Abelian varieties and decidability in number theory
Open Access
- Author:
- Springer, Caleb
- Graduate Program:
- Mathematics
- Degree:
- Doctor of Philosophy
- Document Type:
- Dissertation
- Date of Defense:
- April 12, 2021
- Committee Members:
- Martin Fürer, Outside Unit & Field Member
Mihran Papikian, Major Field Member
Kirsten Eisentraeger, Chair & Dissertation Advisor
Carina Curto, Program Head/Chair
Linda Westrick, Major Field Member - Keywords:
- finite field
abelian variety
arithmetic geometry
number theory
decidability
definability
algorithm - Abstract:
- This dissertation consists of two parts, both of which are focused upon problems and techniques from algebraic number theory and arithmetic geometry. In the first part, we consider abelian varieties defined over finite fields, which are a key object in cryptography in addition to being inherently intriguing in their own right. First, generalizing a theorem of Lenstra for elliptic curves, we present an explicit description of the group of rational points of a simple abelian variety over a finite field as a module over its endomorphism ring, under some technical conditions. Next, we present an algorithm for computing endomorphism rings in the case of ordinary abelian varieties of dimension 2, again under certain conditions, building on the work of Bisson and Sutherland. We prove the algorithm has subexponential running time by exploiting ideal class groups and class field theory. In the second part, we turn our attention to questions of decidability and definability for algebraic extensions of the rational numbers, in the vein of Hilbert’s Tenth Problem and its generalizations. First, we show that a key technique for proving undecidability results fails for “most” subfields of the algebraic closure of the rational numbers. More specifically, we view the set of subfields of the algebraic closure of the rational numbers as a topological space, and prove there is a meager subset containing all subfields for which the ring of integers is existentially or universally definable. Finally, we present explicit families of infinite algebraic extensions of the rational numbers whose first-order theory is undecidable. This is achieved by leveraging the unit groups of totally imaginary quadratic extensions of totally real fields, building on the work of Martínez-Ranero, Utreras and Videla.