Izogenije eliptičkih krivulja

master's thesis

• Author: Petar Orlić
• Mentor: Filip Najman (mentor)
• Committee member: Filip Najman (predsjednik povjerenstva)
• Committee member: Slaven Kožić (član povjerenstva)
• Committee member: Marko Erceg (član povjerenstva)
• Committee member: Marko Vrdoljak (član povjerenstva)
• Granter: University of Zagreb; Faculty of Science; (Department of Mathematics); Zagreb
• Defense date and country: 2021-07-22, Croatia
• Scientific / art field, discipline and subdiscipline: NATURAL SCIENCES; Mathematics

Abstract

U ovom radu bavili smo se izogenijama eliptičkih krivulja. Prvo smo definirali eliptičke krivulje i operaciju zbrajanja, a zatim se posvetili izogenijama. Nakon uvodnih definicija i primjera naveli smo i dokazali neka njihova osnovna svojstva. Nakon toga prešli smo na modularne krivulje. Definirali smo modularne krivulje \(X_0\left( N\right)\) i \(X_1\left( N\right)\) preko kongruencijskih podgrupa modularne grupe SL\(_2\left(\mathbb{Z}\right)\). Zatim smo iskazali tvrdnju da su to algebarske krivulje definirane nad \(\mathbb{Q}\), a nakon toga smo dokazali teoreme koji daju vezu točaka na modularnoj krivulji i eliptičkih krivulja. Takoder smo dali primjere kako se modularne krivulje mogu koristiti u dokazivanju tvrdnji za eliptičke krivulje. Onda smo krenuli na Mazurov teorem. Naveli smo slučajeve u kojima krivulja \(X_0\left( N\right)\) ima racionalne nekaspidalne točke i rekli da teorem kaže da su to svi slučajevi kad je \(N\) prost, a nakon toga smo počeli obradivati pojmove i razultate potrebne za dokaz teorema. Koristeći polje \(p\)-adskih brojeva \(\mathbb{Q}_p\) uveli smo pojmove potencijalno dobre i potencijalno multiplikativne redukcije i, uz pomoć Atkin-Lehnerove involucije, kojoj smo takođr posvetili malo vremena, dokazali važnu pomoćnu tvrdnju. Zatim smo definirali karakter izogenije, ciklotomski karakter i mod \(N\)-Galoisove reprezentacije eliptičkih krivulja. Pokazali smo neke veze izmedu tih pojmova i iskazali rezultat o faktorizaciji karaktera izogenije. Iza toga smo definirali Frobeniusov trag, naveli neka njegova svojstva i iskazali vezu s karakterom izogenije. Naposlijetku, uz pomoć rezultata iz ovog rada te nekoliko teorema iz algebarske teorije brojeva dokazali smo Mazurov teorem..

Abstract (english)

In this thesis we considered isogenies of elliptic curves. We first defined elliptic curves and the addition operation, after which we turned to isogenies. After some introductory definitions and examples, we stated and proved some of their basic properties. After that, we considered modular curves. We defined modular curves \(X_0\left( N\right)\) and \(X_1\left( N\right)\) using the congruence subgroups of the modular group SL\(_2\left(\mathbb{Z}\right)\). Then we stated the fact that these are algebraic curves defined over \(\mathbb{Q}\) and proved the theorems that give the connection between points on a modular curve and elliptic curves. We also gave examples of how modular curves can be used to prove statements about elliptic curves. Then we moved on to Mazur's theorem. We listed the cases when the curve \(X_0\left( N\right)\) has rational non-cuspidal points and mentioned that the theorem says that these are all the cases when \(N\) is a prime number. After that, we started considering notions and results needed for the proof of the theorem. Using the field of \(p\)-adic numbers \(\mathbb{Q}_p\) we introduced the notions of potential good and potential multiplicative reduction and, using the Atkin-Lehner involution, which we also considered for a while, proved an important auxiliary statement. We then defined the isogeny character, the cyclotomic character and mod \(N\)-Galois representations of elliptic curves. We proved some connections between these notions and stated a result about the factorization of the isogeny character. Next, we defined the trace of Frobenius, stated its basic properties and linked it with the isogeny character. Finally, using the results from this paper and several algebraic number theory theorems we proved Mazur's theorem.

• Keywords: modularne krivulje; algebarske krivulje; Mazurov teorem; Atkin-Lehnerova involucija; Frobeniusov trag
• Keywords (english): modular curves; algebraic curves; Mazur's theorem; Atkin-Lehner involution; trace of Frobenius
• Language: croatian
• URN:NBN: urn:nbn:hr:217:534598
• 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: 2021-09-16 10:33:31