Point counting on curves using a gonality preserving lift

Point Counting on Elliptic and Hyperelliptic Curves

On the gonality of curves in Pn

Here we study the gonality of several projective curves which arise in a natural way (e.g. curves with maximal genus in Pn, curves with given degree d and genus g for all possible d, g if n = 3 and with large g for arbitrary (d, g, n)).

A linear lower bound on the gonality of modular curves

0.2. Remarks. The proof, which was included in the author’s thesis [א], follows closely a suggestion of N. Elkies. In the exposition here many details were added to the argument in [א]. We utilize the work [L-Y] of P. Li and S. T. Yau on conformal volumes, as well as the known bound on the leading nontrivial eigenvalue of the non-euclidean Laplacian λ1 ≥ 21 100 [L-R-S]. If Selberg’s eigenvalue ...

Curves, dynamical systems, and weighted point counting.

Suppose X is a (smooth projective irreducible algebraic) curve over a finite field k. Counting the number of points on X over all finite field extensions of k will not determine the curve uniquely. Actually, a famous theorem of Tate implies that two such curves over k have the same zeta function (i.e., the same number of points over all extensions of k) if and only if their corresponding Jacobi...

Point Counting in Families of Hyperelliptic Curves

Let EΓ be a family of hyperelliptic curves defined by Y 2 = Q(X,Γ), where Q is defined over a small finite field of odd characteristic. Then with γ in an extension degree n field over this small field, we present a deterministic algorithm for computing the zeta function of the curve Eγ by using Dwork deformation in rigid cohomology. The complexity of the algorithm is O(n) and it needs O(n) bits...