Some heuristics about elliptic curves

We give some heuristics for counting elliptic curves with certain properties. In particular, we re-derive the Brumer-McGuinness heuristic for the number of curves with positive/negative discriminant up to $X$, which is an application of lattice-point counting. We then introduce heuristics (with refinements from random matrix theory) that allow us to predict how often we expect an elliptic curve $E$ with even parity to have $L(E,1)=0$. We find that we expect there to be about $c_1X^{19/24}(\log X)^{3/8}$ curves with $|\Delta|<X$ with even parity and positive (analytic) rank; since Brumer and McGuinness predict $cX^{5/6}$ total curves, this implies that asymptotically almost all even parity curves have rank 0. We then derive similar estimates for ordering by conductor, and conclude by giving various data regarding our heuristics and related questions

Rank distribution in a family of cubic twists

In 1987, Zagier and Kramarz published a paper in which they presented evidence that a positive proportion of the even-signed cubic twists of the elliptic curve $x^3+y^3=1$ should have positive rank. We extend their data, showing that it is more likely that the proportion goes to zero

Some remarks on Heegner point computations

We explain how to find a rational point on a rational elliptic curve of rank 1 using Heegner points. We give some examples, and list new algorithms that are due to Cremona and Delaunay. These are notes from a short course given at the Institut Henri Poincare in December 2004

Explicit lower bounds on the modular degree of an elliptic curve

We derive an explicit zero-free region for symmetric square L-functions of elliptic curves, and use this to derive an explicit lower bound for the modular degree of rational elliptic curves. The techniques are similar to those used in the classical derivation of zero-free regions for Dirichlet L-functions, but here, due to the work of Goldfield-Hoffstein-Lieman, we know that there are no Siegel zeros, which leads to a strengthened result

Cross-middleware Interoperability in Distributed Concurrent Engineering

Secure, distributed collaboration between different organizations is a key challenge in Grid computing today. The GDCD project has produced a Grid-based demonstrator Virtual Collaborative Facility (VCF) for the European Space Agency. The purpose of this work is to show the potential of Grid technology to support fully distributed concurrent design, while addressing practical considerations including network security, interoperability, and integration of legacy applications. The VCF allows domain engineers to use the concurrent design methodology in a distributed fashion to perform studies for future space missions. To demonstrate the interoperability and integration capabilities of Grid computing in concurrent design, we developed prototype VCF components based on ESA’s current Excel-based Concurrent Design Facility (a non-distributed environment), using a STEP-compliant database that stores design parameters. The database was exposed as a secure GRIA 5.1 Grid service, whilst a .NET/WSE3.0-based library was developed to enable secure communication between the Excel client and STEP database

Secondary terms in the number of vanishings of quadratic twists of elliptic curve L-functions

We examine the number of vanishings of quadratic twists of the L-function associated to an elliptic curve. Applying a conjecture for the full asymptotics of the moments of critical L-values we obtain a conjecture for the first two terms in the ratio of the number of vanishings of twists sorted according to arithmetic progressions.Comment: 16 pages, many figure