Od kongruencijskih brojeva do eliptičkih krivulja

master's thesis

• Author: Borna Cicvarić
• Mentor: Marcela Hanzer (mentor)
• Committee member: Marcela Hanzer (predsjednik povjerenstva)
• Committee member: Mirko Primc (član povjerenstva)
• Committee member: Pavle Pandžić (član povjerenstva)
• Committee member: Franka Miriam Bruckler (član povjerenstva)
• Granter: University of Zagreb; Faculty of Science; (Department of Mathematics); Zagreb
• Defense date and country: 2015-07-13, Croatia
• Scientific / art field, discipline and subdiscipline: NATURAL SCIENCES; Mathematics

Abstract

Ukratko, ovaj rad započeli smo problemom kongruentnih brojeva, te smo problem povezali s eliptičkom krivuljom \(y^2=x^3-n^2x\). Potom smo definirali eliptičke krivulje i eliptičke funkcije. Pokazali smo da se svaka eliptička funkcija može prikazati kao racionalna funkcija Weierstrassove \(\wp\) funkcije, te da preko nje možemo napraviti bijekciju između \(\mathbb{C} / L\) i točaka na eliptičkoj krivulji u \(\mathbb{P}_{\mathbb{C}}^2\). Preko te bijekcije definirali smo zbrajanje na eliptičkoj krivulji i objasnili njegovu geometrijsku interpretaciju. Sada kad možemo zbrajati točke na eliptičkoj krivulji (tj. taj skup točaka ima svojstvo grupe) promatramo točke konačnog reda, koji tvore podgrupu. Na kraju promotrimo eliptičke krivulje \(y^2 = x^3-n^2x\) (koje ćemo označavati \(E_n\)) nad \(\mathbb{Q}\). Pomoću redukcije modulo \(p\), gdje je \(p \in \mathbb{N}\) prost, dokažemo da se torzijska podgrupa od \(E_n(\mathbb{Q})\) sastoji od samo 4 elementa, te da svaki drugi element iz \(E_n(\mathbb{Q})\) ima beskonačan red. Na kraju naše rezultate povežemo s problemom kongruentnih brojeva i dobijemo krajnji teorem: Teorem. \(n \in \mathbb{N}\) je kongruentan broj ako i samo ako u \(E_n(\mathbb{Q})\) postoji točka beskonačnog reda (tj. ako postoje \(P, Q \in E_n(\mathbb{Q})\) i \(P \neq 0\) takvi da je \(P = 2Q\)).

Abstract (english)

To summarize, we began with the congruent number problem and related it to the elliptic curve \(y^2=x^3-n^2x\). Next, we defined elliptic curves and elliptic functions. We showed that every elliptic function can be written as a rational function of the Weierstrass \(\wp\) function, and used it to construct a bijection between \(\mathbb{C} / L\) and points of an elliptic curve in \(\mathbb{P}_{\mathbb{C}}^2\). Using that bijection, we defined addition of points of an elliptic curve and explained its geometrical interpretation. Now the points of an elliptic curve have the properties of a group, so we examined the points of finite order, which form a subgroup. In the end we focus on the curve \(y^2 = x^3-n^2x\), (which we will denote \(E_n\)) over \(\mathbb{Q}\). Using reduction modulo \(p\), where \(p \in \mathbb{N}\) is a prime, we prove that the torsion subgroup of \(E_n(\mathbb{Q})\) consists of only 4 elements, and all the other elements of \(E_n(\mathbb{Q})\) have infinite order. At the end we use our results and go back to the congruent number problem to prove the following theorem: Teorem. \(n \in \mathbb{N}\) is a congruent number if and only if \(E_n(\mathbb{Q})\) contains a point of infinite order (in other words, if there exist \(P, Q \in E_n(\mathbb{Q})\), with \(P \neq 0\) such that \(P = 2Q\)).

• Keywords: problem kongruentnih brojeva; eliptička krivulja; eliptička funkcija; Weierstrassova funkcija
• Keywords (english): congruent number problem; elliptic curve; elliptic functions; Weierstrass function
• Language: croatian
• URN:NBN: urn:nbn:hr:217:987792
• Study programme: Title: Mathematical Statistics; 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: 2019-01-07 11:13:34