We study the Seiberg-Witten curves for N = 2 SUSY gauge theories arising from type IIA string configurations with two orientifold sixplanes. Such theories lift to elliptic models in M-theory. We express the M-theory background for these models as a nontrivial elliptic fibration over C. We discuss singularities of this surface, and write the Seiberg-Witten curve for several theories as a subvari...

This paper presents explicit equations over Q for 32 extremal semistable elliptic K3 surfaces. They are realized as pull-back of non-semi-stable extremal rational elliptic surfaces via base change. Together with work of J. Top and N. Yui which exhibited the same procedure for the semi-stable extremal rational elliptic surfaces, this exhausts this approach to produce extremal semi-stable ellipti...

The security of most elliptic curve cryptosystems is based on the intractability of the Elliptic Curve Discrete Logarithm Problem (ECDLP). Such a problem turns out to be computationally unfeasible when elliptic curves are suitably chosen. This paper provides an algorithm to obtain cryptographically good elliptic curves from a given one. The core of such a procedure lies on the usage of successi...

It has been suggested that a major obstacle in finding an index calculus attack on the elliptic curve discrete logarithm problem lies in the difficulty of lifting points from elliptic curves over finite fields to global fields. We explore the possibility of circumventing the problem of explicitly lifting points by investigating whether partial information about the lifting would be sufficient f...

This paper is concerned with the conjectural correspondence between Galois representations and modular forms. Although such relation has been fully established in the case the Galois representation is realizable on the `-adic Tate module of an elliptic curve over Q (cf. [24],[20],[5]), very little is known in general. As a first step towards the understanding of more complicated cases, we consi...

Let E be an elliptic curve over Q, with L-function LE(s). For any primitive Dirichlet character χ, let LE(s, χ) be the L-function of E twisted by χ. In this paper, we use random matrix theory to study vanishing of the twisted L-functions LE(s, χ) at the central value s = 1. In particular, random matrix theory predicts that there are infinitely many characters of order 3 and 5 such that LE(1, χ)...

For q odd, it was shown in [3] that the elliptic quadric Q−(7, q) possesses a unique 1-system, the so-called classical 1-system. Here, the same result will be obtained for even q. 1 Basic properties of 1-systems of Q−(7, q) A 1-system M of the elliptic quadric Q−(7, q) is a set {L0, L1, . . . , Lq4} of q4+1 lines of Q−(7, q) with the property that every plane of Q−(7, q) containing a line Li of...