Related papers: Towards a classification of $p^2$-discriminant ide…

We show that every elliptic curve over a finite field of odd characteristic whose number of rational points is divisible by 4 is isogenous to an elliptic curve in Legendre form, with the sole exception of a minimal respectively maximal…

Let $K_i$ be a number field for all $i \in \mathbb{Z}_{> 0}$ and let $\mathcal{E}$ be a family of elliptic curves containing infinitely many members defined over $K_i$ for all $i$. Fix a rational prime $p$. We give sufficient conditions for…

We show that for all odd primes $p$, there exist ordinary elliptic curves over $\bar{\mathbb{F}}_p(x)$ with arbitrarily high rank and constant $j$-invariant. This shows in particular that there are elliptic curves with arbitrarily high rank…

Let $p$ be a prime and $K$ a number field of degree $p$. We count the number of elliptic curves, up to $\bar{K}$-isomorphism, having a prescribed property, where this property is either that the curve contains a fixed torsion group as a…

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…

We list the elliptic curves defined over $Q(\sqrt 5)$ with good reduction away from 2. There are 368 isomorphism classes. We give a global minimal model for each class.

Let $\ell$ and $p \geq 3$ be distinct prime numbers. Let $E/\mathbb{Q}_{\ell}$ be an elliptic curve with $p$-torsion module $E_p$. Let $\mathbb{Q}_{\ell}(E_p)$ be the $p$-torsion field of $E$. We provide a complete description of the degree…

Let $E$ be an elliptic curve defined over a number field $K$. We say that a prime number $p$ is exceptional for $(E,K)$ if $E$ admits a $p$-isogeny defined over $K$. The so-called exceptional set of all such prime numbers is finite if and…

We investigate how various invariants of elliptic curves, such as the discriminant, Kodaira type, Tamagawa number and real and complex periods, change under an isogeny of prime degree p. For elliptic curves over l-adic fields, the…

We prove two theorems concerning isogenies of elliptic curves over function fields. The first one describes the variation of the height of the $j$-invariant in an isogeny class. The second one is an "isogeny estimate", providing an explicit…

Let $\ell$ be an odd prime, and suppose $E$ is an elliptic curve defined over the rational numbers $\mathbb{Q}$. If $E$ has an $\ell$-torsion point, then there has been significant work done on characterizing the $\ell$-divisibility of the…

Let $E$ and $E'$ be 2-isogenous elliptic curves over $\Q$. Following \cite{ck}, we call a good prime $p$ \emph{anomalous} if $E(\F_p) \simeq E'(\F_p)$ but $E(\F_{p^2}) \not \simeq E'(\F_{p^2})$. Our main result is an explicit formula for…

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…

In this paper, we introduce the study of minimal torsion curves within a fixed geometric isogeny class. For a $\overline{\mathbb{Q}}$-isogeny class $\mathcal{E}$ of elliptic curves and $N \in \mathbb{Z}^+$, we wish to determine the least…

Isogenies occur throughout the theory of elliptic curves. Recently, the cryptographic protocols based on isogenies are considered as candidates of quantum-resistant cryptographic protocols. Given two elliptic curves $E_1, E_2$ defined over…

Boix, De Stefani and Vanzo have characterized ordinary/supersingular elliptic curves over $\mathbb{F}_p$ in terms of the level of the defining cubic homogenous polynomial. We extend their study to arbitrary genus, in particular we prove…

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…

For a prime number p, we characterize the groups that may arise as torsion subgroups of an elliptic curve with complex multiplication defined over a number field of degree 2p. In particular, our work shows that a classification in the…

We classify elliptic curves over the rationals whose N\'eron model over the integers is semi-abelian, with good reduction at p=2, and whose Mordell--Weil group contains an element of order two that stays non-trivial at p=2. Furthermore, we…

Genus 5 curves can be hyperelliptic, trigonal, or non-hyperelliptic non-trigonal, whose model is a complete intersection of three quadrics in $\mathbb{P}^4$. We present and explain algorithms we used to determine, up to isomorphism over…