Modularna metoda za rješavanje diofantskih jednadžbi

master's thesis

• Author: Marta Dujella
• Mentor: Filip Najman (mentor)
• Committee member: Filip Najman (predsjednik povjerenstva)
• Committee member: Marko Tadić (član povjerenstva)
• Committee member: Pavle Pandžić (član povjerenstva)
• Committee member: Nenad Antonić (član povjerenstva)
• Granter: University of Zagreb; Faculty of Science; (Department of Mathematics); Zagreb
• Defense date and country: 2018-09-26, Croatia
• Scientific / art field, discipline and subdiscipline: NATURAL SCIENCES; Mathematics

Abstract

Andrew Wiles dokazao je 1995. godine posljednji Fermatov teorem, koristeći vezu između eliptičkih krivulja i modularnih formi. Zapravo, Wiles je dokazao poseban slučaj teorema o modularnosti. Cilj ovog rada bio je pokazati primjene teorema o modularnosti i teorije eliptičkih krivulja i modularnih formi na diofantske jednadžbe. U prvom poglavlju dajemo pregled osnovnih pojmova vezanih uz eliptičke krivulje. Posebno se bavimo sa redukcijom eliptičkih krivulja modulo \(p\): opisujemo slučajeve i podslučajeve dobre i loše redukcije te definiramo konduktor eliptičke krivulje. Na primjeru pokazujemo traženje minimalne jednadžbe i konduktora. Nadalje, definiramo izogenije eliptičkih krivulja i dajemo nekoliko bitnih primjera izogenija. Na kraju poglavlja iskazujemo Mazurov teorem o \(p\)-izogenijama. U drugom poglavlju definiramo modularnu grupu i njene kongruencijske podgrupe, a zatim definiramo i modularne forme obzirom na kongruencijske podgrupe. U nastavku definiramo newforme: posebnu klasu modularnih formi, koje su posebno bitne za teorem o modularnosti i dajemo pregled osnovnih svojstava newformi. Na kraju poglavlja iskazujemo Ribetov teorem i teorem o modularnosti, koji daju osnovu za primjenu ove teorije na rješavanje diofantskih jednadžbi. U posljednjem poglavlju pokazujemo na primjerima kako se teorem o modularnosti i Ribetov teorem mogu primjeniti na rješavanje diofantskih jednadžbi. Prvi primjer je Fermatova jednadžba \(a^p+b^p+c^p=0\), tj. pokazujemo da iz teorema o modularnosti i Ribetovog teorema slijedi posljednji Fermatov teorem. Nakon toga se bavimo jednadžbom \(x^p+L^ry^p+z^p=0\) i na tom primjeru pokazujemo nekoliko metoda koje se mogu primjeniti na takve jednadžbe. Jedan od pristupa je tzv. Krausova metoda, za čiju primjenu dajemo još jedan primjer.

Abstract (english)

Andrew Wiles proved Fermat’s last theorem in 1995, using the connection between elliptic curves and modular forms. Actually, Wiles proved a special case of the modularity theorem. The goal of this thesis was to demonstrate application of the modularity theorem and theories of elliptic curves and modular forms to Diophantine equations. In the first chapter we give an overview of basic terms related to elliptic curves. We especially deal with reduction of elliptic curves modulo \(p\): we describe the cases and subcases of good and bad reduction and define the conductor of an elliptic curve. We show an example of finding the minimal equation and conductor. Furthermore, we define isogenies of elliptic curves and give a few important examples of isogenies. The chapter ends with the statement of Mazur’s theorem on \(p\)-isogenies. In the second chapter we define the modular group and its congruence subgroups and then also define modular forms with respect to congruence subgroups. Subsequently, we define newforms: a special class of modular forms, which are especially important for the modularity theorem, and we give an overview of basic properties of newforms. In the end of this chapter we state Ribet’s theorem and the modularity theorem, which give the basis for application of this theory on solving Diophantine equations. In the final chapter we demonstrate the application of the modularity and Ribet’s theorem to solving Diophantine equations. The first example is Fermat’s eqautions \(a^p+b^p+c^p=0\), i.e. we show that Fermat’s last theorem follows from modularity theorem and Ribet’s theorem. After that we look at the equation \(x^p+L^ry^p+z^p=0\) and in that example we show a few methods that can be applied to such equations. One approach is the so called method of Kraus, and for this method we give one more example.

• Keywords: Wiles; Fermat; teorem o modularnosti; teorija eliptičkih krivulja; modularne forme; diofantske jednadžbe; Mazurov teorem; newforme; Ribetov teorem; Krausova metoda
• Keywords (english): Wiles; Fermat; modularity theorem; theory of elliptic curves; modular forms; Diophantine equations; Mazur’s theorem; newforms; Ribet’s theorem; method of Kraus
• Language: croatian
• URN:NBN: urn:nbn:hr:217:289489
• 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: 2019-03-28 12:10:29