• algebraic tori
• arithmetic geometry
• Manin-Mumford
• torsion points

The Manin-Mumford conjecture states that if X is a proper, integral curve over C of geometric genus at least 2, embedded in a C-abelian variety A, then X contains at most finitely many C-torsion points of A.
The purpose of this thesis is to adapt Raynaud’s proof [Inventiones Mathematicae, n.71 (1983), pp. 207-233] of the Manin-Mumford conjecture to the toric case. We define a torsion coset of a group to be a left translate of a subgroup by a torsion element.
We will prove the following: if X is a smooth closed algebraic curve in (C^*)^n and none of its connected components is a torsion coset, then X contains at most finitely many torsion points of (C^*)^n .
Many proofs of this result are known, but, as far as we know, none of them
relies on Raynaud’s approach.