Birch i Swinnerton-Dyerova slutnja

master's thesis

• Author: Marko Sikirić
• Mentor: Filip Najman (mentor)
• Committee member: Filip Najman (predsjednik povjerenstva)
• Committee member: Dragutin Svrtan (član povjerenstva)
• Committee member: Tomislav Pejković (član povjerenstva)
• Committee member: Mirko Primc (član povjerenstva)
• Granter: University of Zagreb; Faculty of Science; (Department of Mathematics); Zagreb
• Defense date and country: 2016-09-23, Croatia
• Scientific / art field, discipline and subdiscipline: NATURAL SCIENCES; Mathematics

Abstract

U ovom radu smo probali što preciznije objasniti ovu važnu matematičku slutnju. Zbog matematičke dubine najvažniji rezultati su samo iskazani. Ono što smo pokušali vidjeti u radu je što točnija definicija glavnih pojmova i njihovo objašnjenje. Na početku rada dajemo općenitu sliku eliptičkih krivulja te njihovih osnovnih svojstava. Kasnije se trudimo da što preciznije pojasnimo pojmove L-funkcija i Tate-Šafarevičeve grupe, gdje ulazimo kratko u pojašnjenje p-adskih brojeva i Galoisove kohomologije. Na kraju dajemo iskaz Birch i Swinnerton-Dyerove slutnje, te navodimo glavne rezultate i posljedice. Svakako rad treba shvatiti kao uvod u ovo duboko područje matematike.

Abstract (english)

In this work we tried, as precise as possible, to define and explain this extraordinary conjecture. The deepest results are just stated. We tried to show, as exactly as possible, the main objects and their explanation. First we showed the main properties of elliptic curves and the mathematical theory around them. Then we define and explain L-functions and the very elusive Tate-Šafarevič group, where we to come to p-adic numbers and briefly touch on the subject of Galois Cohomology. At the end we state the Birch and Swinnerton-Dyer conjecture and try to understand where such a strong idea comes from. We show the main results and some consequences of this conjecture. This work is to be understood as an introduction to this wide and deep mathematical domain.

• Keywords: Birch i Swinnerton-Dyerove slutnja; eliptičke krivulje; L-funkcija; Tate-Šafarevičeve grupe; p-adski brojevi; Galoisove kohomologije
• Keywords (english): Birch and Swinnerton-Dyer conjecture; elliptic curves; L-functions; Tate-Šafarevič group; p-adic numbers; Galois Cohomology
• Language: croatian
• URN:NBN: urn:nbn:hr:217:369567
• Study programme: Title: Computer Science and Mathematics; Study programme type: university; Study level: graduate; Academic / professional title: magistar/magistra računarstva i matematike (magistar/magistra računarstva i matematike)
• Type of resource: Text
• File origin: Born digital
• Access conditions: Open access
• Created on: 2019-02-06 15:06:30