#### Date of Award

Spring 5-22-2021

#### Degree Type

Dissertation

#### Degree Name

Doctor of Philosophy (PhD)

#### Department

Mathematics

#### Advisor(s)

Steven Diaz

#### Keywords

elliptic curves, Galois field, nonic field, odd degree, torsion subgroups

#### Subject Categories

Physical Sciences and Mathematics

#### Abstract

The Mordell-Weil Theorem states that if K is a number field and E/K is an elliptic curve that the group of K-rational points E(K) is a finitely generated abelian group, i.e. E(K) = Z^{r_K} ⊕ E(K)_tors, where r_K is the rank of E and E(K)_tors is the subgroup of torsion points on E. Unfortunately, very little is known about the rank r_K. Even in the case of K = Q, it is not known which ranks are possible or if the ranks are bounded. However, there have been great strides in determining the sets E(K)_tors. Progress began in 1977 with Mazur’s classification of the possible torsion subgroups E(Q)_tors for rational elliptic curves, and there has since been an explosion of classifications.

Inspired by work of Chou, González Jiménez, Lozano-Robledo, and Najman, the purpose of this work is to classify the set Φ_Q^{Gal}(9), i.e. the set of possible torsion subgroups for rational elliptic curves over nonic Galois fields. We not only completely determine the set Φ_Q^{Gal}(9), but we also determine the possible torsion subgroups based on the isomorphism type of Gal(K/Q). We then determine the possibilities for the growth of torsion from E(Q)_tors to E(K)_tors, i.e. what the possibilities are for E(K)_tors ⊇ E(Q)_tors given a fixed torsion subgroup E(Q)_tors. Extending the techniques used in the classification of Φ_Q^{Gal}(9), we then determine the possible structures over all odd degree Galois fields. Finally, we explicitly determine the sets Φ_Q^{Gal}(d) for all odd d based on the prime factorization for d while proving a number of other related results.

#### Access

Open Access

#### Recommended Citation

Mcwhorter, Caleb, "Torsion Subgroups Of Rational Elliptic Curves Over Odd Degree Galois Fields" (2021). *Dissertations - ALL*. 1322.

https://surface.syr.edu/etd/1322