29 citations · 73 across the 8 of their papers we have counts for
9 papers · 1 filter
Elliptic Curves x^3 + y^3 = k of High Rank
Noam D. Elkies, Nicholas F. Rogers
We use rational parametrizations of certain cubic surfaces and an explicit formula for descent via 3-isogeny to construct the first examples of elliptic curves E_k: x^3 + y^3 = k o…
Elliptic curves of large rank and small conductor
Noam D. Elkies, Mark Watkins
For r=6,7,...,11 we find an elliptic curve E/Q of rank at least r and the smallest conductor known, improving on the previous records by factors ranging from 1.0136 (for r=6) to ov…
The conjugate dimension of algebraic numbers
Neil Berry, Arturas Dubickas, Noam D. Elkies +2
We find sharp upper and lower bounds for the degree of an algebraic number in terms of the -dimension of the space spanned by its conjugates. For all but seven nonnegative integ…
Curves D y^2 = x^3 - x of odd analytic rank
Noam D. Elkies
For nonzero rational D, which may be taken to be a squarefree integer, let E_D be the elliptic curve Dy^2=x^3-x over Q arising in the "congruent number" problem. It is known that t…
Mordell-Weil lattices in characteristic 2, III: A Mordell-Weil lattice of rank 128
Noam D. Elkies
In the first paper of this series, we constructed a family of lattices in dimensions 2^{n+1} for positive integers n, and proved that the associated lattice packings of spheres equ…
Explicit towers of Drinfeld modular curves
Noam D. Elkies
We give explicit equations for the simplest towers of Drinfeld modular curves over any finite field, and observe that they coincide with the asymptotically optimal towers of curves…