Sharpening 'Manin-Mumford' for certain algebraic groups in dimension 2

Abstract

The present paper arises from the extensions of the Manin-Mumford conjecture, where we shall focus on the case of (complex connected) commutative algebraic groups $G$ of dimension $2$. This context predicts finiteness for the set of torsion points in an algebraic curve inside $G$, unless the curve is 'special', i.e. a translate of an algebraic subgroup of $G$. Here we shall consider not merely the set of torsion points, but its topological closure in $G$ (which equals the maximal compact subgroup). In the case of abelian varieties this closure is the whole space, but this is not so for other groups $G$; actually, we shall prove that in certain cases (where a natural dimensional condition is fulfilled) the intersection of this larger set with a non-special curve must still be a finite set. Beyond this, in the paper we shall briefly review some of the basic algebraic theory of group extensions of an elliptic curve by the additive group ${\mathbb{G}}_{\mathrm{a}}$, which are especially relevant in the said result. We shall conclude by stating some general questions in the same direction and discussing some simple examples. The paper concludes with the reproduction of a letter of Serre (whom we thank for his permission) to the second author, explaining how to obtain explicit projective embeddings of the said group extensions.