Related papers: Mazur's isogeny theorem

Let $E/\mathbb{Q}$ be an elliptic curve, let $p>2$ be a prime of good reduction for $E$, and assume that $E$ admits a rational $p$-isogeny with kernel $\mathbb{F}_p(\phi)$. In this paper we prove the cyclotomic Iwasawa main conjecture for…

The Frey--Mazur conjecture states that an elliptic curve over $\mathbb{Q}$ is determined up to isogeny by its $p$-torsion Galois representation for $p\geq 17$. We study a geometric analog of this conjecture, and show that the map from…

Let $K$ be a quadratic field which is not an imaginary quadratic field of class number one. We describe an algorithm to compute the primes $p$ for which there exists an elliptic curve over $K$ admitting a $K$-rational $p$-isogeny. This…

Let $E$ be an elliptic curve over $\mathbb{Q}$ such that $\mathrm{End}_{\bar{\mathbb{Q}}}(E)=\mathbb{Z}$ and which admits a non-trivial cyclic $\mathbb{Q}$-isogeny. We prove that, for $p>37$, the residual mod $p$ Galois representation…

Let $q > 5$ be a prime and $K$ a quadratic number field. In this article we extend a previous result of Najman and the author and prove that if $E/K$ is an elliptic curve with potentially multiplicative reduction at all primes $\mathfrak q…

The main goal of this paper is to determine for which prime numbers $r\geq 3$ can an elliptic curve~$E$ defined over $\mathbb Q$ have an $r$-isogeny over $\mathbb Q(\zeta_r)$. We study this question under various assumptions on the…

We show that if $E$ is an elliptic curve over $\mathbf{Q}$ with a $\mathbf{Q}$-rational isogeny of degree 7, then the image of the 7-adic Galois representation attached to $E$ is as large as allowed by the isogeny, except for the curves…

We say that two elliptic curves $E$ and $F$ over $\mathbb{Q}$ are congruent modulo a prime $p$ if their $p$-torsion Galois modules (over the algebraic closure of $\mathbb{Q}$) are isomorphic. Such a congruence is called trivial if there is…

We study this subject by first proving that the p-primary subgroup of the classical Selmer group for an elliptic curve with good, ordinary reduction at a prime p has a very simple and elegant description which involves just the Galois…

This article deals with the Galois representation attached to elliptic curves with an isogeny of prime degree over a number field. We first determine uniform criteria for the irreducibility of Galois representations attached to elliptic…

Consider a rational point on an elliptic curve under an isogeny. Suppose that the action of Galois partitions the set of its pre-images into n orbits. It is shown that all such points above a certain height have their denominator divisible…

We provide an explicit and algorithmic version of a theorem of Momose classifying isogenies of prime degree of elliptic curves over number fields, which we implement in Sage and PARI/GP. Combining this algorithm with recent work of…

Mazur's theorem states that there are exactly 15 possibilities for the torsion subgroup of an elliptic curve over the rational numbers. We determine how often each of these groups actually occurs. Precisely, if $G$ is one of these 15…

Given two CM elliptic curves over a number field and a natural number $m$, we establish a polynomial lower bound (in terms of $m$) for the number of rational primes $p$ such that the reductions of these elliptic curves modulo a prime above…

Using Galois representations, we analyze fields of definition of cyclic isogenies on elliptic curves to prove the following uniformity result: for any number field $F$ which has no rational CM, under GRH there exists an effectively…

Let $p$ be an odd prime and $E/\mathbb{Q}$ be a rational elliptic curve. There is a smooth affine curve $Y_E(p)$ whose rational points parametrize elliptic curves $F/\mathbb{Q}$ such that $F[p]$ and $E[p]$ are isomorphic Galois modules.…

Let E be an elliptic curve over a number field K which admits a cyclic p-isogeny with p odd and semistable at primes above p. We determine the root number and the parity of the p-Selmer rank for E/K, in particular confirming the parity…

Given a complex quasiprojective curve $B$ and a non-isotrivial family $\mathcal{E}$ of elliptic curves over $B$, the $p$-torsion $\mathcal{E}[p]$ yields a monodromy representation $\rho_\mathcal{E}[p]:\pi_1(B)\rightarrow…

Let $p$ be an odd prime. We study Mazur's conjecture on the growth of the Mordell--Weil ranks of an elliptic curve $E/\mathbb{Q}$ over $\mathbb{Z}_p$-extensions of an imaginary quadratic field, where $p$ is a prime of good reduction for…

We determine all possible degrees of cyclic isogenies of non-CM elliptic curves with rational $j$-invariant over number fields of degree $p$, where $p$ is an odd prime. The question had been answered for $p=2$, so this paper completes the…