Abstract
In this thesis we study questions from a branch of mathematics called
arithmetic geometry: the research on number theoretic properties of
geometric objects described by algebraic equations.
The first topic concerns socalled maximal curves. A curve is described by an
algebraic equation in two unknowns. Solutions to the equation are called
points. In our case the curve has only finitely many points. The work of Hasse,
Weil and Serre provides bounds on the number of points. For a maximal curve the
number of points is equal to this upper bound. We study constructions of
maximal curves such that the curve remains the same, but the points are
considered over varying fields.
The second topic is on the Hesse pencil. This is a set of (elliptic) curves
that have nine flex points in common. The set is provided with a geometric
structure. We show that the structure on the flex points determines whether or
not a given elliptic curve belongs to the Hesse pencil.
The last topic is related to a curve introduced by Mestre. We split the
socalled Jacobian variety of this curve in simple factors. In doing so,
we adapt methods by Faltings and by van Wamelen to new situations.
arithmetic geometry: the research on number theoretic properties of
geometric objects described by algebraic equations.
The first topic concerns socalled maximal curves. A curve is described by an
algebraic equation in two unknowns. Solutions to the equation are called
points. In our case the curve has only finitely many points. The work of Hasse,
Weil and Serre provides bounds on the number of points. For a maximal curve the
number of points is equal to this upper bound. We study constructions of
maximal curves such that the curve remains the same, but the points are
considered over varying fields.
The second topic is on the Hesse pencil. This is a set of (elliptic) curves
that have nine flex points in common. The set is provided with a geometric
structure. We show that the structure on the flex points determines whether or
not a given elliptic curve belongs to the Hesse pencil.
The last topic is related to a curve introduced by Mestre. We split the
socalled Jacobian variety of this curve in simple factors. In doing so,
we adapt methods by Faltings and by van Wamelen to new situations.
Original language  English 

Qualification  Doctor of Philosophy 
Awarding Institution 

Supervisors/Advisors 

Award date  5Dec2016 
Place of Publication  [Groningen] 
Publisher  
Print ISBNs  9789036793681 
Electronic ISBNs  9789036793674 
Publication status  Published  2016 