arXiv subjects

Petar Orlić

Publications and source records attributed to Petar Orlić.

11 recordsLinked to original sources

arXiv2026

$D$-elliptic modular curves $X_0(N)$

We present a method for determining whether there exists a degree $D$ rational map from $X_0(N)$ to some elliptic curve $E/\mathbb{Q}$. Moreover, we explain how to obtain a quadratic form that represents all possible degrees of such maps using the degree pairing method. This method previously required odd analytic rank and some additional technical conditions to be satisfied. In this paper, we extended the method to even rank curves as well and remove all technical conditions. We also show how to prove or disprove the existence of degree $D$ maps to elliptic curves by a lifting criterion on the lattices $H_1(-,\mathbb{Z})$ As an application of this method, we determine all $D$-elliptic curves $X_0(N)$ for all $D\leq 100$. Interestingly, in some cases we found degree $D$ rational maps that we could not explain by degeneracy maps, quotient maps, and modular parameterisation.

math.NT↗

arXiv2026

Modular curves $X_0(N)$ of density degree $5$

We determine all modular curves $X_0(N)$ with density degree $5$, i.e. all curves $X_0(N)$ with infinitely many points of degree $5$ and only finitely many points of degree $d\leq4$. As a consequence, the problem of determining all curves $X_0(N)$ with infinitely many points of degree $5$ remains open for only $30$ levels $N$.

math.NT↗

arXiv2026

Morphisms on the modular curve $X_0(p)$ and degree $6$ points

Let $p$ be a prime. We study non-constant morphisms $f:X_0(p)_\mathbb \to Y$, where $Y/\mathbb Q$ is a curve of genus $\geq 2$. We prove that for $p<3000$ such an $f$ of degree $d>1$ must be isomorphic to the quotient map $X_0(p)\to X_0^+(p)$. Supported by computational and theoretical evidence, we also conjecture that this is true for all primes $p$. These results allow us to classify all points of degree $\leq 25$ on $X_0(p)$ that come from a map to some curve of genus $\geq 2$. As an application, we were able to determine all curves $X_0(p)$ with infinitely many points of degree $6$ over $\mathbb Q$ except for $p=193$, continuing the previous results on small degree points on $X_0(N)$.

math.AG↗

arXiv2025

Tetragonal modular quotients of $X_0(N)$

Let $N$ be a positive integer. For every $d\mid N$ such that $(d,N/d)=1$ there exists an Atkin-Lehner involution $w_d$ of the modular curve $X_0(N)$. Let $B(N)$ be the group of all such involutions. In this paper we determine all $\mathbb C$ and $\mathbb Q$-tetragonal quotient curves $X_0(N)/W_N$, where $W_N\subseteq B(N)$ such that $4\leq|W_N|\leq 2^{ω(N)-1}$, thus completing the classification of all $\mathbb C$-tetragonal quotients of $X_0(N)$ by Atkin-Lehner involutions.

math.NT↗

arXiv2025

Tetragonal modular quotients $X_0^*(N)$

Let $N$ be a positive integer. For every $d\mid N$ such that $(d,N/d)=1$ there exists an Atkin-Lehner involution $w_d$ of the modular curve $X_0(N)$. The curve $X_0^*(N)$ is a quotient curve of $X_0(N)$ by $B(N)$, the group of all involutions $w_d$. In this paper we determine all quotient curves $X_0^*(N)$ whose $\mathbb C$-gonality is equal to $4$. We also determine all curves $X_0^*(N)$ whose $\mathbb Q$-gonality is equal to $4$ with the exception of level $N=378$.

math.NT↗

arXiv2025

Intermediate modular curves with infinitely many quartic points

For every group $\{\pm1\}\subseteq Δ\subseteq (\mathbb Z/N\mathbb Z)^\times$, there exists an intermediate modular curve $X_Δ(N)$. In this paper we determine all curves $X_Δ(N)$ with infinitely many points of degree $4$ over $\mathbb Q$. To do that, we developed a method to compute possible degrees of rational morphisms from $X_Δ(N)$ to an elliptic curve.

math.NT↗

arXiv2025

Tetragonal modular quotients $X_0^{+d}(N)$

Let $N$ be a positive integer. For every $d | N$ such that $(d, N/d) = 1$ there exists an Atkin-Lehner involution $w_d$ of the modular curve $X_0(N)$. In this paper we determine all quotient curves $X_0(N)/w_d$ whose $\mathbb{Q}$-gonality is equal to $4$ and all quotient curves $X_0(N)/w_d$ whose $\mathbb{C}$-gonality is equal to $4$.

math.NT↗

arXiv2024

Tetragonal intermediate modular curves

For every group $\{\pm1\}\subseteq Δ\subseteq (\mathbb{Z}/N\mathbb{Z})^\times$, there exists an intermediate modular curve $X_Δ(N)$. In this paper we determine all curves $X_Δ(N)$ whose $\mathbb{Q}$-gonality is equal to $4$, all curves $X_Δ(N)$ whose $\mathbb{C}$-gonality is equal to $4$, and all curves $X_Δ(N)$ whose $\mathbb{Q}$-gonality is equal to $5$. We also determine the $\mathbb{Q}$-gonality of all curves $X_Δ(N)$ for $N\leq 40$ and $\{\pm1\}\subsetneq Δ\subsetneq (\mathbb{Z}/N\mathbb{Z})^\times$.

math.NT↗

arXiv2024

Tetragonal modular quotients $X_0^+(N)$

In this paper we determine all quotient curves $X_0^+(N)$ whose $\mathbb{Q}$ or $\mathbb{C}$-gonality is equal to $4$. As a consequence, we find several new cases when the modular curve $X_0(N)$ has $\mathbb{Q}$-gonality equal to $8$.

math.NT↗

arXiv2024

Modular curves $X_0(N)$ with infinitely many quartic points

We determine all modular curves $X_0(N)$ with infinitely many quartic points. To do this, we define a pairing that induces a quadratic form representing all possible degrees of a rational morphism from $X_0(N)$ to a positive rank elliptic curve.

math.NT↗

arXiv2023

Gonality of the modular curve $X_0(N)$

In this paper we determine the $\mathbb Q$-gonalities of the modular curves $X_0(N)$ for all $N<145$. We determine the $\mathbb C$-gonality of many of these curves and the $\mathbb Q$-gonalities and $\mathbb C$-gonalities for many larger values of $N$. Using these results and some further work, we determine all the modular curves $X_0(N)$ of gonality $4$, $5$ and $6$ over $\mathbb Q$. We also find the first known instances of pentagonal curves $X_0(N)$ over $\mathbb C$.

math.NT↗