Modularity of CM elliptic curves over division fields

Let E be a CM elliptic curve defined over an algebraic number field F. In general E will not be modular over F. In this paper, we determine extensions of F, contained in suitable division fields of E, over which E is modular. Under some weak assumptions on E, we construct a minimal subfield of division fields over which E is modular.     

A Formula for the Number of Elliptic Curves with Exceptional Primes

We prove a conjecture of Duke on the number of elliptic curves over the rationals of bounded height which have exceptional primes.     

Generating More MNT Elliptic Curves

In their seminal paper, Miyaji et al. [13] describe a simple method for the creation of elliptic curves of prime order with embedding degree 3, 4, or 6. Such curves are important for the realisation of pairing-based cryptosystems on ordinary (non-supersingular) elliptic curves. We provide an alternative derivation of their results, and extend them to allow for the generation of many more suitable curves. Research supported by Enterprise Ireland grant IF/2002/0312/N.     

3-Selmer groups for curves <Emphasis Type=Italic>y</Emphasis><Superscript>2</Superscript> = <Emphasis Type=Italic>x</Emphasis><Superscript>3</Superscript> + <Emphasis Type=Italic>a</Emphasis>

We explicitly perform some steps of a 3-descent algorithm for the curves y ² = x ³ + a, a a nonzero integer. In general this will enable us to bound the order of the 3-Selmer group of such curves.     

The modularity of some Q-curves

A Q-curve is an elliptic curve, defined over a number field, that is isogenous to each of its Galois conjugates. Ribet showed that Serre's conjectures imply that such curves should be modular. Let E be an elliptic curve defined over a quadratic field such that E is 3-isogenous to its Galois conjugate. We give an algorithm for proving any such E is modular and give an explicit example involving a quotient of J_o (169). As a by-product, we obtain a pair of 19-isogenous elliptic curves, and relate this to the existence of a rational point of order 19 on J₁ (13).     

On the supersingular reduction of elliptic curves

Let a∈Q and denote byE _a the curvey ² = (x ² + l)(x + a). We prove thatE _a(F_p) is cyclic for infinitely many primesp. This fact was known previously only under the assumption of the generalized Riemann hypothesis. Research partially supported by NSERC grant A9418.     

On construction of certain CM elliptic curves

Let E be a CM elliptic curve defined over an algebraic number field F. In the previous paper [N. Murabayashi, On the field of definition for modularity of CM elliptic curves, J. Number Theory 108 (2004) 268-286], we gave necessary and sufficient conditions for E to be modular over F, i.e. there exists a normalized newform f of weight two on Γ₁(N) for some N such that HomF(E,Jf)≠{0}. We also determined the multiplicity of E as F-simple factor of Jf when HomF(E,Jf)≠{0}. In this process we separated into the three cases. In this paper we construct certain CM elliptic curves which satisfy the conditions of each case. In other words, we show that all three cases certainly occur.     

Heights of CM Points on Complex Affine Curves

In this note we show that, assuming the generalized Riemann hypothesis for quadratic imaginary fields, an irreducible algebraic curve in is modular if and only if it contains a CM point of sufficiently large height. This is an effective version of a theorem of Edixhoven.     

Average Root Numbers for a Nonconstant Family of Elliptic Curves

We give some examples of families of elliptic curves with nonconstant j-invariant where the parity of the (analytic) rank is not equidistributed among the fibres.     

Twists of elliptic curves

If E is an elliptic curve over , then let E(D) denote theD-quadratic twist of E. It is conjectured that there are infinitely many primesp for which E(p) has rank 0, and that there are infinitely many primes for which has positive rank. For some special curvesE we show that there is a set S of primes p with density for which if is a squarefree integer where , then E(D) has rank 0. In particular E(p) has rank 0 for every . As an example let E1 denote the curve .Then its associated set of primes S₁ consists of the prime11 and the primes p for which the order of the reduction ofX₀(11) modulo p is odd. To obtain the general result we show for primes that the rational factor of L(E(p),1) is nonzero which implies thatE(p) has rank 0. These special values are related to surjective Galois representations that are attached to modularforms. Another example of this result is given, and we conclude with someremarks regarding the existence of positive rank prime twists via polynomialidentities.     

Computing the Rational Torsion of an Elliptic Curve Using Tate Normal Form

It is a classical result (apparently due to Tate) that all elliptic curves with a torsion point of order n(4?n?10, or n=12) lie in a one-parameter family. However, this fact does not appear to have been used ever for computing the torsion of an elliptic curve. We present here an extremely down-to-earth algorithm using the existence of such a family.     

Going in Circles: Variations on the Money-Coutts Theorem

Given a polygon A ₁,...,A _n, consider the chain of circles: S ₁ inscribed in the angle A ₁, S ₂ inscribed in the angle A ₂ and tangent to S ₁, S ₃ inscribed in the angle A ₃ and tangent to S ₂, etc. We describe a class of n-gons for which this process is 2n-periodic. We extend the result to the case when the sides of a polygon are arcs of circles. The case of triangles is known as the Money-Coutts theorem.     

Elliptic Curves Suitable for Pairing Based Cryptography

For pairing based cryptography we need elliptic curves defined over finite fields whose group order is divisible by some prime with where k is relatively small. In Barreto et al. and Dupont et al. [Proceedings of the Third Workshop on Security in Communication Networks (SCN 2002), LNCS, 2576, 2003; Building curves with arbitrary small Mov degree over finite fields, Preprint, 2002], algorithms for the construction of ordinary elliptic curves over prime fields with arbitrary embedding degree k are given. Unfortunately, p is of size .We give a method to generate ordinary elliptic curves over prime fields with p significantly less than which also works for arbitrary k. For a fixed embedding degree k, the new algorithm yields curves with where or depending on k. For special values of k even better results are obtained.We present several examples. In particular, we found some curves where is a prime of small Hamming weight resp. with a small addition chain.AMS classification: 14H52, 14G50     

Root Numbers of Non-Abelian Twists of Elliptic Curves

We study the global root number of the complex L-function oftwists of elliptic curves over Q by real Artin representations.We obtain examples of elliptic curves over Q which, while nothaving any rational points of infinite order, conjecturallymust have points of infinite order over the fields for every cube-free m > 1. We describe analogousphenomena for elliptic curves over the fields , and in the towers and , where r 3 is prime.2000 Mathematics Subject Classification 11G40, 11G05.