Title Generalizirani Wronskiani i modularne krivulje
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 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 \(X(1)\).
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 \(\mathfrak{R}_{\Gamma}\). 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.
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
Terms of use
Created on 2022-03-15 13:56:23