The arithmetic of maximal curves, the Hesse pencil and Mestre curve

Ane Schelte Izaäk Anema

Research output: ThesisThesis fully internal (DIV)

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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 so-called 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
so-called Jacobian variety of this curve in simple factors. In doing so,
we adapt methods by Faltings and by van Wamelen to new situations.
Original languageEnglish
QualificationDoctor of Philosophy
Awarding Institution
  • University of Groningen
Supervisors/Advisors
  • Top, Jakob, Supervisor
  • Dokchitser, T., Assessment committee, External person
  • Siksek, S., Assessment committee, External person
  • Waalkens, Holger, Assessment committee
Award date5-Dec-2016
Place of Publication[Groningen]
Publisher
Print ISBNs978-90-367-9368-1
Electronic ISBNs978-90-367-9367-4
Publication statusPublished - 2016

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