Related papers: Exp function for Edwards curves over local fields

We consider the expansion of the real field by the group of rational points of an elliptic curve over the rational numbers. We prove a completeness result, followed by a quantifier elimination result. Moreover we show that open sets…

On a symplectic manifold, there is a natural elliptic complex replacing the de Rham complex. It can be coupled to a vector bundle with connection and, when the curvature of this connection is constrained to be a multiple of the symplectic…

Let $E/\mathbb{Q}_p$ be an elliptic curve whose mod $p$ Galois image is contained in the normaliser of a non-split Cartan. We classify the possible $p$-adic images of $E$ using tools from $p$-adic Hodge theory via a careful analysis of the…

Given an elliptic curve $E$ over a local field $K$ with residue characteristic $3$, we investigate the action of the absolute Galois group of $K$ in the case of potentially good reduction. In particular the only not completely known case is…

We present two algorithms to compute the endomorphism ring of an ordinary elliptic curve E defined over a finite field F_q. Under suitable heuristic assumptions, both have subexponential complexity. We bound the complexity of the first…

We study the existence of rational points on modular curves of $\cal{D}$-elliptic sheaves over local fields and the structure of special fibres of these curves. We discuss some applications which include finding presentations for arithmetic…

In this note is we exhibit an elementary method to construct explicitly curves over finite fields with many points. Despite its elementary character the method is very efficient and can be regarded as a partial substitute for the use of…

We consider a pointed curve $(X,P)$ which is given by the Weierstrass normal form, $y^r + A_{1}(x) y^{r-1} + A_{2}(x) y^{r-2} +\cdots + A_{r-1}(x) y + A_{r}(x)$ where $x$ is an affine coordinate on $\mathbb{P}^1$, the point $\infty$ on $X$…

This paper proposes new explicit formulas for the doubling and addition step in Miller's algorithm to compute the Tate pairing. For Edwards curves the formulas come from a new way of seeing the arithmetic. We state the first geometric…

Given an elliptic curve $E$ and a finite Abelian group $G$, we consider the problem of counting the number of primes $p$ for which the group of points modulo $p$ is isomorphic to $G$. Under a certain conjecture concerning the distribution…

We prove that all elliptic curves defined over real quadratic fields are modular.

We introduce an analogue of the Mertens conjecture for elliptic curves over finite fields. Using a result of Waterhouse, we classify the isogeny classes of elliptic curves for which this conjecture holds in terms the size of the finite…

For a fixed number field and an elliptic curve defined and semi-stable over this number field, we consider the set of prime numbers p such that the Galois representation attached to the p-torsion points of the elliptic curve is reducible.…

We give a characterization of the codomain $[\ell]E(k)$ of the multiplication-by-$\ell$ map $[\ell]$ in the case of elliptic curves over a field $k$ of characteristic $\ne 2,3$ with $\ell$-torsion $E[\ell]=\langle W_1,W_2 \rangle$ fully…

Let E be an elliptic curve defined over Q. We study the relationship between the torsion subgroup E(Q)_tors and the torsion subgroup E(K)_tors, where K is a quadratic number field.

In this paper we present an iterative construction of irreducible polynomials over finite fields based upon repeated applications of transforms induced by endomorphisms of odd prime degree of ordinary elliptic curves.

Consider a pair of ordinary elliptic curves $E$ and $E'$ defined over the same finite field $\mathbb{F}_q$. Suppose they have the same number of $\mathbb{F}_q$-rational points, i.e. $|E(\mathbb{F}_q)|=|E'(\mathbb{F}_q)|$. In this paper we…

We study the p-adic valuation of the values of normalised Hecke eigenforms attached to non-isotrivial elliptic curves defined over function fields of transcendence degree one over finite fields of characteristic p. We derive upper bounds on…

We construct Abel maps for a stable curve $X$. Namely, for each one-parameter deformation of $X$ with regular total space, and every integer $d>0$, we construct by specialization a map $\alpha^d_X$ from the smooth locus of $X^d$ to the…

For an elliptic curve $E/\Q$, we determine the maximum number of twists $E^d/\Q$ it can have such that $E^d(\Q)_{tors}\supsetneq E(\Q)[2]$. We use these results to determine the number of distinct quadratic fields $K$ such that…