Clay Mathematics Institute

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Relative Manin-Mumford in additive extensions

Harry Schmidt (University of Basel)

Abstract: We will discuss the relative Manin-Mumford conjecture for families of two dimensional commutative algebraic groups. These will depend on one complex parameter λ and we are especially interested in the case of an additive extension of the Legendre family Eλ. We then have an exact sequence 0 → Ga Gλ → Eλ → 0 where Ga is the additive group (C, +). In this context the relative Manin-Mumford conjecture states that the intersection of a curve in Gλ with the set of torsion points is at most finite unless it is contained in a smaller family of algebraic subgroups in Gλ. It is possible to prove this by following the strategy employed by Masser and Zannier in their proof of the relative Manin-Mumford conjecture for the product of two Legendre families.