arXiv subjects

Maarten Derickx

Publications and source records attributed to Maarten Derickx.

28 records · Page 2Linked to original sources

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 Trbović. 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.

math.NT↗

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).

math.NT↗

arXiv2021

Torsion points on elliptic curves over number fields of small degree

We determine the set $S(d)$ of possible prime orders of $K$-rational points on elliptic curves over number fields $K$ of degree $d$, for $d = 4$, $5$, $6$, and $7$.

math.NT↗

arXiv2021

Elliptic curves with a point of order 13 defined over cyclic cubic fields

We show that there is essentially a unique elliptic curve $E$ defined over a cubic Galois extension $K$ of $\mathbb Q$ with a $K$-rational point of order 13 and such that $E$ is not defined over $\mathbb Q$.

math.NT↗

arXiv2019

Groups of generalized $G$-type and applications to torsion subgroups of rational elliptic curves over infinite extensions of $\mathbb{Q}$

Recently there has been much interest in studying the torsion subgroups of elliptic curves base-extended to infinite extensions of $\mathbb{Q}$. In this paper, given a finite group $G$, we study what happens with the torsion of an elliptic curve $E$ over $\mathbb{Q}$ when changing base to the compositum of all number fields with Galois group $G$. We do this by studying a group theoretic condition called generalized $G$-type, which is a necessary condition for a number field with Galois group $H$ to be contained in that compositum. In general, group theory allows one to reduce the original problem to the question of finding rational points on finitely many modular curves. To illustrate this method we completely determine which torsion structures occur for elliptic curves defined over $\mathbb{Q}$ and base-changed to the compositum of all fields whose Galois group is $A_4$.

math.NT↗

arXiv2019

Elliptic Curves over Totally Real Cubic Fields are Modular

We prove that all elliptic curves defined over totally real cubic fields are modular. This builds on previous work of Freitas, Le Hung and Siksek, who proved modularity of elliptic curves over real quadratic fields, as well as recent breakthroughs due to Thorne and to Kalyanswamy.

math.NT↗

arXiv2018

Torsion of elliptic curves over cyclic cubic fields

We determine all the possible torsion groups of elliptic curves over cyclic cubic fields, over non-cyclic totally real cubic fields and over complex cubic fields.

math.NT↗

arXiv2016

Torsion subgroups of elliptic curves over quintic and sextic number fields

Let $Φ^\infty(d)$ denote the set of finite abelian groups that occur infinitely often as the torsion subgroup of an elliptic curve over a number field of degree $d$. The sets $Φ^\infty(d)$ are known for $d\le 4$. In this article we determine $Φ^\infty(5)$ and $Φ^\infty(6)$.

math.NT↗

arXiv2014

Gonality of the modular curve X1(N)

In this paper we compute the gonality over Q of the modular curve X1(N) for all N <= 40 and give upper bounds for each N <= 250. This allows us to determine all N for which X1(N) has infinitely points of degree <= 8. We conjecture that the modular units of Q(X1(N)) are freely generated by f_2,...,f_{[N/2]+1} where f_k is obtained from the equation for X1(k).

math.NT↗

arXiv2014

Computing Galois representations and equations for modular curves $X_H(\ell)$

We construct plane models of the modular curve $X_H(\ell)$, and use their explicit equations to compute Galois representations associated to modular forms for values of $\ell$ that are significantly higher than in prior works.

math.NT↗