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Maarten Derickx

Publications and source records attributed to Maarten Derickx.

At least 19 recordsLinked to original sources
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

Computing class groups and gonalities of algebraic curves over finite fields

We give practical algorithms for computing the divisor class group and the gonality of a curve over a finite field, achieving several orders of magnitude speedup over existing methods for sufficiently large genus or residue field. The approach relies on introducing a precomputation step involving power series-expansions, which allows for an efficient amortized computation of large numbers of Riemann-Roch spaces.

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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

Class groups of imaginary quadratic points on $X_1(16)$

The main result is to show that if $K \ncong \mathbb Q(\sqrt{-15})$ is an imaginary quadratic field and $E$ is an elliptic curve over $K$ with a torsion point of order 16, then the class number of $K$ is divisible by 10. This gives an affirmative answer to a 12 year old question by David Krumm. This is done by setting up a more general framework for studying divisibility of class groups of imaginary quadratic points on hyper-elliptic curves and applying it to $X_1(16)$.

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arXiv2025

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 $X_0(N)$.

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arXiv2025

Prime order torsion on elliptic curves over number fields. Part I: Asymptotics

We study the asymptotics of the set $S(d)$ of possible prime orders of $K$-rational points on elliptic curves over number fields $K$ of degree $d$ as $d$ tends to infinity. Assuming some conjectures on the sparsity of newforms of weight $2$ and prime level with unexpectedly high analytic rank, we show that $\max S(d) \le 3d + 1$ for sufficiently large even $d$ and $\max S(d) = o(d)$ for odd $d$.

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arXiv2025

Intermediate modular curves with infinitely many quartic points

For every group $\{\pm1\}\subseteq \Delta\subseteq (\mathbb Z/N\mathbb Z)^\times$, there exists an intermediate modular curve $X_\Delta(N)$. In this paper we determine all curves $X_\Delta(N)$ 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 $X_\Delta(N)$ to an elliptic curve.

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arXiv2025

Contracting low degree points on curves

The main result of this article is that all but finitely many points of small enough degree on a curve can be written as a pullback of a smaller degree point. The main theorem has several corollaries that yield improvements on results of Kadets and Vogt, Khawaja and Siksek, and Vojta under a slightly stronger assumption on the degree of the points.

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arXiv2025

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 $N$.

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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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arXiv2024

Large degree primitive points on curves

A number field $K$ is called primitive if $\mathbb Q$ and $K$ are the only subfields of $K$. Let $X$ be a nice curve over $\mathbb Q$ of genus $g$. A point $P$ of degree $d$ on $X$ is called primitive if the field of definition $\mathbb Q(P)$ of the point is primitive. In this short note we prove that if $X$ has a divisor of degree $d> 2g$, then $X$ has infinitely many primitive points of degree $d$. This complements the results of Khawaja and Siksek that show that points of low degree are not primitive under certain conditions.

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arXiv2024

Torsion subgroups of elliptic curves over quadratic fields and a conjecture of Granville

We study the problem of determining the groups that can arise as the torsion subgroup of an elliptic curve over a fixed quadratic field, building on work of Kamienny-Najman, Krumm, and Trbovi\'c. By employing techniques to study rational points on curves developed by Bruin and Stoll, we determine the possible torsion subgroups of elliptic curves over quadratic fields $\mathbb{Q}(\sqrt{d})$ for all squarefree $d$ with $|d| < 800$, improving on the previously known range of $-5 < d < 26$. We use our computations to study the validity of a conjecture of Granville concerning how many twists of a given hyperelliptic curve admit a nontrivial rational point.

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arXiv2023

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.

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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

Towards strong uniformity for isogenies of prime degree

Let $E$ be an elliptic curve over a number field $k$ of degree $d$ that admits a $k$-rational isogeny of prime degree $p$. We study the question of finding a uniform bound on $p$ that depends only on $d$, and obtain, under a certain condition on the signature of the isogeny, such a uniform bound by explicitly constructing nonzero integers that $p$ must divide. As a corollary we find a uniform bound on torsion points defined over unramified extensions of the base field, generalising Merel's Uniform Boundedness result for torsion.

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arXiv2022

Computing classical modular forms

We discuss practical and some theoretical aspects of computing a database of classical modular forms in the L-functions and Modular Forms Database (LMFDB).

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arXiv2022

Explicit isogenies of prime degree over number fields

We provide an explicit and algorithmic version of a theorem of Momose classifying isogenies of prime degree of elliptic curves over number fields, which we implement in Sage and PARI/GP. Combining this algorithm with recent work of Box-Gajovi\'c-Goodman we determine the first instances of isogenies of prime degree for cubic number fields, as well as for several quadratic fields not previously known. While the correctness of the general algorithm relies on the Generalised Riemann Hypothesis, the algorithm is unconditional for the restricted class of semistable elliptic curves.

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