This is an exposition of some of the main features of the theory of elliptic curves and modular forms.

This paper analyzes the exact extent to which 0 and ∞ cause trouble in Montgomery’s fast branchless formulas for x-coordinate scalar multiplication on elliptic curves of the form by = x + ax + x. The analysis shows that some multiplications and branches can be eliminated from elliptic-curve primality proofs and from elliptic-curve cryptography.

This paper provides an overview of elliptic curves and their use in cryptography. The purpose of this paper is an in-depth examination of the Elliptic Curve Discrete Logarithm (ECDLP) including techniques in attacking cryptosystems dependent on the ECDLP. The paper includes properties of elliptic curve and methods for various attacks.

We show that the Hankel determinants of a generalized Catalan sequence satisfy the equations of the elliptic sequence. As a consequence, the coordinates of the multiples of an arbitrary point on the elliptic curve are expressed by the Hankel determinants. PACS numbers: 02.30.Ik, 02.30.Gp, 02.30.Lt

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 over 100 (for r = 10 and r = 11). We describe our search methods, and tabulate, for each r = 5, 6, . . . , 11, the five curves of lowest conductor, and (except for r = 11) also the five of lowest absolute disc...

Manjul Bhargava has recently made a great advance in the arithmetic theory of elliptic curves. Together with his student, Arul Shankar, he determines the average order of the Selmer group Sel(E,m) for an elliptic curve E over Q, when m = 2, 3, 4, 5. We recall that the Selmer group is a finite subgroup of H(Q, E[m]), which is defined by local conditions. Their result (cf. [1, 2]) is that the ave...

We discuss the idea of a “family of L-functions” and describe various methods which have been used to make predictions about L-function families. The methods involve a mixture of random matrix theory and heuristics from number theory. Particular attention is paid to families of elliptic curve L-functions. We describe two random matrix models for elliptic curve families: the Independent Model an...

The normal form x2+y2 = a2+a2x2y2 for elliptic curves simplifies formulas in the theory of elliptic curves and functions. Its principal advantage is that it allows the addition law, the group law on the elliptic curve, to be

We present a lower bound for the exponent of the group of rational points of an elliptic curve over a finite field. Earlier results considered finite fields Fqm where either q is fixed or m = 1 and q is prime. Here we let both q and m vary and our estimate is explicit and does not depend on the elliptic curve.

For an abelian variety A over a number field k we discuss the divisibility in H(k,A) of elements of the subgroup X(A/k). The results are most complete for elliptic curves over Q.