Generators for Cubic Surfaces with two Skew Lines over Finite Fields

Let S be a smooth cubic surface defined over a field K. As observed by Segre and Manin, there is a secant and tangent process on S that generates new K-rational points from old. It is natural to ask for the size of a minimal generating set for S(K). In a recent paper, for fields K with at least 13 elements, Siksek showed that if S contains a skew pair of K-lines then S(K) can be generated from one point. In this paper we prove the corresponding version of this result for fields K having at least 4 elements, and slightly milder results for #K=2 or 3.Comment: 10 pages. arXiv admin note: text overlap with arXiv:1012.1838 by other author

SS-unit equations and the asymptotic Fermat conjecture over number fields

Recent attempts at studying the Fermat equation over number fields have uncovered an unexpected and powerful connection with $S$-unit equations. In this expository paper we explain this connection and its implications for the asymptotic Fermat conjecture

Elliptic Curves over Real Quadratic Fields are Modular

We prove that all elliptic curves defined over real quadratic fields are modular.Comment: 38 pages. Magma scripts available as ancillary files with this arXiv versio

Perfect powers expressible as sums of two fifth or seventh powers

We show that the generalized Fermat equations with signatures (5,5,7), (5,5,19), and (7,7,5) (and unit coefficients) have no non-trivial primitive integer solutions. Assuming GRH, we also prove the nonexistence of non-trivial primitive integer solutions for the signatures (5,5,11), (5,5,13), and (7,7,11). The main ingredients for obtaining our results are descent techniques, the method of Chabauty-Coleman, and the modular approach to Diophantine equations.Comment: The current version incorporates minor comments of the refere

On the complexity of computing the 2-Selmer group of an elliptic curve

In this paper we give an algorithm for computing the 2-Selmer group of an elliptic curve Y-2 = X-3 + AX + B which has complexity O(L-D(0.5, c(1))), where D is the absolute discriminant of the curve. Our algorithm is unconditional but the complexity estimate assumes the GRH and a standard conjecture on the distribution of smooth reduced ideals. This improves on the corresponding algorithm of Birch and Swinnerton-Dyer, which has complexity of O(root D)

A fast Diffie-Hellman protocol in genus 2

In this paper it is shown how the multiplication by M map on the Kummer surface of a curve of genus 2 defined over F-q can be used to construct a Diffie-Hellman protocol. We show that this map can be computed using only additions and multiplications in F-q. In particular we do not use any divisions, polynomial arithmetic, or square root functions in F-q, hence this may be easier to implement than multiplication by M on the Jacobian. In addition we show that using the Kummer surface does not lead to any loss in security

Shifted powers in binary recurrence sequences

Let {

Explicit 4-descents on an elliptic curve

Abstract. It is shown that the obvious method of descending from an element of the 2-Selmer group of an elliptic curve, E, will indeed give elements of order 1, 2 or 4 in the Weil-Chatelet group of E. Explicit algorithms for such a method are given. 1

On the diophantine equation x(2) + C=2y(n)

In this paper, we study the Diophantine equation x(2) + C = 2y(n) in positive integers x, y with gcd(x, y) = 1, where n >= 3 and C is a positive integer. If C = 1 (mod 4), we give a very sharp bound for prime values of the exponent n; our main tool here is the result on existence of primitive divisors in Lehmer sequences due to Bilu, Hanrot and Voutier. We illustrate our approach by solving completely the equations x(2) + 17(a1) = 2y(n), x(2) + 5(a1)13(a2) = 2y(n) and x(2) + 3(a1)11(a2) = 2y(n)