Torzijske grupe eliptičkih krivulja

master's thesis

• Author: Borna Vukorepa
• Mentor: Filip Najman (mentor)
• Committee member: Filip Najman (predsjednik povjerenstva)
• Committee member: Zlatko Drmač (član povjerenstva)
• Committee member: Goran Muić (član povjerenstva)
• Committee member: Ljiljana Arambašić (član povjerenstva)
• Granter: University of Zagreb; Faculty of Science; (Department of Mathematics); Zagreb
• Defense date and country: 2018-07-18, Croatia
• Scientific / art field, discipline and subdiscipline: NATURAL SCIENCES; Mathematics

Abstract

U ovom radu promatrali smo torzijske grupe elitpičkih krivulja. Nakon uvodnih definicija i malih rezultata o točkama malog reda, promatrali smo situaciju najprije za polje kompleksnih brojeva. Prateći ranije dokazane rezultate, odredili smo kako može izgledati grupa kompleksnih točaka i zaključili smo da je za svaku racionalnu eliptičku krivulju \(E(\mathbb{C})_{tors}\) uvijek ista, točnije, izomorfna je torusu u aditivnom smislu. Koristeći taj zaključak, promatrali smo kako može izgledati grupa realnih točaka racionalne eliptičke krivulje i zaključili da postoje dvije mogućnosti te smo odredili uz koji uvjet se dešava koji slučaj. Samim time, odredili smo i kako može izgledati \(E(\mathbb{R})_{tors}\). Dokazali smo Nagell-Lutzov teorem koji se može na očit način koristiti za određivanje torzijske grupe specifične eliptičke krivulje. Uz tu očitu metodu, obradili smo još jednu koja koristi eliptičke krivulje nad poljem \(\mathbb{F}_p\), za \(p\) prikladno odabran prost broj. Također smo iskazali i Mordell-Weilov i Mazurov teorem. Nakon toga, prešli smo na polja algebarskih brojeva. Najprije smo pokazali da točke konačnog reda na racionalnoj eliptičkoj krivulji zaista imaju algebarske brojeve za svoje koordinate. Definirali smo Weilovo sparivanje i Galoisovu reprezentaciju i dokazali nekoliko važnih rezultata koji se povezuju s njima, a mogu biti od velike pomoći pri klasifikaciji torzijskih grupa nad Galoisovim proširenjima. Dotaknuli smo se i izogenija eliptičkih krivulja i pokazali jednu značajnu tvrdnju koja ih povezuje s torzijom. Također smo naveli neke bitne, ranije pokazane poznate rezultate vezane za torziju nad nekim specifičnim poljima. Na kraju, upotrijebili smo sve razvijene alate, iskazane i dokazane tvrdnje kako bismo (djelomično) klasificirali torzijske grupe racionalnih eliptičkih krivulja nad \((\mathbb{Q}(\zeta_{11}))\).

Abstract (english)

In this paper we considered torsion groups of elliptic curves. After some introductory definitions and some results for points of small order, we considered the situation for the field of complex numbers. Following known results, we determined the group of complex points and concluded that for every rational elliptic curve \(E(\mathbb{C})_{tors}\)is always the same. More precisely, it is isomorphic to a torus as an additive group. Using that conclusion, we considered the group of real points on a rational elliptic curve and concluded that there are only two options. We also determined the conditions under which each of the two options arises. With that, we have determined the structure of \(E(\mathbb{C})_{tors}\). We proved the Nagell-Lutz theorem which can help in determining the torsion group structure for some specific elliptic curve. In addition to that obvious method, we gave one more method which uses elliptic curves over \(\mathbb{F}_p\), for a suitable prime \(p\). We also stated the Mordell-Weil theorem and Mazur’s theorem. After that, we moved on to number fields. First we proved that the coordinates of points of finite order on rational elliptic curves are algebraic numbers. We defined the Weil pairing and Galois representation and proved several results which can be helpful in classifying torsion groups of elliptic curves over Galois extensions. We also considered isogenies of elliptic curves and proved one important proposition which connects them to torsion groups. Additionally, we stated several important well-known results concerning torsion over some specific fields. Finally, we used all the developed tools and results we quoted and/or proved to (partially) classify torsion groups of rational elliptic curves over \((\mathbb{Q}(\zeta_{11}))\).

• Keywords: torzijske grupe; elitpičke krivulje; Nagell-Lutzov teorem; Mordell-Weilov teorem; Mazurov teorem; Weilovo sparivanje; Galoisova reprezentacija; izogenije eliptičkih krivulja
• Keywords (english): torsion groups; elliptic curves; Nagell-Lutz theorem; Mordell-Weil theorem; Mazur’s theorem; Weil pairing; Galois representation; isogenies of elliptic curves
• Language: croatian
• URN:NBN: urn:nbn:hr:217:941743
• Study programme: Title: Pure Mathematics; Study programme type: university; Study level: graduate; Academic / professional title: magistar/magistra matematike (magistar/magistra matematike)
• Type of resource: Text
• File origin: Born digital
• Access conditions: Open access
• Created on: 2019-07-09 08:27:21