Lokalna rješivost kvadratnih zavrtaja krivulja genusa 1

master's thesis

• Title (english): Local solvability of square screws of genus 1 curves
• Author: Lukas Novak
• Mentor: Matija Kazalicki (mentor)
• Committee member: Matija Kazalicki (predsjednik povjerenstva)
• Committee member: Rudi Mrazović (član povjerenstva)
• Committee member: Pavle Pandžić (član povjerenstva)
• Committee member: Igor Ciganović (član povjerenstva)
• Granter: University of Zagreb; Faculty of Science; (Department of Mathematics); Zagreb
• Defense date and country: 2022-09-28, Croatia
• Scientific / art field, discipline and subdiscipline: NATURAL SCIENCES; Mathematics

Abstract

U ovom radu određujemo asimptotiku kvadratno slobodnih brojeva \(q \in \mathbb{Z}\) za koje je krivulja \(qy^2 = (x^2-x-3)(x^2+2x-12) \) svugdje lokalno rješiva, tj. ima rješenje u \(\mathbb{Q}p\) za svaki prosti broj \(p\). Postupak određivanja tražene asimptotike prati metode klasičnog rezultata o asimptotici brojeva koji se mogu prikazati kao suma dva kvadrata. Rješenje određivanja tražene asimptotike je podijeljeno u nekoliko koraka. Prvi korak se sastoji u određivanju nužnih i dovoljnih uvjeta za \(q\) za koje će promatrana krivulja biti svugdje lokalno rješiva. U drugom koraku se definira određen niz brojeva pomoću kojeg se zatim definira brojaća funkcija za tražene \(q\)-ove i pripadni Dirichletov red. Nakon toga se proučavaju svojsta tog Dirichletovog reda : apsolutna i uniformna konvergencija na području \(\operatorname{Re}(s)\geq c>1\), analitičko proširenje na \(\operatorname{Re}(s)>\frac{1}{2}\), reziduum u \(s=1\) te gornja ograda na \(\operatorname{Re}(s)=\frac{1}{2}+\delta\). U zadnjem koraku se primjenjuje Perronova formula te se uvodi kontura ključanice (key-hole contour) koja omogućuje ocijeniti integral dobiven u Perronovoj formuli. Iz te ocjene napokon dobijemo željenu asimptotiku.

Abstract (english)

In this paper we study the asymptotic behaviour of the number of square-free integers \(q \in \mathbb{Z}\) for which the curve \(qy^2 = (x^2-x-3)(x^2+2x-12) \) is everywhere locally solvable, i.e. for which it has a solution in \(\mathbb{Q}p\) for every prime number \(p\). The process for obtaining this asymptotic follows the methods used in the classical problem of determining the asymptotic of numbers that can be written as the sum of two squares. The solution for obtaining the wanted asymptotic is divided in several steps. The first step consists of finding the necessary and sufficient conditions on \(q\) for which the given curve is everywhere locally solvable. In the second step we define a certain sequence of numbers by which we then define the counting function for our \(q\)’s and a Dirichlet’s series that belongs to that sequence of numbers. After that we study the properties of this Dirichlet’s series : absolute and uniform convergence in region \(\operatorname{Re}(s)\geq c>1\), analytical continuation to \(\operatorname{Re}(s)>\frac{1}{2}\), residue in \(s=1\) and upper bound on \(\operatorname{Re}(s)=\frac{1}{2}+\delta\). In the last step we use Perron’s formula and introduce the key-hole contour which helps us to assess the integral obtained from Perron’s formula. From there we finally get the wanted asymptotic behaviour.

• Keywords: asimptotika kvadratno slobodnih brojeva; Dirichletov red; apsolutna konvergencija; uniformna konvergencija; Perronova formula; kontura ključanice
• Keywords (english): asymptotic behaviour of the number of square-free integers; Dirichlet's series; absolute convergence; uniform convergence; Perron's formula; key-hole contour
• Language: croatian
• URN:NBN: urn:nbn:hr:217:782764
• 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: 2022-11-03 12:40:04