A gap theorem for abelian varieties over differential fields
Let \(\mathcal F\) be a differential field of characteristic zero with derivation \(\delta\) and field of constants \(\mathcal C\). Assume that \(\mathcal F\) is differentially closed. Let \(\mathcal F\{y_1, \dots, y_N\}\) denote the ring of differential polynomials over \(\mathcal F\) (ordinary polynomials in indeterminates \(\delta^iy_j\) which are differentiated as the notation implies). A subset \(\Sigma\) of the affine space \(\mathcal F^N\) is closed in the \(\delta\) topology if it is the set of common zeros of a subset of \(\mathcal F\{y_1, \dots, y_N\}\). If \(\Sigma\) is irreducible and if \(I(\Sigma)\) is the ideal of differential polynomials vanishing on it, the absolute dimension \(a(\Sigma)\) is the transcendence degree of the quotient field of \(\mathcal F\{y_1, \dots, y_N\}/I(\Sigma)\). The notions of \(\delta\)-closed subset and absolute dimension generalize in an obvious way to arbitrary algebraic varieties over \(\mathcal F\). The main result of this paper is the following theorem: Let \(A\) be an abelian variety over \(\mathcal F\) of dimension \(g\) which contains no abelian subvariety that descends to \(\mathcal C\). Let \(\Gamma \subset A\) be a \(\delta\)-closed subgroup of finite absolute dimension. Then any \(\delta\)-closed subset \(\Sigma\) of \(\Gamma\) Zariski dense in \(A\) has absolute dimension at least \(g+1\). Let \(A^\#\) denote the \(\delta\)-closure of the torsion points of \(A\). As a consequence of the above theorem, and previous work of the first author, the authors obtain the following results of \textit{E. Hrushovski} and \textit{Z. Sokolovic} [``Minimal subsets of differential closed fields'', Trans. Am. Math. Soc. (to appear)]. Suppose \(A\) is simple and does not descend to \(\mathcal C\). Then: Any proper \(\delta\)-closed subset of \(A^\#\) is finite; any \(\delta\)-closed subset of \(A^\# \times A^\#\) is a finite union of translates of \(\delta\)-closed subgroups; if \(B\) is another simple abelian variety over \(\mathcal F\) which does not descend to \(\mathcal C\) and is not isogenous to \(A\) then any proper \(\delta\)-closed subset of \(A^\# \times B^\#\) is a finite union of points or points times one of the factors. The proof of the main result depends on the first author's earlier work on \(D\) schemes, and in particular on an analysis of the universal \(D\) schemes associated to \(\Gamma\) and \(\Sigma\).
simple abelian variety over differential field
scientific article; zbMATH DE number 1053504