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The general philosophy of this talk is to provide a link between the regular inverse Galois theory- in particular modular towers theory- and the theory of abelian varieties. Fix a prime number p, a p-perfect finite group G and a r-tuple C of p’-conjugacy classes of G. From this data, M. Fried constructs in a canonical way atower of reduced Hurwitz spaces called the modular tower associated with (G, p, C). When G is the dihedral group D2p and C is four copies of the conjugacy class of involutions in G, the resulting modular tower is the usual tower of modular curves (Y1(p n+1) → Y1(p n))n≥1. Fried’s conjectures generalize the theorems of Manin, Mazur and Merel for the tower of modular curves to any modular towers. I will begin by constructing a variant of Fried’s modular towers I called abelianized modular towers. Abelianized modular towers are finite quotients of Fried’s modular towers and their arithmetic properties are strongly connected with torsion on abelian varieties via class field theory for function fields. In particular, the conjectural generalization of Merel’s theorem for abelian varieties of fixed dimension g implies the disappearance of rational points o

Year: 2006

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