Related papers: Note On Elliptic Groups of Prime Orders

Let $K$ be a number field. For positive integers $m$ and $n$ such that $m\mid n$, we let $\mathscr{S}_{m,n}$ be the set of elliptic curves $E/K$ defined over $K$ such that $E(K)_{\operatorname{tors}}\supseteq \mathscr{T}\cong…

In 1988, Koblitz conjectured the infinitude of primes p for which |E(F_p)| is prime for elliptic curves E over Q, drawing an analogy with the twin prime conjecture. He also proposed studying the primality of |E(F_{p^l})| / |E(F_p)|, in…

Let $\ell$ be a prime number and let $F$ be a number field and $E/F$ a non-CM elliptic curve with a point $\alpha \in E(F)$ of infinite order. Attached to the pair $(E,\alpha)$ is the $\ell$-adic arboreal Galois representation…

Let $E/\mathbb{Q}$ be a fixed elliptic curve. For each prime $p$ of good reduction, write $E(\mathbb{F}_p) \cong \mathbb{Z}/d_p \mathbb{Z} \oplus \mathbb{Z}/e_p \mathbb{Z}$, where $d_p \mid e_p$. Kowalski proposed investigating the average…

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…

Let $p$ be a prime number and $E_{p}$ denote the elliptic curve $y^2=x^3+px$. It is known that for $p$ which is congruent to $1, 9$ modulo $16$, the rank of $E_{p}$ over $\mathbb{Q}$ is equal to $0, 2$. Under the condition that the Birch…

A recent paper of Shekhar compares the ranks of elliptic curves $E_1$ and $E_2$ for which there is an isomorphism $E_1[p] \simeq E_2[p]$ as $\mathrm{Gal}(\bar{\mathbf{Q}}/\mathbf{Q})$-modules, where $p$ is a prime of good ordinary reduction…

In this article, we consider a family of elliptic curves defined by $E_{m}: y^2= x^3 -m^2 x + (pqr)^2 $ where $m $ is a positive integer and $p, q, ~\text{and}~ r$ are distinct odd primes and study the torsion as well the rank of…

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…

Let $E$ be an elliptic curve defined over $\mathbb{Q}$. In this article, we classify all groups that can arise as $E(\mathbb{Q}(\zeta_p))_{\text{tors}}$ up to isomorphism for any prime $p$. When $p - 1$ is not divisible by small integers…

Let $p, q$ be twin prime numbers with $q-p=2$ . Consider the elliptic curves E=E_\sigma: y^2 = x (x+\sigma p)(x+\sigma q) . (\sigma =\pm 1). E=E_\sigma is also denoted as E_+ or E_- when \sigma = +1or $-1.Here the Mordell-Weil group and the…

Let \beta be a real number. Then for almost all irrational \alpha>0 (in the sense of Lebesgue measure) \limsup_{x\to\infty}\pi_{\alpha,\beta}^*(x)(\log x)^2/x>=1, where \pi_{\alpha,\beta}^*(x)={p<=x: both p and [\alpha p+\beta] are primes}.

Let $E/\mathbb{Q}$ be an elliptic curve with complex multiplication (CM), and for each prime $p$ of good reduction, let $a_E(p) = p + 1 - \#E(\mathbb{F}_p)$ denote the trace of Frobenius. By the Hasse bound, $a_E(p) = 2\sqrt{p} \cos…

Let E be an elliptic curve over Q with prime conductor p. For each non-negative integer n we put K_n:=Q(E[p^n]). The aim of this paper is to estimate the order of the p-Sylow group of the ideal class group of K_n. We give a lower bounds in…

This article considers the family of elliptic curves given by $E_{pq}: y^2=x^3-5pqx$ and certain conditions on odd primed $p$ and $q$. More specifically, we have proved that if $p \equiv 33 \pmod {40}$ and $ q \equiv 7 \pmod {40}$, then the…

For an elliptic curve $E$ over $\ratq$ and an integer $r$ let $\pi_E^r(x)$ be the number of primes $p\le x$ of good reduction such that the trace of the Frobenius morphism of $E/\fie_p$ equals $r$. We consider the quantity $\pi_E^r(x)$ on…

Let $E$ be an elliptic curve defined over $\mathbb Q$. Let $\Gamma$ be a subgroup of $E(\mathbb Q)$ and $P\in E(\mathbb Q)$. In [1], it was proved that if $E$ has no nontrivial rational torsion points, then $P\in\Gamma$ if and only if $P\in…

For a given elliptic curve $E$ over a finite local ring, we denote by $E^{\infty}$ its subgroup at infinity. Every point $P \in E^{\infty}$ can be described solely in terms of its $x$-coordinate $P_x$, which can be therefore used to…

Let $p>3$ be a prime and $E$ be a supersingular elliptic curve defined over $\mathbb{F}_{p^2}$. Let $c$ be a prime with $c < 3p/16$ and $G$ be a subgroup of $E[c]$ of order $c$. The pair $(E,G)$ is called a supersingular elliptic curve with…

Let K be a number field, and let E be an elliptic curve over K. A famous result by Faltings of 1983 can be reformulated for elliptic curves as follows: if S is a set of primes of good reduction for E having density one, then the K-isogeny…