Title Torsion of elliptic curves with rational j-invariant over number fields Title (croatian) Torzija eliptičkih krivulja s racionalnom j-invarijantom nad poljima algebarskih brojeva Author Tomislav Gužvić Mentor Filip Najman (mentor)Committee member Matija Kazalicki (predsjednik povjerenstva)Committee member Filip Najman (član povjerenstva)Committee member Nikola Adžaga (član povjerenstva)Committee member Andrej Dujella (član povjerenstva)Granter University of Zagreb Defense date and country 2021-02-19, Croatia Scientific / art field, NATURAL SCIENCES Universal decimal classificationUDC ) 51 - Mathematics Abstract 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 \(\mathbb{Q}\). 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 \(C_3 \bigoplus C_{18}\), that there exist an elliptic curve \(E / \mathbb{Q}\) and a sextic number field \(K\) such that \(E(K)_{tors} \cong G\). Additionally, we provide a partial result regarding the group \(C_3 \bigoplus C_{18}\). For a positive integer \(d\), define \(\Phi(d)\) 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 \(\Phi_{\mathbb{Q}}(d)\) 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 \(\mathbb{Q}\). Define \(\Phi_{j \in \mathbb{Q}}(d)\) 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 \(j(E) \in \mathbb{Q}\). With the help of the previously mentioned result, we are able to completely determine the sets \(\Phi_{j \in \mathbb{Q}}(p)\), 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 \(p \geq 7\), then \(E(K)_{tors} \in \Phi(1)\). 2. If \(p = 3\) or \(p = 5\), then \(E(K)_{tors} \in \Phi_{\mathbb{Q}}(p)\). 3. If \(p = 2\), then \(E(K)_{tors} \in \Phi_{\mathbb{Q}}(2)\) or \(E(K)_{tors} \cong \mathbb{Z}/13\mathbb{Z}\). In the sixth chapter, we are able to determine all the sets \(\Phi_{\mathbb{Q}}(pq)\), 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 \(\Phi_{\mathbb{Q}}(pq) = \Phi_{\mathbb{Q}}(p) \cup \Phi_{\mathbb{Q}}(q)\). A detailed description of the sets \(\Phi_{\mathbb{Q}}(pq)\) 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.
Abstract (croatian) U ovoj disertaciji odredit ćemo moguće torzijske strukture eliptičkih krivulja s racionalnom \(j\)-invarijantom definiranih nad nekim poljem algebarskih brojeva. Prvo ćemo promatrati eliptičke krivulje definirane nad \(\mathbb{Q}\). Neka je \(K\) sekstično polje. Odredit ćemo sve mogućnosti \(G\) za \(E(K)_{tors}\) i dokazati da za svaku moguću grupu \(G\) osim \(C_3 \bigoplus C_{18}\) postoji eliptička krivulja \(E / \mathbb{Q}\) i sekstično polje \(K\) takvo da je \(E(K)_{tors} \cong G\). Nadalje, dokazat ćemo parcijalni rezultat za grupu \(C_3 \bigoplus C_{18}\). Za prirodan broj \(d\) definiramo \(\Phi(d)\) kao skup mogućih klasa izomorfizama grupa \(E(K)_{tors}\), gdje \(K\) varira po svim poljima algebarskih brojeva \(K\) stupnja \(d\) i \(E\) varira po svim eliptičkim krivuljama nad \(K\). Za prirodan broj \(d\) definiramo \(\Phi_{\mathbb{Q}}(d)\) kao skup mogućih klasa izomorfizama grupa \(E(K)_{tors}\), gdje \(K\) varira po svim poljima algebarskih brojeva \(K\) stupnja \(d\) i \(E\) varira po svim eliptičkim krivuljama nad \(\mathbb{Q}\). Za prirodan broj \(d\) definiramo \(\Phi_{j \in \mathbb{Q}}(d)\) kao skup mogućih klasa izomorfizama grupa \(E(K)_{tors}\), gdje \(K\) varira po svim poljima algebarskih brojeva \(K\) stupnja \(d\) i \(E\) varira po svim eliptičkim krivuljama nad \(K\), te \(j(E) \in \mathbb{Q}\). Uz pomoć prethodnog rezultata u mogućnosti smo u potpunosti odrediti skupove \(\Phi_{j \in \mathbb{Q}}(p)\), gdje je \(p\) prost broj. Preciznije, naši rezultati su sljedeći. Neka je \(K\) polje algebarskih brojeva takvo da je \([K : Q] = p\) i \(E / K\) eliptička krivulja s racionalnom \(j\) invarijantom. Tada 1. Ako je \(p \geq 7\), tada \(E(K)_{tors} \in \Phi(1)\). 2. Ako je \(p = 3\) ili \(p = 5\), tada \(E(K)_{tors} \in \Phi_{\mathbb{Q}}(p)\). 3. Ako je \(p = 2\), tada \(E(K)_{tors} \in \Phi_{\mathbb{Q}}(2)\) ili \(E(K)_{tors} \cong \mathbb{Z}/13\mathbb{Z}\). U šestom poglavlju odredit ćemo sve skupove \(\Phi_{\mathbb{Q}}(pq)\), gdje su \(p\) i \(q\) prosti brojevi. Mnoge takve skupove ćemo odrediti koristeći već poznate rezultate, te rezultate dokazane u drugom i trećem poglavlju. U većini slučajeva vrijedit će \(\Phi_{\mathbb{Q}}(pq) = \Phi_{\mathbb{Q}}(p) \cup \Phi_{\mathbb{Q}}(q)\). Detaljniji opis skupova \(\Phi_{\mathbb{Q}}(pq)\) može se pronaći u petom poglavlju. Dokazi nekih rezultata u ovoj disertaciji temelje se na računanju u Magmi [3]. Svi programi i izračuni korišteni u dokazima mogu se pronaći u posljednjem poglavlju.
Keywords
torsion structures
elliptic curves
number fields
Keywords (croatian)
torzijske strukture
eliptičke krivulje
polje algebarskih brojeva
Language english URN:NBN urn:nbn:hr:217:445466 Study programme Title: Mathematics Type of resource Text Extent vii, 119 str. File origin Born digital Access conditions Open access Terms of use Created on 2022-01-24 13:30:44
