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doctoral thesis
Gonality of modular curves and their quotients
Zagreb: University of Zagreb, Faculty of Science, 2025. urn:nbn:hr:217:587029
Orlić, Petar
University of Zagreb
Faculty of Science
Department of Mathematics

Institutional repository: Repository of the Faculty of Science

Cite this document

Orlić, P. (2025). Gonality of modular curves and their quotients (Doctoral thesis). Zagreb: University of Zagreb, Faculty of Science. Retrieved from https://urn.nsk.hr/urn:nbn:hr:217:587029

Orlić, Petar. "Gonality of modular curves and their quotients." Doctoral thesis, University of Zagreb, Faculty of Science, 2025. https://urn.nsk.hr/urn:nbn:hr:217:587029

Orlić, Petar. "Gonality of modular curves and their quotients." Doctoral thesis, University of Zagreb, Faculty of Science, 2025. https://urn.nsk.hr/urn:nbn:hr:217:587029

Orlić, P. (2025). 'Gonality of modular curves and their quotients', Doctoral thesis, University of Zagreb, Faculty of Science, accessed 22 March 2025, https://urn.nsk.hr/urn:nbn:hr:217:587029

Orlić P. Gonality of modular curves and their quotients [Doctoral thesis]. Zagreb: University of Zagreb, Faculty of Science; 2025 [cited 2025 March 22] Available at: https://urn.nsk.hr/urn:nbn:hr:217:587029

P. Orlić, "Gonality of modular curves and their quotients", Doctoral thesis, University of Zagreb, Faculty of Science, Zagreb, 2025. Available at: https://urn.nsk.hr/urn:nbn:hr:217:587029

Metadata
TitleGonality of modular curves and their quotients
Title (croatian)Gonalnost modularnih krivulja i njihovih kvocijenata
AuthorPetar Orlić
MentorFilip Najman (mentor)
Committee memberMatija Kazalicki (predsjednik povjerenstva)
Committee memberFilip Najman (član povjerenstva)
Committee memberNikola Adžaga (član povjerenstva)
Committee memberAndrej Dujella (član povjerenstva)
GranterUniversity of Zagreb
Faculty of Science
(Department of Mathematics)
Zagreb
Defense date and country2025-01-29, Croatia
Scientific / art field,
discipline and subdiscipline		NATURAL SCIENCES
Mathematics
Universal decimal classification
(UDC)		51 - Mathematics
Abstract
Modular curves are one of the significant objects in number theory. One of their important properties is that points on modular curves represent classes of elliptic curves with some additional structure. Therefore, we can answer some questions regarding elliptic curves by studying modular curves. The central topic of this thesis is the modular curve X0(N) and its quotient curves. Points on the curve X0(N) parametrize classes of elliptic curves together with a cyclic subgroup of order... More N (or equivalently, a cyclic isogeny of degree N). For an algebraic curve defined over a field k, its k-gonality is the minimal possible degree of a morphism from that curve to the projective line P1. Problems concerning determining the Q-gonality and C-gonality of curves defined over Q are particularly interesting. One of the reasons for that is that preimages of degree d rational morphisms to P1 are the source of degree d points on curves. All cases when the C-gonality of the curve X0(N) is 2,3, or 4 were determined by Ogg, Hasegawa and Shimura, and Jeon and Park in [37, 53, 80]. All Q-trigonal curves X0(N) were also determined by Hasegawa and Shimura in the same paper. The author’s paper in collaboration with Filip Najman [76] determines all cases when the Q-gonality of the curve X0(N) is 4,5, or 6. The Q-gonality of all curves X0(N) for N≤144 is also determined there, along with the C-gonality for many of these curves. These results can be found in Section 2.1. For every divisor d of N such that gcd(d,N/d)=1 there exists an involution wd on X0(N) defined over Q, called an Atkin-Lehner involution. The curve X+d0(N) is a quotient curve of the modular curve X0(N) by wd. If d=N, we denote this curve as X+0(N). All cases when the C-gonality of the curve X+d0(N) is 2 or 3 were determined by Furumoto and Hasegawa and Hasegawa and Shimura in [32, 38]. The author’s papers [81, 83] determine all cases when the Q-gonality of the curve X+d0(N) is 3 or 4 and all cases when the C-gonality of the curve X+d0(N) is equal to 4. These results can be found in Section 2.2. For every group {±1}⊊Δ⊊(Z/NZ)×, there exists an intermediate modular curve XΔ(N) between the curves X1(N) and X0(N). Since the Q-gonality of curves X1(N) has been determined for N≤40 in [23] and we know the Q-gonality of curves X0(N) for N≤144, the question of determininng the Q-gonality of intermediate curves XΔ(N) naturally arises. All cases when the C-gonality of the curve XΔ(N) is 2 or 3 were determined by Ishii and Momose and Jeon and Kim in [43, 48]. The author's paper [82] determines all cases when the Q-gonality of the curve XΔ(N) is 4 or 5 and all cases when the C-gonality of the curve XΔ(N) is equal to 4. These results can be found in Section 2.3. The existence of rational maps to P1 and elliptic curves is closely related to the existence of infinitely many points of a certain degree over Q. Bars [6] determined all cases when the curve X0(N) has infinitely quadratic points and Jeon [45] determined all cases when the curve X0(N) has infinitely many cubic points. The author's paper in collaboration with Maarten Derickx [21] determines all cases when the curve X0(N) has infinitely many quartic points. These results can be found in Chapter 3. A lot of the results in this thesis rely on Magma [11] and Sage computations. The codes that verify all computations in this thesis can be found on https://github.com/orlic1 and https://github.com/koffie/mdsage/tree/main/articles/derickx_ orlic-quartic_X0. Less
Abstract (croatian)
Modularne krivulje jedan su od značajnijih predmeta istraživanja u teoriji brojeva. Jedno od njihovih važnijih svojstava je to da točke na modularnim krivuljama reprezentiraju klase eliptičkih krivulja s nekom dodatnom strukturom. Stoga na neka pitanja koja se tiču eliptičkih krivulja možemo odgovoriti proučavajući modularne krivulje. Središnji objekt ove disertacije je modularna krivulja X0(N) i njezine kvocijentne krivulje. Točke na krivulji X0(N) parametriziraju klase eliptičkih... More krivulja zajedno s cikličkom podgrupom reda N (ili ekvivalentno, s cikličkom izogenijom stupnja N). Za algebarsku krivulju definiranu nad poljem k, njezina k-gonalnost je minimalni mogući stupanj preslikavanja iz te krivulje u projektivni pravac P1. Osobito su zanimljivi problemi određivanja Q-gonalnosti i C-gonalnosti krivulja definiranih nad Q. Jedan od razloga za to sto su praslike racionalnih preslikavanja stupnja d u P1 izvor točaka stupnja d nad Q na krivuljama. Sve slučajeve kada je C-gonalnost krivulje X0(N) jednaka 2,3 ili 4 odredili su Ogg, Hasegawa i Shimura te Jeon i Park u [37, 53, 80]. Hasegawa i Shimura su u istom članku odredili i sve slučajeve kad je krivulja X0(N) Q-trigonalna. Autorov članak u suradnji s Filipom Najmanom [76] određuje sve slučajeve kada je Q-gonalnost krivulje X0(N) jednaka 4,5 ili 6. Q-gonalnost krivulja X0(N) za N≤144 je također određena u tom članku, kao i C-gonalnost mnogih od tih krivulja. Ovi rezultati se mogu naći u Poglavlju 2.1. Za svaki djelitelj d od N za koji je M(d,N/d)=1 postoji involucija wd krivulje X0(N) definirana nad Q koju zovemo Atkin-Lehnerova involucija. Krivulja X+d0(N) je kvocijentna krivulja modularne krivulje X0(N) po wd. Ako je d=N, označavamo ovu krivulju s X+0(N). Sve slučajeve kada je C-gonalnost krivulje X+d0(N) jednaka 2 ili 3 odredili su Furuvmoto i Hasegawa te Hasegawa i Shimura u [32, 38]. Autorovi članci [81, 83] određuju sve slučajeve kada je Q-gonalnost krivulje X+d0(N)) jednaka 3 ili 4 te sve slučajeve kada je C-gonalnost krivulje X+d0(N) jednaka 4. Ovi rezultati se mogu naći u Poglavlju 2.2. Za svaku grupu {±1}⊊Δ⊊(Z/NZ)× postoji modularna međukrivulja XΔ(N) koja se nalazi izmedu krivulja X1(N) i X0(N). Kako je Q-gonalnost krivulja X1(N) određena za N≤40 u [23] i kako znamo kolika je Q-gonalnost krivulja X0(N) za N≤144, prirodno se postavlja pitanje odredivanja Q-gonalnosti krivulja XΔ(N). Sve slučajeve kada je C-gonalnost krivulje XΔ(N) jednaka 2 ili 3 odredili su Ishii i Momose te Jeon i Kim u [43, 48]. Autorov članak [82] određuje sve slučajeve kada je Q-gonalnost krivulje XΔ(N) jednaka 4 ili 5 te sve slučajeve kada je C-gonalnost krivulje XΔ(N) jednaka 4. Ovi rezultati se mogu naći u Poglavlju 2.3. Postojanje racionalnog preslikavanja u P1 i eliptičke krivulje je usko povezano s postojanjem beskonačno mnogo točaka određenog stupnja nad Q. Bars [6] je odredio sve slučajeve kada krivulja X0(N) ima beskonačno mnogo kvadratnih točaka, a Jeon [45] je odredio sve slučajeve kada krivulja X0(N) ima beskonačno kubičnih točaka. Autorov članak u suradnji s Maartenom Derickxom [21] određuje sve slučajeve kada krivulja X0(N) ima beskonačno mnogo kvartičnih točaka. Ovi rezultati se mogu naći u Poglavlju 3. Dosta rezultata u ovoj disertaciji se oslanja na izračune u računalnim sustavima Magma [11] i Sage. Kodovi koji opravdavaju sve izračune u ovoj disertaciji se nalaze na https://github.com/orlic1 i https://github.com/koffie/mdsage/tree/main/articles/derickx_orlic-quartic_X0. Less
Keywords
Keywords (croatian)
Languageenglish
URN:NBNurn:nbn:hr:217:587029
ProjectNumber: IP-2022-10-5008
Title: Teorija brojeva i aritmetička geometrija
Title: Number theory and arithemtic geometry
Acronym: TEBAG
Leader: Filip Najman
Jurisdiction: Croatia
Funder: Hrvatska zaklada za znanost
Funding stream: Research Projects
Study programmeTitle: Doctoral study
Study programme type: university
Study level: postgraduate
Academic / professional title: doktor/doktorica u području prirodnih znanosti (doktor/doktorica u području prirodnih znanosti)
Type of resourceText
Extentviii, 174 str.
File originBorn digital
Access conditionsOpen access
Terms of useLicence In copyright icon
Created on2025-03-04 12:13:27
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