Counting hyperelliptic curves on an Abelian surface with quasi-modular forms

Counting Curves with Modular Forms

We consider the type IIA string compactified on the Calabi-Yau space given by a degree 12 hypersurface in the weighted projective space P(1,1,2,2,6). We express the prepotential of the low-energy effective supergravity theory in terms of a set of functions that transform covariantly under PSL(2,ZZ) modular transformations. These functions are then determined by monodromy properties, by singular...

Counting hyperelliptic curves

We find a closed formula for the number hyp(g) of hyperelliptic curves of genus g over a finite field k = Fq of odd characteristic. These numbers hyp(g) are expressed as a polynomial in q with integer coefficients that depend on g and the set of divisors of q − 1 and q + 1. As a by-product we obtain a closed formula for the number of self-dual curves of genus g. A hyperelliptic curve is self-du...

Point Counting on Elliptic and Hyperelliptic Curves

On Hyperelliptic Modular Curves over Function Fields

We prove that there are only finitely many modular curves of Delliptic sheaves over Fq(T ) which are hyperelliptic. In odd characteristic we give a complete classification of such curves.

Modular equations for hyperelliptic curves

We define modular equations describing the `-torsion subgroups of the Jacobian of a hyperelliptic curve. Over a finite base field, we prove factorization properties that extend the well-known results used in Atkin’s improvement of Schoof’s genus 1 point counting algorithm.