Modularne krivulje

master's thesis

• Author: Antonela Trbović
• Mentor: Filip Najman (mentor)
• Granter: University of Zagreb; Faculty of Science; (Department of Mathematics); Zagreb
• Defense date and country: 2017-09-26, Croatia
• Scientific / art field, discipline and subdiscipline: NATURAL SCIENCES; Mathematics

Abstract

Centralni objekti koje proučavamo u ovom radu su modularne krivulje. Modularnu krivulju definiramo kao kvocijent gornje poluravnine s obzirom na djelovanje neke kongruencijske grupe, odnosno podgrupe modularne grupe \(SL_2(\mathbb{Z})\). U teoriji brojeva važnu ulogu igra činjenica da su modularne krivulje takoder prostori parametara, tj. svaka točka na modularnoj krivulji parametrizira eliptičku krivulju s nekim dodatnim svojstvom. Objasniti ovu činjenicu bio je glavni cilj ovoga rada, a to smo učinili u teoremu 3.2.2. Kako bismo došli do definicije modularne krivulje, prvo smo definirali modularne forme, slabo modularne funkcije koje zadovoljavaju neke uvjete holomorfnosti, koje i same mogu biti zanimljiv predmet proučavanja, te smo naveli neke osnovne primjere kao što su Eisensteinovi redovi i diskriminanta, koji su se pojavljivali kroz cijeli rad. Zatim smo definirali kompleksne toruse i eliptičke krivulje te pokazali na koji način možemo uspostaviti bijekciju između ta dva objekta. Pokazali smo i da postoji korespondencija između kompleksnih torusa i eliptičkih krivulja do na izomorfizam. Nadalje, definirali smo kongruencijske podgrupe od \(SL_2(\mathbb{Z}),\) a s time smo došli i do definicije modularne krivulje, skupa svih orbita pri djelovanju neke kongruencijske podgrupe na gornju poluravninu. Zatim smo u glavnom teoremu ovoga rada pokazali da postoji bijekcija između nekih modularnih krivulja i skupa eliptičkih krivulja sa određenim torzijskim svojstvom. Naposlijetku smo definirali dobru i lošu redukciju eliptičke krivulje te \(L-\) funkciju pridruženu eliptičkoj krivulji. Naveli smo i dvije različite verzije iskaza Teorema o modularnosti.

Abstract (english)

The central objects of this thesis are modular curves. We define a modular curve as the quotient of the complex upper half plane with respect to a congruence subgroup, i.e. a subgroup of the modular group \(SL_2(\mathbb{Z})\). In number theory, the fact that modular curves are also moduli spaces, i.e. every point on a modular curve parameterizes an elliptic curve with the associated torsion data, is very important. The main goal of this thesis was to prove this fact, which we did in the theorem 3.2.2. Before the definition of a modular curve, we first defined modular forms, weakly modular functions that satisfy some holomorphy conditions, that are also interesting subjects of study in themselves, and we gave some basic examples, as Eisenstein series and the discriminant function, which we encounter throughout the entire thesis. Furthermore, we defined complex tori and elliptic curves and showed that there exists a bijection between these two objects. Also, we proved that there exists a correspondence between complex tori and elliptic curves up to isomorphism. After the definition of a congruence subgroup of \(SL_2(\mathbb{Z}),\)we defined a modular curve, a set of all orbits under the action of some congruence subgroup on the complex upper half plane. In the main theorem of this thesis we proved that there exists a bijection between modular curves and the set of elliptic curves with the associated torsion data. In the end, we defined good and bad reductions of elliptic curves and the \(L-\) function associated to an elliptic curve. We also state two different versions of The Modularity Theorem.

• Keywords: modularna krivulja; kongruencijska grupa; modularna forma; kompleksni torus; eliptička krivulja; Teorem o modularnosti
• Keywords (english): modular curve; congruence subgroup; modular forms; complex tori; complex torus; elliptic curve; Modularity Theorem
• Language: croatian
• URN:NBN: urn:nbn:hr:217:373347
• 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: 2018-05-04 10:48:00