Related papers: Mazur's isogeny theorem

Using methods from analytic number theory, for $m > 5$ and for $m = 4$, we obtain asymptotics with power-saving error terms for counts of elliptic curves with a cyclic $m$-isogeny up to quadratic twist over the rational numbers. For $m >…

Let $\mathbb{F}_r$ be a finite field of characteristic $p>3$. For any power $q$ of $p$, consider the elliptic curve $E=E_{q,r}$ defined by $y^2=x^3 + t^q -t$ over $K=\mathbb{F}_r(t)$. We describe several arithmetic invariants of $E$ such as…

In 1996, Merel showed there exists a function $B\colon \mathbb{Z}^+\rightarrow \mathbb{Z}^+$ such that for any elliptic curve $E/F$ defined over a number field of degree $d$, one has the torsion group bound $\# E(F)[\textrm{tors}]\leq…

We consider the structure of rational points on elliptic curves in Weierstrass form. Let x(P)=A_P/B_P^2 denote the $x$-coordinate of the rational point P then we consider when B_P can be a prime power. Using Faltings' Theorem we show that…

We study the prime number race for elliptic curves over the function field of a proper, smooth and geometrically connected curve over a finite field. This constitutes a function field analogue of prior work by Mazur, Sarnak and the second…

We prove an explicit surjectivity result for products of non-isotrivial, non-isogenous elliptic curves over a function field of arbitrary characteristic. This is by way of an isogeny degree bound in this setting, generated from bounds for…

We give several generalisations of the Deuring reduction criterion for elliptic curves to abelian varieties of higher dimension. In particular the Newton polygon of the reduction of an abelian variety A with complex multiplication by F at a…

We prove the $p$-parity conjecture for elliptic curves over global fields of characteristic $p > 3$. We also present partial results on the $\ell$-parity conjecture for primes $\ell \neq p$.

Let $\mathsf{E}/\mathbb{Q}$ be an elliptic curve. By the modularity theorem, it admits a surjection from a modular curve $X_0(N) \to \mathsf{E}$, and the minimal degree among such maps is called the modular degree of $\mathsf{E}$. By the…

Let p > 2 be prime. We state and prove (under mild hypotheses on the residual representation) a geometric refinement of the Breuil-M\'ezard conjecture for 2-dimensional mod p representations of the absolute Galois group of Qp. We also state…

Let $ p $ be a prime lager than 3. Let $k$ be a number field, which does not contain the subfield of $\mathbb{Q} (\zeta_{p^2})$ of degree $p$ over $\mathbb{Q}$. Suppose that $\mathcal{E}$ is an elliptic curve defined over $k$. We prove that…

It is proved, as was conjectured by Eisenbud-Koh-Stillman, that for a finitely generated graded module $M$ over the symmetric algebra $S(V)$, if the Koszul group ${\cal K}_{p,0}(M,V)\ne 0$, then the set of rank 1 relations in $M_0\otimes V$…

Let k be an algebraically closed field of characteristic p. Let X(p^e;N) be the curve parameterizing elliptic curves with full level N structure (where p does not divide N) and full level p^e Igusa structure. By modular curve, we mean a…

Let p be a prime and K be a number field. Let rho_{E,p}:G_K \longrightarrow Aut(T_p E)\cong GL_2(Z_p) be the Galois representation given by the Galois action on the p-adic Tate module of an elliptic curve E over K. Serre showed that the…

Let $M$ be the moduli space of rank $2$ stable bundles with fixed determinant of degree $1$ on a smooth projective curve $C$ of genus $g\ge 2$. When $C$ is generic, we show that any elliptic curve on $M$ has degree (respect to…

Caraiani and Newton have proven that if $F$ is an imaginary quadratic number field such that $X_0(15)$ has rank $0$ over $F$, then every elliptic curve over $F$ is modular. This paper is concerned with the quadratic fields…

We discuss the $\ell$-adic case of Mazur's "Program B" over $\mathbb{Q}$, the problem of classifying the possible images of $\ell$-adic Galois representations attached to elliptic curves $E$ over $\mathbb{Q}$, equivalently, classifying the…

Following work of Mazur-Tate and Satoh, we extend the definition of division polynomials to arbitrary isogenies of elliptic curves, including those whose kernels do not sum to the identity. In analogy to the classical case of division…

We show that if $E/\mathbb{Q}$ is an elliptic curve without complex multiplication and for which there is a prime $q$ such that the image of $\bar{\rho}_{E,q}$ is contained in the normaliser of a split Cartan subgroup of…

Consider two elliptic curves $E,E'$ defined over the finite field $\mathbb{F}_q$, and suppose that there exists an isogeny $\psi$ between $E$ and $E'$. We propose an algorithm that determines $\psi$ from the knowledge of $E$, $E'$ and of…