Explicit classification for torsion subgroups of rational points of elliptic curves

Given an elliptic curve $E/\mathbb{Q}$ with torsion subgroup $G = E(\mathbb{Q})_{\rm tors}$ we study what groups (up to isomorphism) can occur as the torsion subgroup of $E$ base-extended to $K$, a degree 6 extension of $\mathbb{Q}$. We also determine which groups $H = E(K)_{\rm tors}$ can occur infinitely often and which ones occur for only finitely many curves. This article is a first step towards a complete classification of torsion growth of over sextic fields.

Let K be a number field and let $\mathcal{E}$ be an elliptic curve defined over $K$. Let $m$ be a positive integer. We denote by $K(\mathcal{E}[m])$ the number fields obtained by adding to $K$ the coordinates of the $m$-torsion points of $\mathcal{E}$. We look for small (sometimes "minimal") set of generators of $K(\mathcal{E}[m])$. For $m=3$ and $m=4$, we describe explicit generators, degree and Galois groups of the extensions $K(\mathcal{E}[m])/K$.

The possible torsion groups of elliptic curves induced by Diophantine triples over quadratic fields, which do not appear over Q, are Z/2Z x Z/10Z, Z/2Z x Z/12Z and Z/4Z x Z/4Z. In this paper, we show that all these torsion groups indeed appear over some quadratic field. Moreover, we prove that there are infinitely many Diophantine triples over quadratic fields which induce elliptic curves with these torsion groups.

In this paper we classify the complex elliptic curves $E$ for which there exist cyclic subgroups $C\leq (E,+)$ of order $n$ such that the elliptic curves $E$ and $E/C$ are isomorphic, where $n$ is a positive integer. Important examples are provided in the last section. Moreover, we answer the following question: given a complex elliptic curve E, when can one find a cyclic subgroup $C$ of order $n$ of $(E,+)$ such that $(E,C)\sim(\frac{E}{C},\frac{E[n]}{C})$, $E[n]$ being the ...

Let $E/\mathbb{Q}$ be an elliptic curve and $p \in \{5,7,11 \}$ be a prime. We determine the possibilities for $E(\mathbb{Q}(\zeta_{p}))_{tors}$. Additionally, we determine all the possibilities for $E(\mathbb{Q}(\zeta_{16}))_{tors}$ and $E(\mathbb{Q}(\zeta_{27}))_{tors}$. Using these results we are able to determine the possibilities for $E(\mathbb{Q}(\mu_{p^{\infty}}))_{tors}$.

We give a classification of the cuspidal automorphic representations attached to rational elliptic curves with a non-trivial torsion point of odd order. Such elliptic curves are parameterizable, and in this paper, we find the necessary and sufficient conditions on the parameters to determine when split or non-split multiplicative reduction occurs. Using this and the known results on when additive reduction occurs for these parametrized curves, we classify the automorphic repr...

In this note we present the main details of the construction of an elliptic curve over $\mathbb{Q}(u)$ with torsion $\mathbb{Z}/4\mathbb{Z}$ and rank 6. Previously only rank 5 examples for such curves were known.

Given an elliptic curve over a field of characteristic different from 2,3, its discriminant defines a $\mu_{12}$-torsor over the field. In this paper, we give an explicit description of this $\mu_{12}$-torsor in terms of the 3-torsion points and of the 4-torsion points on the given elliptic curve. %In addition, we show that such a description involves the Weil pairing in a certain way. As an application, we generalize a result of Coates on the 12-th root of the discriminant o...

By focusing on the family $E:y^2=x^3+a$, we present strategies for determining the structure of the torsion subgroup of the Mordell-Weil group of an elliptic curve, $E(K)$, over quadratic field $K$. Generalizations of the Nagell-Lutz theorem and Mazur's theorem to curves defined over quadratic fields allows us to determine the full torsion subgroup of $E(K)$ as one of at most three possibilities when $a$ is a square.

Mordell curves over a number field $K$ are elliptic curves of the form $ y^2 = x^3 + c$, where $c \in K \setminus \{ 0 \}$. Let $p \geq 5$ be a prime number, $K$ a number field such that $[K:\mathbb{Q}] \in \{ 2p, 3p \}$ and let $E$ be a Mordell curve defined over $K$. We classify all the possible torsion subgroups $E(K)_{\text{tors}}$ for all Mordell curves $E$ defined over $\mathbb{Q}$ when $[K: \mathbb{Q}] \in \{2p, 3p \}$.