Mordell-Weilove grupe i izogenije familija eliptičkih krivulja

doctoral thesis

• Title (english): Mordell-Weil groups and isogenies of the families of elliptic curves
• Author: Miljen Mikić
• Mentor: Andrej Dujella (mentor)
• Mentor: Filip Najman (komentor)
• Committee member: Ivica Gusić (predsjednik povjerenstva)
• Committee member: Andrej Dujella (član povjerenstva)
• Committee member: Filip Najman (član povjerenstva)
• Committee member: Alan Filipin (član povjerenstva)
• Committee member: Matija Kazalicki (član povjerenstva)
• Granter: University of Zagreb; Faculty of Science; (Department of Mathematics); Zagreb
• Defense date and country: 2014-12-02, Croatia
• Scientific / art field, discipline and subdiscipline: NATURAL SCIENCES; Mathematics
• Universal decimal classification (UDC): 51 - Mathematics

Abstract

U ovom radu dokazano je da torzijska grupa eliptičkih krivulja induciranih \(D(4)\)-trojkama može biti ili \(\mathbb{Z}/2\mathbb{Z} \times \mathbb{Z}/2\mathbb{Z}\) ili \(\mathbb{Z}/2\mathbb{Z} \times \mathbb{Z}/6\mathbb{Z}\). Time je ujedno (kao specijalni slučaj) dobiveno i da torzijska grupa eliptičkih krivulja induciranih Diofantovim trojkama može biti jedna od navedenih. Promatrane su i familije eliptičkih krivulja generiranih Diofantovim trojkama oblika \(\{k−1, k+1, c_{l}(k)\}\) te su određeni torzijska grupa i rang (a time i Mordell-Weilova grupa) koje mogu imati takve krivulje za veliki broj vrijednosti od \(k\) i \(l\). Konačno, promatranjem modularnih krivulja \(X_0(n)\) prebrojano je koliko ima eliptičkih krivulja s cikličkom izogenijom stupnja \(n\) nad raznim kvartičnim poljima.

Abstract (english)

In this work it is proved that the torsion group of elliptic curves induced by \(D(4)\)-triples can be either \(\mathbb{Z}/2\mathbb{Z} \times \mathbb{Z}/2\mathbb{Z}\) or \(\mathbb{Z}/2\mathbb{Z} \times \mathbb{Z}/6\mathbb{Z}\). Therefore, as a special case it is proved that the torsion group of elliptic curves induced by Diophantine triples can be one of these two, as well. Families of elliptic curves induced by Diophantine triples of the form \(\{k−1, k+1, c_{l}(k)\}\) are examined and the possible form of the torsion group and rank (i.e. Mordell-Weil group) of these curves is determined for the large number of values of \(k\) and \(l\). Finally, by studying modular curves \(X_0(n)\) the number of elliptic curves with cyclic isogeny of degree \(n\) over various quartic fields is found.

• Keywords: torzijska grupa; eliptičke krivulje; Diofantove trojske; Mordell-Weilova grupa
• Keywords (english): torsion group; elliptic curves; Diophantine triples; Mordell-Weil group
• Language: croatian
• URN:NBN: urn:nbn:hr:217:252457
• Study programme: Title: Mathematics; Study programme type: university; Study level: postgraduate; Academic / professional title: doktor/doktorica znanosti, područje prirodnih znanosti, polje matematika (doktor/doktorica znanosti, područje prirodnih znanosti, polje matematika)
• Type of resource: Text
• Extent: ii, 84 str.
• File origin: Born digital
• Access conditions: Open access
• Created on: 2019-03-19 10:00:32