Diophantine problems: inequalities and abelian varieties

by Florian Maximilian Munkelt
Doctoral thesis
Date of Examination:2023-06-09
Date of issue:2023-11-10
Advisor:Prof. Dr. Damaris Schindler
Referee:Prof. Dr. Damaris Schindler
Referee:Prof. Dr. Jörg Brüdern
Referee:Dr. Davide Lombardo
Persistent Address: http://dx.doi.org/10.53846/goediss-10191

 

 

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Abstract

English

In this thesis we consider the density of rational points near manifolds and a bound for the torsion on a simple abelian variety of type IV over a number field. In the first part the main result is an asymptotic for the number of rational points close to a compact parametrized manifold under a significantly weaker curvature condition than previous authors considered. In the second part, which is joined work with Victoria Cantoral-Farfan, we study the torsion subgroup of the Mordell-Weil group of a simple abelian variety of type IV. An optimal exponent is established to give a bound for the order of the torsion subgroup for a finite extension of the base field in terms of the extension degree. As a consequence we obtain a lower bound for the degree of an extension generated by a torsion point.
Keywords: Number theory; Diophantine equations; Diophantine inequalities; Abelian varieties; Uniform Boundedness Conjecture
 
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