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Two problems in arithmetic geometry. Explicit Manin-Mumford, and arithmetic Bernstein-Kusnirenko

Abstract : In the first part of this thesis we present sharp bounds on the number of maximal torsion cosets in a subvariety of a complex algebraic torus (\mathbb{C}^{\times})^n and of an Abelian variety. In both cases, we give an explicit bound in terms of the degree of the defining polynomials and the ambient variety. Moreover, the dependence on the degree of the polynomials is sharp. In the case of the complex torus, we also give an effective bound in terms of the toric degree of the subvariety. As a consequence of the latter result, we prove the conjectures of Ruppert, and Aliev and Smyth on the number of isolated torsion points of a hypersurface. These conjectures bound this number in terms of the multidegree and the volume of the Newton polytope of a polynomial defining the hypersurface, respectively.In the second part of the thesis, we present an upper bound for the height of isolated zeros, in the torus, of a system of Laurent polynomials over an adelic field satisfying the product formula. This upper bound is expressed in terms of the mixed integrals of the local roof functions associated to the chosen height function and to the system of Laurent polynomials. We also show that this bound is close to optimal in some families of examples. This result is an arithmetic analogue of the classical Bern\v{s}tein-Ku\v{s}nirenko theorem.
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Submitted on : Friday, December 15, 2017 - 4:52:58 PM
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César Martinez Metzmeier. Two problems in arithmetic geometry. Explicit Manin-Mumford, and arithmetic Bernstein-Kusnirenko. Algebraic Geometry [math.AG]. Normandie Université, 2017. English. ⟨NNT : 2017NORMC224⟩. ⟨tel-01665314⟩

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