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

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).Comment: 17 pages. In this version, Theorem 3 is extended from d <= 6 to d <=

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.Comment: 19 pages, comments welcom

Hyperelliptic and trigonal modular curves in characteristic pp

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 $p$.Comment: 16 page

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.Comment: 14 page

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 $d = 4,5$ and $6$

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).Comment: 62 pages; minor edit

Torsion points on elliptic curves over quintic and sextic number fields

The determination of which finite abelian groups can occur as the torsion subgroup of an elliptic curve over a number field has a long history starting with Barry Mazur who proved that there are exactly 15 groups that can occur as the torsion subgroup of an elliptic curve over the rational numbers. It is a theorem due to Loïc Merel that for every integer d the set of isomorphism classes of groups occurring as the torsion subgroup of a number field of degree d is finite. If a torsion subgroup occurs for a certain degree, then one can also ask for how many distinct pairwise non-isomorphic elliptic curves this happens. The question which torsion groups can occur for infinitely many non-isomorphic elliptic curves of a fixed degree is studied during this talk. The main result is a complete classification of the torsion subgroups that occur infinitely often for degree 5 and 6. This is joint work with Andrew Sutherland and heavily builds on previous joint work with Mark van Hoeij.Non UBCUnreviewedAuthor affiliation: Universiteit LeidenGraduat