MathOverflow most recent 30 from http://mathoverflow.net 2013-05-23T01:39:04Z http://mathoverflow.net/feeds/question/77644 http://www.creativecommons.org/licenses/by-nc/2.5/rdf http://mathoverflow.net/questions/77644/are-there-nonobvious-cases-where-equations-have-finitely-many-algebraic-integer-s Are there nonobvious cases where equations have finitely many algebraic integer solutions? David Speyer 2011-10-10T02:07:28Z 2011-10-10T07:28:01Z <p>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. </p> <p>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?</p> <p>This question is inspired by <a href="http://mathoverflow.net/questions/77604" rel="nofollow">this one</a>; I have no application in mind.</p> http://mathoverflow.net/questions/77644/are-there-nonobvious-cases-where-equations-have-finitely-many-algebraic-integer-s/77670#77670 Answer by ACL for Are there nonobvious cases where equations have finitely many algebraic integer solutions? ACL 2011-10-10T07:28:01Z 2011-10-10T07:28:01Z <p>This is basically true, in view of a density theorem due to Robert Rumely (<em>Arithmetic over the ring of all algebraic integers</em>, J. reine u. angew. Math. <strong>368</strong>, 1986, p. 127-133). It relies on Rumely's capacity theory, and his extension of the theorem of Fekete-Szegö.</p> <p>For a generalization, and an algebraic proof, see also Laurent Moret-Bailly, <em>Groupes de Picard et problèmes de Skolem. II.</em> Annales scientifiques de l'École Normale Supérieure, Sér. 4, 22 no. 2 (1989), p. 181-194. (Numdam, <a href="http://www.numdam.org/item?id=ASENS_1989_4_22_2_181_0" rel="nofollow">http://www.numdam.org/item?id=ASENS_1989_4_22_2_181_0</a>)</p> <p>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...).</p>