Counting curves on elliptic ruled surface

Counting Curves of Any Genus on Rational Ruled Surfaces

In this paper we study the geometry of the Severi varieties parametrizing curves on the rational ruled surface Fn. We compute the number of such curves through the appropriate number of fixed general points on Fn (Theorem 1.1), and the number of such curves which are irreducible (Theorem 1.3). These numbers are known as Severi degrees; they are the degrees of unions of components of the Hilbert...

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.

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 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...