Analitičke metode modularnih formi

master's thesis

• Author: Sonja Žunar
• Mentor: Goran Muić (mentor)
• Committee member: Goran Muić (predsjednik povjerenstva)
• Committee member: Marcela Hanzer (član povjerenstva)
• Committee member: Matija Kazalicki (član povjerenstva)
• Committee member: Ljiljana Arambašić (član povjerenstva)
• Granter: University of Zagreb; Faculty of Science; (Department of Mathematics); Zagreb
• Defense date and country: 2014-07-11, Croatia
• Scientific / art field, discipline and subdiscipline: NATURAL SCIENCES; Mathematics

Abstract

Ovaj je rad uvod u analitičku teoriju brojeva i teoriju modularnih formi. U prvom dijelu rada dokazujemo Dirichletov teorem o prostim brojevima u aritmetičkim nizovima. U drugom dijelu proučavamo modularne forme: konstruiramo osnovne primjere modularnih funkcija i modularnih formi (Eisensteinov red, modularnu diskriminantu i modularnu invarijantu) i istražujemo njihova svojstva; određujemo dimenzije i konstruiramo baze prostora modularnih formi; uvodimo Heckeove operatore kao korespondencije na skupu rešetki u \(\mathbb{C}\) i istražujemo svojstva njihovih svojstvenih funkcija.

Abstract (english)

This thesis is an introduction to analytic number theory and the theory of modular forms. In the first part of the thesis, we prove the Dirichlet’s theorem on primes in arithmetic progressions. In the second part, we study modular forms: we construct basic examples of modular functions and modular forms (the Eisenstein series, the modular discriminant and the modular invariant) and investigate their properties; we find the dimensions and construct bases of spaces of modular forms; we introduce the Hecke operators as correspondences on the set of lattices in \(\mathbb{C}\) and investigate properties of their eigenfunctions.

• Keywords: Dirichletov teorem o prostim brojevima u aritmetičkim nizovima; modularne forme; Eisensteinov red; modularna diskriminanta; modularna invarijanta; Heckeove operator
• Keywords (english): Dirichlet’s theorem on primes in arithmetic progressions; Eisenstein series; modular discriminant; modular invariant; Hecke operators
• Language: croatian
• URN:NBN: urn:nbn:hr:217:707990
• 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: 2019-02-20 15:34:07