FPGAs are an attractive platform for elliptic curve cryptography hardware. Since field multiplication is the most critical operation in elliptic curve cryptography, we have studied how efficient several field multipliers can be mapped to lookup table based FPGAs. Furthermore we have compared different curve coordinate representations with respect to the number of required field operations, and ...

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...

Introduction. Although it has occupied a central place in number theory for almost a century, the arithmetic of elliptic curves is still today a subject which is rich in conjectures, but sparse in definitive theorems. In this lecture, I will only discuss one main topic in the arithmetic of elliptic curves, namely the conjecture of Birch and Swinnerton-Dyer. We briefly recall how this conjecture...

Let k be an algebraically closed field of characteristic p. Let X(p;N) be the curve parameterizing elliptic curves with full level N structure (where p N) and full level p Igusa structure. By modular curve, we mean a quotient of any X(p;N) by any subgroup of ((Z/peZ)× × SL2(Z/NZ)) /{±1}. We prove that in any sequence of distinct modular curves over k, the k-gonality tends to infinity. This exte...

This article explains the Hasse principle and gives a self-contained development of certain counterexamples to this principle. The counterexamples considered are similar to the earliest counterexample discovered by Lind and Reichardt. This type of counterexample is important in the theory of elliptic curves: today they are interpreted as nontrivial elements in Tate– Shafarevich groups.

We consider the expansion of the real field by a subgroup of a one-dimensional definable group satisfying a certain diophantine condition. The main example is the group of rational points of an elliptic curve over a number field. We prove a completeness result, followed by a quantifier elimination result. Moreover we show that open sets definable in that structure are semialgebraic.

— We construct examples of elliptic fibrations of orbifold general type (in the sense of Campana) which have no étale covers dominating a variety of general type.