Gonality of modular curves and their quotients

doctoral thesis

• Title (croatian): Gonalnost modularnih krivulja i njihovih kvocijenata
• Author: Petar Orlić
• Mentor: Filip Najman (mentor)
• Committee member: Matija Kazalicki (predsjednik povjerenstva)
• Committee member: Filip Najman (član povjerenstva)
• Committee member: Nikola Adžaga (član povjerenstva)
• Committee member: Andrej Dujella (član povjerenstva)
• Granter: University of Zagreb; Faculty of Science; (Department of Mathematics); Zagreb
• Defense date and country: 2025-01-29, 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 $X_0(N)$ and its quotient curves. Points on the curve $X_0(N)$ 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 $\mathbb{P}^1$. Problems concerning determining the $\mathbb{Q}$-gonality and $\mathbb{C}$-gonality of curves defined over $\mathbb{Q}$ are particularly interesting. One of the reasons for that is that preimages of degree $d$ rational morphisms to $\mathbb{P}^1$ are the source of degree $d$ points on curves. All cases when the $\mathbb{C}$-gonality of the curve $X_0(N)$ is 2,3, or 4 were determined by Ogg, Hasegawa and Shimura, and Jeon and Park in [37, 53, 80]. All $\mathbb{Q}$-trigonal curves $X_0(N)$ 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 $\mathbb{Q}$-gonality of the curve $X_0(N)$ is 4,5, or 6. The $\mathbb{Q}$-gonality of all curves $X_0(N)$ for $N \leq 144$ is also determined there, along with the $\mathbb{C}$-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 $w_d$ on $X_0(N)$ defined over $\mathbb{Q}$, called an Atkin-Lehner involution. The curve $X^{+d}_0 (N)$ is a quotient curve of the modular curve $X_0(N)$ by $w_d$. If $d = N$, we denote this curve as $X_0^+(N)$. All cases when the $\mathbb{C}$-gonality of the curve $X^{+d}_ 0 (N)$ 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 $\mathbb{Q}$-gonality of the curve $X^{+d}_ 0 (N)$ is 3 or 4 and all cases when the $\mathbb{C}$-gonality of the curve $X^{+d}_ 0 (N)$ is equal to 4. These results can be found in Section 2.2. For every group $\{ \pm 1 \} \subsetneq \Delta \subsetneq (\mathbb{Z}/N\mathbb{Z})^{\times}$, there exists an intermediate modular curve $X_{\Delta}(N)$ between the curves $X_1(N)$ and $X_0(N)$. Since the $\mathbb{Q}$-gonality of curves $X_1(N)$ has been determined for $N \leq 40$ in [23] and we know the $\mathbb{Q}$-gonality of curves $X_0(N)$ for $N \leq 144$, the question of determininng the $\mathbb{Q}$-gonality of intermediate curves $X_{\Delta} (N)$ naturally arises. All cases when the $\mathbb{C}$-gonality of the curve $X_{\Delta} (N)$ 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 $\mathbb{Q}$-gonality of the curve $X_{\Delta} (N)$ is 4 or 5 and all cases when the $\mathbb{C}$-gonality of the curve $X_{\Delta} (N)$ is equal to 4. These results can be found in Section 2.3. The existence of rational maps to $\mathbb{P}^1$ and elliptic curves is closely related to the existence of infinitely many points of a certain degree over $\mathbb{Q}$. Bars [6] determined all cases when the curve $X_0(N)$ has infinitely quadratic points and Jeon [45] determined all cases when the curve $X_0(N)$ has infinitely many cubic points. The author's paper in collaboration with Maarten Derickx [21] determines all cases when the curve $X_0(N)$ 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 $X_0(N)$ i njezine kvocijentne krivulje. Točke na krivulji $X_0(N)$ 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 $\mathbb{P}^1$. Osobito su zanimljivi problemi određivanja $\mathbb{Q}$-gonalnosti i $\mathbb{C}$-gonalnosti krivulja definiranih nad $\mathbb{Q}$. Jedan od razloga za to sto su praslike racionalnih preslikavanja stupnja $d$ u $\mathbb{P}^1$ izvor točaka stupnja $d$ nad $\mathbb{Q}$ na krivuljama. Sve slučajeve kada je $\mathbb{C}$-gonalnost krivulje $X_0(N)$ 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 $X_0(N)$ $\mathbb{Q}$-trigonalna. Autorov članak u suradnji s Filipom Najmanom [76] određuje sve slučajeve kada je $\mathbb{Q}$-gonalnost krivulje $X_0(N)$ jednaka 4,5 ili 6. $\mathbb{Q}$-gonalnost krivulja $X_0(N)$ za $N \leq 144$ je također određena u tom članku, kao i $\mathbb{C}$-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 $w_d$ krivulje $X_0(N)$ definirana nad $\mathbb{Q}$ koju zovemo Atkin-Lehnerova involucija. Krivulja $X^{+d}_0 (N)$ je kvocijentna krivulja modularne krivulje $X_0(N)$ po $w_d$. Ako je $d = N$, označavamo ovu krivulju s $X_0^+(N)$. Sve slučajeve kada je $\mathbb{C}$-gonalnost krivulje $X^{+d}_ 0 (N)$ 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 $\mathbb{Q}$-gonalnost krivulje $X^{+d}_ 0 (N)$) jednaka 3 ili 4 te sve slučajeve kada je $\mathbb{C}$-gonalnost krivulje $X^{+d}_ 0 (N)$ jednaka 4. Ovi rezultati se mogu naći u Poglavlju 2.2. Za svaku grupu $\{ \pm 1 \} \subsetneq \Delta \subsetneq (\mathbb{Z}/N\mathbb{Z})^{\times}$ postoji modularna međukrivulja $X_{\Delta}(N)$ koja se nalazi izmedu krivulja $X_1(N)$ i $X_0(N)$. Kako je $\mathbb{Q}$-gonalnost krivulja $X_1(N)$ određena za $N \leq 40$ u [23] i kako znamo kolika je $\mathbb{Q}$-gonalnost krivulja $X_0(N)$ za $N \leq 144$, prirodno se postavlja pitanje odredivanja $\mathbb{Q}$-gonalnosti krivulja $X_{\Delta}(N)$. Sve slučajeve kada je $\mathbb{C}$-gonalnost krivulje $X_{\Delta} (N)$ jednaka 2 ili 3 odredili su Ishii i Momose te Jeon i Kim u [43, 48]. Autorov članak [82] određuje sve slučajeve kada je $\mathbb{Q}$-gonalnost krivulje $X_{\Delta} (N)$ jednaka 4 ili 5 te sve slučajeve kada je $\mathbb{C}$-gonalnost krivulje $X_{\Delta} (N)$ jednaka 4. Ovi rezultati se mogu naći u Poglavlju 2.3. Postojanje racionalnog preslikavanja u $\mathbb{P}^1$ i eliptičke krivulje je usko povezano s postojanjem beskonačno mnogo točaka određenog stupnja nad $\mathbb{Q}$. Bars [6] je odredio sve slučajeve kada krivulja $X_0(N)$ ima beskonačno mnogo kvadratnih točaka, a Jeon [45] je odredio sve slučajeve kada krivulja $X_0(N)$ ima beskonačno kubičnih točaka. Autorov članak u suradnji s Maartenom Derickxom [21] određuje sve slučajeve kada krivulja $X_0(N)$ 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: Elliptic curve; Modular curves; Gonality
• Keywords (croatian): Eliptičke krivulje; Modularne krivulje; Gonalnost
• Language: english
• URN:NBN: urn:nbn:hr:217:587029
• Project: Number: 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 programme: Title: 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 resource: Text
• Extent: viii, 174 str.
• File origin: Born digital
• Access conditions: Open access
• Created on: 2025-03-04 12:13:27