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

  • Pietro Corvaja

    Università di Udine, Italy
  • David Masser

    Universität Basel, Switzerland
  • Umberto Zannier

    Scuola Normale Superiore, Pisa, Italy

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 GG of dimension 22. This context predicts finiteness for the set of torsion points in an algebraic curve inside GG, unless the curve is 'special', i.e. a translate of an algebraic subgroup of GG. Here we shall consider not merely the set of torsion points, but its topological closure in GG (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 GG; 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 Ga\mathbb G_{\rm 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.

Cite this article

Pietro Corvaja, David Masser, Umberto Zannier, Sharpening 'Manin-Mumford' for certain algebraic groups in dimension 2. Enseign. Math. 59 (2013), no. 3/4, pp. 225–269

DOI 10.4171/LEM/59-3-2