Teorem o modularnosti

master's thesis

• Title (english): Modularity theorem
• Author: Ivan Novak
• Mentor: Filip Najman (mentor)
• Committee member: Filip Najman (predsjednik povjerenstva)
• Committee member: Andrej Dujella (član povjerenstva)
• Committee member: Goran Radunović (član povjerenstva)
• Committee member: Vanja Wagner (član povjerenstva)
• Granter: University of Zagreb; Faculty of Science; (Department of Mathematics); Zagreb
• Defense date and country: 2023-07-19, Croatia
• Scientific / art field, discipline and subdiscipline: NATURAL SCIENCES; Mathematics

Abstract

U ovom radu bavili smo se teoremom o modularnosti i objektima vezanim za njega; eliptičkim krivuljama, modularnim formama i modularnim krivuljama. Definirali smo modularne forme i odredili neka njihova osnovna svojstva. Nakon toga smo definirali eliptičke krivulje nad poljem kompleksnih brojeva kao kompleksne toruse, te odredili kako izgledaju holomorfne funkcije na njima. Također smo definirali modularne krivulje kao Riemannove plohe i dokazali da one parametriziraju klase eliptičkih krivulja zajedno s nekom informacijom o torziji. Iskazali smo teorem o modularnosti preko uvedenih kompleksno-analitičkih objekata. Pomoću rezultata iz kompleksne analize o kompaktnim Riemannovim plohama smo vidjeli da su prostori modularnih formi za kongruencijske podgrupe konačnodimenzionalni, te iskazali formule za dimenzije. Nakon toga smo definirali Heckeove operatore, normalne operatore na prostorima \(\Gamma_1(N)\) koji međusobno komutiraju, uveli pojam svojstvenih formi te novih i starih formi. Definirali smo i L-funkcije modularnih formi te eliptičkih krivulja, te iskazali teorem o modularnosti preko Fourierovih koeficijenata odnosno L-funkcija. Na kraju smo, uz pomoć niza ranije poznatih rezultata o eliptičkim krivuljama, skicirali dokaz Velikog Fermatovog teorema te time ustanovili važnost teorema o modularnosti kao povijesno najtežeg i posljednjeg koraka u dokazu.

Abstract (english)

In this paper, we explored the modularity theorem and the mathematical objects related to it: elliptic curves, modular forms, and modular curves. We defined modular forms and discussed some of their basic properties. Afterwards, we defined elliptic curves over the field of complex numbers as complex tori and determined their function fields. We defined modular curves as Riemann surfaces and proved that they parameterize isomorphism classes of elliptic curves together with some information about their torsion. We stated the modularity theorem in terms of those complex-analytic objects. Using results from complex analysis about compact Riemann surfaces, it follows that the spaces of modular forms for congruence subgroups are finite-dimensional and we provided the dimension formulas for those spaces. Next, we defined Hecke operators, which are a family of commuting normal operators on the spaces \(\Gamma_1(N)\). We introduced eigenforms as common eigenvectors for all Hecke operators and we explained the concept of newforms. We defined L-functions for modular forms and elliptic curves and stated the modularity theorem in terms of equality of their Fourier coefficients and, equivalently, their L-functions. Finally, with the help of a series of previously known results about elliptic curves, we gave an outline of a proof of Fermat's last Theorem, thereby establishing the significance of the modularity theorem as the historically most difficult and final step in the proof.

• Keywords: eliptičke krivulje; modularne forme; modularne krivulje; kompleksni torusi; Riemannove plohe
• Keywords (english): lliptic curves; modular forms; modular curves; Riemann surfaces
• Language: croatian
• URN:NBN: urn:nbn:hr:217:149590
• Study programme: Title: Theoretical Mathematics; Study programme type: university; Study level: graduate; Academic / professional title: sveučilišni magistar matematike (sveučilišni magistar matematike)
• Type of resource: Text
• File origin: Born digital
• Access conditions: Open access
• Created on: 2024-02-09 13:47:56