Generalizirani Wronskiani i modularne krivulje

doctoral thesis

• Title (english): Generalized Wronskians and modular curves
• Author: Damir Mikoč
• Mentor: Goran Muić (mentor)
• Committee member: Matija Kazalicki (predsjednik povjerenstva)
• Committee member: Goran Muić (član povjerenstva)
• Committee member: Iva Kodrnja (član povjerenstva)
• Committee member: Marko Tadić (član povjerenstva)
• Granter: University of Zagreb; Faculty of Science; (Department of Mathematics); Zagreb
• Defense date and country: 2022-02-15, Croatia
• Scientific / art field, discipline and subdiscipline: NATURAL SCIENCES; Mathematics
• Universal decimal classification (UDC): 51 - Mathematics

Abstract

Fokus ove teze su modularne krivulje i modularne forme za neku Fuchsovu grupu prve vrste $\Gamma$, posebno za grupu $\Gamma_0(N)$. Proučavamo Weierstrassove i $n$–Weierstrassove točke, $n \in \mathbb{N}$, na krivulji $\mathfrak{R}_{\Gamma}$ u jeziku modularnih formi. Za danu modularnu krivulju $\mathfrak{R}_{\Gamma}$ i paran cijeli broj $m \geq 4$ želimo dati efektivan algoritam za provjeru jeli kasp $a_{\infty}, m/2$–Weierstrassova točka na $\mathfrak{R}_{\Gamma}$. Uvodimo... More prirodno poopćenje pojma Wronskiana kaspidalnih modularnih forma. Proučavamo Wronskiane kanonskih baza prostora $M_m(SL_2(\mathbb{Z}))$. Proučavamo biracionalna preslikavanja $X(1) \longrightarrow \mathbb{P}^2$ i računamo jednadžbe dobivenih krivulja. Razvijen je algoritam u SAGE-u koji funkcionira za sve krivulje tipa $X_0(N)$, genusa $g \geq 3$ koje nisu hipereliptičke. Kao posljedicu tog algoritma izračunali smo jednadžbe svih kanonskih krivulja tipa $X_0(N)$, genusa $3 \leq g \leq 5$, koje nisu hipereliptičke. Razvijen je algoritam za račun Wronskiana linearno nezavisnih modularnih formi. U SAGE-u su izračunati Wronskiani kanonskih baza prostora $M_m(SL_2(\mathbb{Z}))$, za parne $m = 12,14,16,\dots,108,110,120$. Temeljem toga dokazan je teorem o vrijednosti tih Wronskiana za bilo koji parni $m$, do na neku ne–nul konstantu $\lambda$. Za $m = 12t$ iskazana je slutnja o vrijednosti konstante $\lambda$ do na predznak. Dani su numerički primjeri računa u SAGE-u kojima smo dobili jednadžbe ravninskih krivulja $\mathcal{C} \subseteq \mathbb{P}^2$ biracionalno ekvivalentnih krivulji $X(1)$. Less

Abstract (english)

We are interested in modular curves and modular forms for some Fuchsian group of the first kind, especially for the group $\Gamma_0(N)$. We are studying Weierstrass and $n$–Weierstrass points, $n \in \mathbb{N}$, on curve $\mathfrak{R}_{\Gamma}$ in the language of modular forms. For a given modular curve $\mathfrak{R}_{\Gamma}$ and an even integer $m \geq 4$, we want to give an effective algorithm for checking whether cusp $a_{\infty}$ is a Weierstrass point on... More $\mathfrak{R}_{\Gamma}$. We have introduced a natural generalization of the usual notion of the Wronskian of cuspidal modular forms. We are studying the Wronskians of the canonical bases of the spaces $M_m(SL_2(\mathbb{Z}))$. We are studying the birational maps $X(1) \longrightarrow \mathbb{P}^2$ and calculate the equations of the obtained curves. An algorithm has been developed in SAGE that works for all curves of type $X_0(N)$, of the genus $g \geq 3$, that are not hyperelliptic. As a consequence of this algorithm, we calculated the equations of all canonical curves of type $X_0(N)$, genus $3 \leq g \leq 5$, which are not hyperelliptic. An algorithm for the calculation of the Wronskian of linearly independent modular forms has been developed. In SAGE, we have calculated Wronskians of canonical bases for $M_m(SL_2(\mathbb{Z}))$, for even $m = 12,14,16,\dots,108,110,120$. Based on this, the theorem on the value of these Wronskians for any even $m$, up to some non - zero constant $\lambda$, has been proved. For $m = 12t$ we made a conjecture about the value of the constant $\lambda$ up to the sign. Less

• Keywords: Fuchsova grupa prve vrste; Riemannova ploha; gornja poluravnina; modularna grupa; modularna krivulja; Weierstrassova točka; modularna forma; Wronskian; krivulja X(1); biracionalno preslikavanje; krivulja X_0(N); hipereliptička krivulja; kanonska krivulja
• Keywords (english): Fuchsian group of the first kind; Riemann surface; upper half-plane; modular group; modular curve; Weierstrass point; modular form; Wronskian; curve 𝑋(1); birational map; curve 𝑋_0(𝑁); hyperelliptic curve; canonical curve
• Language: croatian
• URN:NBN: urn:nbn:hr:217:437223
• Promotion: 2022
• Study programme: Title: Mathematics; Study programme type: university; Study level: postgraduate; Academic / professional title: doktor/doktorica znanosti, područje prirodnih znanosti, polje matematika (doktor/doktorica znanosti, područje prirodnih znanosti, polje matematika)
• Type of resource: Text
• Extent: vi, 123 str.
• File origin: Born digital
• Access conditions: Open access
• Created on: 2022-03-15 13:56:23