arXiv subjects

Filip Najman

Publications and source records attributed to Filip Najman.

At least 19 recordsLinked to original sources

arXiv2026

Rational torsion on simple genus two Jacobians

We exhibit new subgroups of rational torsion points in geometrically simple Jacobians of genus-two curves over $\mathbb Q$. The largest group, which has order 96 and invariants [2,2,2,12], is realized by curves of the form $y^2 = x(x-a^2)(x-b^2)(x-c^2)(x-u^2)(x-v^2)$ where $a,b,c,u,v$ are positive integers that satisfy $a^2 + b^2 + c^2 = u^2 + v^2$ and $a^4 + b^4 + c^4 = u^4 + v^4$. We also find realizations of the groups [2,2,20], [2,2,4,4], [2,2,2,8], [2,4,8], and [6,6]. Finally, we record, to the best of our knowledge, all known subgroups that arise in genus-two Jacobians over $\mathbb Q$, in the geometrically simple case and in general.

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arXiv2026

Rational points on modular curves: parameterization and geometric explanations

We show that, conditional on Zywina's effective version of the Serre uniformity conjecture, there is a natural way to parameterize non-CM $\mathbb{Q}$-rational points on all modular curves in terms of the rational points on finitely many modular curves. Our proof refines Zywina's work to give a (conditional) parameterization of the images of adelic Galois representations of elliptic curves. In particular, we show that there are 41 $j$-invariants of elliptic curves whose associated Galois image does not vary in an infinite family. Using our explicit parameterization, we show that all rational points on all modular curves arise from the geometry of modular curves in a formal sense, confirming a philosophy of Mazur and Ogg.

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arXiv2026

Torsion groups of elliptic curves that appear infinitely often over septic, octic and nonic fields

We determine the sets $\Phi^\infty(n)$ of abelian groups that appear as torsion groups of infinitely many elliptic curves, up to $\overline \Q$-isomorphism, over number fields of degree $n=7,8$ and $9$. The proof translates the problem into one about low-degree points on modular curves $X_1(m,n)$. We construct the infinite families using modular units, and eliminate the remaining candidates using finite-field gonality computations, covering arguments, and a specialization argument for $W^0_d$. The most difficult case is $X_1(37)$ in degree $9$, where the Jacobian has positive rank. We handle this case by showing that $W^0_9(X_1(37)_{\F_2})$ contains no translate of the positive-rank elliptic factor induced by the morphism $X_1(37)\to X_0^+(37)$.

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arXiv2025

Sporadic points on $X_0(N)$

We determine all integers $N$ for which the modular curve $X_0(N)$ admits a sporadic CM point (of any degree), as well as all $N$ for which $X_0(N)$ admits a sporadic point, whether CM or non-CM. In a sense, our results generalize the classification of isogenies of elliptic curves over $\Q$ due to Mazur and Kenku: their work determines the $X_0(N)$ with degree 1 sporadic points, whereas we classify all $X_0(N)$ that have a sporadic point of arbitrary degree.

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arXiv2025

Quadratic points on modular curves $X_0(N)$ for $N\leq 100$

We determine the quadratic points on the modular curves $X_0(N)$ for $N\leq 100$ for which this has not been previously done, namely the cases $$N\in\{66,70,78,82,84,86,87,88,90,96,99\}.$$ We accomplish this by improving on the ``going down method," which uses the fact that we have a moduli description of all the (infinitely many) quadratic points on $X_0(n)$ for some divisor $n$ of $N$.

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arXiv2024

Classification of torsion of elliptic curves over quartic fields

Let $E$ be an elliptic curve over a quartic field $K$. By the Mordell-Weil theorem, $E(K)$ is a finitely generated group. We determine all the possibilities for the torsion group $E(K)_{tor}$ where $K$ ranges over all quartic fields $K$ and $E$ ranges over all elliptic curves over $K$. We show that there are no sporadic torsion groups, or in other words, that all torsion groups either do not appear or they appear for infinitely many non-isomorphic elliptic curves $E$. Proving this requires showing that numerous modular curves $X_1(m,n)$ have no non-cuspidal degree $4$ points. We deal with almost all the curves using one of 3 methods: a method for the rank 0 cases requiring no computation; the Hecke sieve, a local method requiring computer-assisted computations; and the global method, an argument for the positive rank cases also requiring no computation. We deal with the handful of remaining cases using ad hoc methods.

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arXiv2023

Towards a classification of isolated $j$-invariants

We develop an algorithm to test whether a non-CM elliptic curve $E/\mathbb{Q}$ gives rise to an isolated point of any degree on any modular curve of the form $X_1(N)$. This builds on prior work of Zywina which gives a method for computing the image of the adelic Galois representation associated to $E$. Running this algorithm on all elliptic curves presently in the $L$-functions and Modular Forms Database and the Stein-Watkins Database gives strong evidence for the conjecture that $E$ gives rise to an isolated point on $X_1(N)$ if and only if $j(E)=-140625/8, -9317,$ $351/4$, or $-162677523113838677$.

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arXiv2023

Quadratic points on $X_0(163)$

We determine all the quadratic points on the genus $13$ modular curve $X_0(163)$, thus completing the answer to a recent question of Banwait, the second-named author, and Padurariu. In doing so, we investigate a curious phenomenon involving a cubic point with complex multiplication on the curve $X_0(163)$. This cubic point prevents us, due to computational restraints, from directly applying the state-of-the-art Atkin--Lehner sieve for computing quadratic points on modular curves $X_0(N)$. To overcome this issue, we introduce a technique which allows us to work with the Jacobian of curves modulo primes by directly computing linear equivalence relations between divisors.

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arXiv2023

On $r$-isogenies over $\mathbb{Q}(\zeta_r)$ of elliptic curves with rational $j$-invariants

The main goal of this paper is to determine for which prime numbers $r\geq 3$ can an elliptic curve~$E$ defined over $\mathbb Q$ have an $r$-isogeny over $\mathbb Q(\zeta_r)$. We study this question under various assumptions on the 2-torsion of $E$. Apart from being a natural question itself, the mod~$r$ representations attached to such $E$ arise in the Darmon program for the generalized Fermat equation of signature $(r,r,p)$, playing a key role in the proof of modularity of certain Frey varieties in the recent work of Billerey, Chen, Dieulefait and Freitas.

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arXiv2023

Hyperelliptic and trigonal modular curves in characteristic $p$

Let $X_\Delta(N)$ be an intermediate modular curve of level $N$, meaning that there exist (possibly trivial) morphisms $X_1(N)\rightarrow X_\Delta(N) \rightarrow X_0(N)$. For all such intermediate modular curves, we give an explicit description of all primes $p$ such that $X_\Delta(N)_{\overline{\mathbb F}_p}$ is either hyperelliptic or trigonal. Furthermore we also determine all primes $p$ such that $X_\Delta(N)_{\mathbb F_p}$ is trigonal. This is done by first using the Castelnuovo-Severi inequality to establish a bound $N_0$ such that if $X_0(N)_{{\overline{\mathbb F}_p}}$ is hyperelliptic or trigonal, then $N \leq N_0$. To deal with the remaining small values of $N$, we develop a method based on the careful study of the canonical ideal to determine, for a fixed curve $X_\Delta(N)$, all the primes $p$ such that the $X_\Delta(N)_{ {\overline{\mathbb F}_p}}$ is trigonal or hyperelliptic. Furthermore, using similar methods, we show that $X_\Delta(N)_{{\overline{\mathbb F}_p}}$ is not a smooth plane quintic, for any $N$ and any $p$.

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arXiv2023

Computing quadratic points on modular curves $X_0(N)$

In this paper we improve on existing methods to compute quadratic points on modular curves and apply them to successfully find all the quadratic points on all modular curves $X_0(N)$ of genus up to $8$, and genus up to $10$ with $N$ prime, for which they were previously unknown. The values of $N$ we consider are contained in the set \[ \mathcal{L}=\{58, 68, 74, 76, 80, 85, 97, 98, 100, 103, 107, 109, 113, 121, 127 \}.\] We obtain that all the non-cuspidal quadratic points on $X_0(N)$ for $N\in \mathcal{L}$ are CM points, except for one pair of Galois conjugate points on $X_0(103)$ defined over $\mathbb{Q}(\sqrt{2885})$. We also compute the $j$-invariants of the elliptic curves parametrised by these points, and for the CM points determine their geometric endomorphism rings.

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arXiv2022

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 $\mathbb C$.

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arXiv2022

Two results on $x^r + y^r = dz^p$

This note proves two theorems regarding Fermat-type equation $x^r + y^r = dz^p$ where $r \geq 5$ is a prime. Our main result shows that, for infinitely many integers~$d$, the previous equation has no non-trivial primitive solutions such that $2 \mid x+y$ or $r \mid x+y$, for a set of exponents $p$ of positive density. We use the modular method with a symplectic argument to prove this result.

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arXiv2022

Cyclic isogenies of elliptic curves over fixed quadratic fields

Building on Mazur's 1978 work on prime degree isogenies, Kenku determined in 1981 all possible cyclic isogenies of elliptic curves over $\mathbb{Q}$. Although more than 40 years have passed, the determination of cyclic isogenies of elliptic curves over a single other number field has hitherto not been realised. In this paper we develop a procedure to assist in establishing such a determination for a given quadratic field. Executing this procedure on all quadratic fields $\mathbb{Q}(\sqrt{d})$ with $|d| < 10^4$ we obtain, conditional on the Generalised Riemann Hypothesis, the determination of cyclic isogenies of elliptic curves over $19$ quadratic fields, including $\mathbb{Q}(\sqrt{213})$ and $\mathbb{Q}(\sqrt{-2289})$. To make this procedure work, we determine all of the finitely many quadratic points on the modular curves $X_0(125)$ and $X_0(169)$, which may be of independent interest.

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arXiv2021

Quadratic points on bielliptic modular curves

Bruin and Najman, Ozman and Siksek, and Box described all the quadratic points on the modular curves of genus $2\leq g(X_0(n)) \leq 5$. Since all the hyperelliptic curves $X_0(n)$ are of genus $\leq 5$ and as a curve can have infinitely many quadratic points only if it is either of genus $\leq 1$, hyperelliptic or bielliptic, the question of describing the quadratic points on the bielliptic modular curves $X_0(n)$ naturally arises; this question has recently also been posed by Mazur. We answer Mazur's question completely and describe the quadratic points on all the bielliptic modular curves $X_0(n)$ for which this has not been done already. The values of $n$ that we deal with are $n=60,62,69,79,83,89,92,94,95,101,119$ and $131$; the curves $X_0(n)$ are of genus up to $11$. We find all the exceptional points on these curves and show that they all correspond to CM elliptic curves. The two main methods we use are Box's relative symmetric Chabauty method and an application of a moduli description of $\Q$-curves of degree $d$ with an independent isogeny of degree $m$, which reduces the problem to finding the rational points on several quotients of modular curves.

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arXiv2021

Sporadic points of odd degree on $X_1(N)$ coming from $\mathbb{Q}$-curves

We say a closed point $x$ on a curve $C$ is sporadic if there are only finitely many points on $C$ of degree at most deg$(x)$. In the case where $C$ is the modular curve $X_1(N)$, most known examples of sporadic points come from elliptic curves with complex multiplication (CM). We seek to understand all sporadic points on $X_1(N)$ corresponding to $\mathbb{Q}$-curves, which are elliptic curves isogenous to their Galois conjugates. This class contains not only all CM elliptic curves, but also any elliptic curve $\overline{\mathbb{Q}}$-isogenous to one with a rational $j$-invariant, among others. In this paper, we show that all non-CM $\mathbb{Q}$-curves giving rise to a sporadic point of odd degree lie in the $\overline{\mathbb{Q}}$-isogeny class of the elliptic curve with $j$-invariant $-140625/8$. In addition, we show that a stronger version of this finiteness result would imply Serre's Uniformity Conjecture.

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arXiv2020

Splitting of primes in number fields generated by points on some modular curves

We study the splitting of primes in number fields generated by points on modular curves. Momose was the first to notice that quadratic points on $X_1(n)$ generate quadratic fields over which certain primes split in a particular way and his results were later expanded upon by Krumm. We prove results about the splitting behaviour of primes in quadratic fields generated by points on the modular curves $X_0(n)$ which are hyperelliptic (except for $n=37$) and in cubic fields generated by points on $X_1(2, 14)$.

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arXiv2020

Counting elliptic curves with prescribed level structures over number fields

Harron and Snowden counted the number of elliptic curves over $\mathbb{Q}$ up to height $X$ with torsion group $G$ for each possible torsion group $G$ over $\mathbb{Q}$. In this paper we generalize their result to all number fields and all level structures $G$ such that the corresponding modular curve $X_G$ is a weighted projective line $\mathbb{P}(w_0,w_1)$ and the morphism $X_G\to X(1)$ satisfies a certain condition. In particular, this includes all modular curves $X_1(m,n)$ with coarse moduli space of genus $0$. We prove our results by defining a size function on $\mathbb{P}(w_0,w_1)$ following unpublished work of Deng, and working out how to count the number of points on $\mathbb{P}(w_0,w_1)$ up to size $X$.

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