Elliptic and Edwards Curves Order Counting Method

Point Counting on Elliptic and Hyperelliptic Curves

Counting Elliptic Curves with Prescribed Torsion

Mazur’s theorem states that there are exactly 15 possibilities for the torsion subgroup of an elliptic curve over the rational numbers. We determine how often each of these groups actually occurs. Precisely, if G is one of these 15 groups, we show that the number of elliptic curves up to height X whose torsion subgroup is isomorphic to G is on the order of X, for some number d = d(G) which we c...

COUNTING POINTS ON ELLIPTIC CURVES OVER F2n,

In this paper we present an implementation of Schoofs algorithm for computing the number of i"2m-P°mts °f an elliptic curve that is defined over the finite field F2m . We have implemented some heuristic improvements, and give running times for various problem instances.

Counting Elliptic Curves in K3 Surfaces

We compute the genus g = 1 family GW-invariants of K3 surfaces for non-primitive classes. These calculations verify Göttsche-Yau-Zaslow formula for non-primitive classes with index two. Our approach is to use the genus two topological recursion formula and the symplectic sum formula to establish relationships among various generating functions. The number of genus g curves in K3 surfaces X repr...

Edwards curves and CM curves

Edwards curves are a particular form of elliptic curves that admit a fast, unified and complete addition law. Relations between Edwards curves and Montgomery curves have already been described. Our work takes the view of parameterizing elliptic curves given by their j-invariant, a problematic that arises from using curves with complex multiplication, for instance. We add to the catalogue the li...