Naslov
Torsion of elliptic curves with rational j-invariant over number fields

Autori
Gužvić, Tomislav

Vrsta, podvrsta i kategorija rada
Ocjenski radovi, doktorska disertacija

Fakultet
Prirodoslovno-matematički fakultet - Matematički odsjek

Mjesto
Zagreb

Datum
19.02

Godina
2021

Stranica
126

Mentor
Najman, Filip

Ključne riječi
torsion structures ; elliptic curves ; number fields

Sažetak
In this thesis we will classify the possible torsion structures of elliptic curves with rational j-invariant defined over number fields. We start with elliptic curves defined over Q. Let K be a sextic number field. We determine all the possibilities G for E(K)tors and we prove that for each such possible group G, with the exception of the group C3⨁C18, that there exist an elliptic curve E/Q and a sextic number field K such that E(K)tors≅G. Additionally, we provide a partial result regarding the group C3⨁C18. For a positive integer d, define Φ(d) to be the set of possible isomorphism classes of groups E(K)tors, where K runs through all number fields K of degree d and E runs through all elliptic curves over K. For a positive integer d, define ΦQ(d) to be the set of possible isomorphism classes of groups E(K)tors, where K runs through all number fields K of degree d and E runs through all elliptic curves over Q. Define Φj∈Q(d) to be the set of possible isomorphism classes of groups E(K)tors, where K runs through all number fields K of degree d and E runs through all elliptic curves over K with j(E)∈Q. With the help of the previously mentioned result, we are able to completely determine the sets Φj∈Q(p), where p is a prime number. More precisely, our result is the following. Let K be a number field such that [K:Q]=p and E/K an elliptic curve with rational j-invariant. The following holds: 1. If p≥7, then E(K)tors∈Φ(1). 2. If p=3 or p=5, then E(K)tors∈ΦQ(p). 3. If p=2, then E(K)tors∈ΦQ(2) or E(K)tors≅Z/13Z. In the sixth chapter, we are able to determine all the sets ΦQ(pq), where p and q are prime numbers. Most of these cases follow easily from previously known results and the results in the first two chapters of this thesis. In most cases we have ΦQ(pq)=ΦQ(p)∪ΦQ(q). A detailed description of the sets ΦQ(pq) can be found in the fifth chapter of this thesis. Some of the proofs in the thesis rely on extensive computations in Magma [3]. All of the programs and calculations used for the proofs can be found in the last chapter

Izvorni jezik
Engleski

Znanstvena područja
Matematika

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