Related papers: Class group statistics for torsion fields generate…

The paper formulates a precise relationship between the Tate-Shafarevich group of an elliptic curve $E$ over ${\mathbb Q}$ with a quotient of the classgroup of ${\mathbb Q}(E[p])$ on which $Gal({\mathbb Q}(E[p]/{\mathbb Q}) = GL_2({\mathbb…

Let $K$ be a non-cylotomic imaginary quadratic field of class number 1 and $E/K$ is an elliptic curve with $E(K)[2]\simeq \mathbb{Z}/2\mathbb{Z} \oplus\mathbb{Z}/2\mathbb{Z}.$ In this article, we determine the torsion groups that can arise…

Let $K=\mathbb{Q}(\sqrt{-p})$ be a quadratic field for an odd prime $p$. We show that there exist infinitely many primes $p$ for which no elliptic curve $E/\mathbb{Q}$ has torsion subgroup $\mathbb{Z}/2\mathbb{Z}\times…

In this article, we study the minimal degree [K(T):K] of a p-subgroup T <= E(\overline{K})_tors for an elliptic curve E/K defined over a number field K. Our results depend on the shape of the image of the p-adic Galois representation…

Let $K$ be a non-cylotomic imaginary quadratic field of class number 1 and $E/K$ is an elliptic curve with $E(K)[2]\simeq \mathbb{Z}_1.$ We determine the odd-order torsion groups that can arise as $E(L)_{\text{tor}}$ where $L$ is a…

For each prime $p$, we show that there exist geometrically simple abelian varieties $A/\mathbb Q$ with non-trivial $p$-torsion in their Tate-Shafarevich groups. Specifically, for any prime $N\equiv 1 \pmod{p}$, let $A_f$ be an optimal…

We determine, for an elliptic curve $E/\mathbb{Q}$, all the possible torsion groups $E(K)_{tors}$, where $K$ is the compositum of all $\mathbb{Z}_{p}$-extensions of $\mathbb{Q}$. Furthermore, we prove that for an elliptic curve…

In this paper it is shown that for every prime p>5 the dimension of the p-torsion in the Tate-Shafarevich group of E/K can be arbitrarily large, where E is an elliptic curve defined over a number field K, with [K:Q] bounded by a constant…

We characterize the possible groups $E(\mathbb{Z}/N\mathbb{Z})$ arising from elliptic curves over $\mathbb{Z}/N\mathbb{Z}$ in terms of the groups $E(\mathbb{F}_p)$, with $p$ varying among the prime divisors of $N$. This classification is…

We show that if F is the rational numbers or a multiquadratic number field, p is 2,3, or 5, and K/F is a Galois extension of degree a power of p, then for elliptic curves E/Q ordered by height, the average dimension of the p-Selmer groups…

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…

Let $\ell$ be an odd prime and $d$ a positive integer. We determine when there exists a degree-$d$ number field $K$ and an elliptic curve $E/K$ with $j(E)\in\mathbb{Q}\setminus\{0,1728\}$ for which $E(K)_{\mathrm{tors}}$ contains a point of…

Let $E$ be an elliptic curve, defined over a quartic extension $K$ of $\mathbb{Q}$, with $j(E) \in \mathbb{Q}$. In this paper, we classify the possible torsion subgroup structures $E(K)_{\text{tors}}$.

We determine, for an elliptic curve $E/\mathbb Q$ and for all $p$, all the possible torsion groups $E(\mathbb Q_{\infty, p})_{tors}$, where $\mathbb Q_{\infty, p}$ is the $\mathbb Z_p$-extension of $\mathbb Q$.

Let $E$ be an elliptic curve over $\mathbb{Q}$. Let $p$ be an odd prime and $\iota: \overline{\mathbb{Q}}\hookrightarrow \mathbb{C}_p$ an embedding. Let $K$ be an imaginary quadratic field and $H_{K}$ the corresponding Hilbert class field.…

This paper is devoted to the study of the $\ell$-adic representations of the absolute Galois group $G$ of ${\mathbb Q}_p$, $p\geq 5$, associated to an elliptic curve over ${\mathbb Q}_p$, as $\ell$ runs through the set of all prime numbers…

Let p be an odd prime number, E an elliptic curve over a number field k, and F/k a Galois extension of degree twice a power of p. We study the Z_p-corank rk_p(E/F) of the p-power Selmer group of E over F. We obtain lower bounds for…

By reformulating and extending results of Elkies, we prove some results on $\mathbb Q$-curves over number fields of odd degree. We show that, over such fields, the only prime isogeny degrees~$\ell$ which an elliptic curve without CM may…

We give an algorithm to explicitly determine all elements of the $q$-torsion (for $q$ an odd prime) of the Brauer group of an elliptic curve over any base field of characteristic different from $q$, containing a primitive $q$-th root of…

For a quadratic field $\mathcal{K}$ without rationally defined CM, we prove that there exists of a prime $p_{\mathcal{K}}$ depending only on $\mathcal{K}$ such that if $d$ is a positive integer whose minimal prime divisor is greater than…