Kummer theory for commutative algebraic groups

Tronto, S. (2022)

Doctoral Thesis

This dissertation is a collection of four research articles devoted to
the study of Kummer theory for commutative algebraic groups. In number
theory, Kummer theory refers to the study of field extensions generated
by n-th roots of some base field. Its generalization to commutative
algebraic groups involves fields generated by the division points of a
fixed algebraic group, such as an elliptic curve or a higher dimensional
abelian variety. Of particular interest in this dissertation is the degree
of such field extensions. In the first two chapter, classical results for
elliptic curves are improved by providing explicitly computable bounds and
uniform and explicit bounds over the field of rational numbers. In the
last two chapters a general framework for the study of similar problems
is developed.

• Number theory
• Field extensions
• Kummer theory
• Elliptic curves
• Algebraic curves
• Galois representations
• Torsion fields
• Cyclotomic fields

• All authors: Tronto, S.
• Supervisor: Stevenhagen, P.
• Co-supervisor: Bruin, P.J.; Perucca, A.
• Committee: Fiocco, M.; Luijk, R.M. van; Voight, J.; Wiese, G.; Salgado Guimaraes da Silva, C.
• Qualification: Doctor (dr.)
• Awarding Institution: Mathematical Institute (MI), Faculty of Science, Leiden University
• Date: 2022-09-08