Related papers: Genus 2 Curves with Complex Multiplication

We describe an algorithm, based on the properties of the characteristic polynomials of Frobenius, to compute $\operatorname{End}_{\overline{K}}(A)$ when $A$ is the Jacobian of a nice genus-2 curve over a number field $K$. We use this…

In this article we give the details of an effective point counting algorithm for genus two curves over finite fields of characteristic three. The algorithm has an application in the context of curve based cryptography. One distinguished…

A set of multi-homogeneous equations for the Jacobian of a genus two curve is given. The approach used is to write down affine equations for the Jacobian minus various tranlations of the Theta-divisor by [2]-division points, and then to…

The "defect" of a curve over a finite field is the difference between the number of rational points on the curve and the Weil-Serre bound for the curve. We present a construction for producing genus-4 double covers of genus-2 curves over…

We bound the running time of an algorithm that computes the genus-two class polynomials of a primitive quartic CM-field K. This is in fact the first running time bound and even the first proof of correctness of any algorithm that computes…

We determine all genus 2 curves, defined over $\mathbb C$, which have simultaneously degree 2 and 3 elliptic subcovers. The locus of such curves has three irreducible 1-dimensional genus zero components in $\mathcal M_2$. For each component…

We compute in a direct (not algorithmic) way the zeta function of all supersingular curves of genus 2 over a finite field k, with many geometric automorphisms. We display these computations in an appendix where we select a family of…

Modular polynomials are an important tool in many algorithms involving elliptic curves. In this article we investigate their generalization to the genus 2 case following pioneering work by Gaudry and Dupont. We prove various properties of…

We give a bound on the primes dividing the denominators of invariants of Picard curves of genus 3 with complex multiplication. Unlike earlier bounds in genus 2 and 3, our bound is based not on bad reduction of curves, but on a very explicit…

We show that a genus $2$ curve over a number field whose jacobian has complex multiplication will usually have stable bad reduction at some prime. We prove this by computing the Faltings height of the jacobian in two different ways. First,…

Curves over finite fields are of great importance in cryptography and coding theory. Through studying their zeta-functions, we would be able to find out vital arithmetic and geometric information about them and their Jacobians, including…

Explicit models of families of genus 2 curves with multiplication by $\sqrt D$ are known for $D= 2, 3, 5$. We obtain generic models for genus 2 curves over $\mathbb Q$ with real multiplication in 12 new cases, including all fundamental…

We compute the class of arithmetic genus two Teichmueller curves in the Picard group of pseudo-Hilbert modular surfaces, distinguished according to their torsion order and spin invariant. As an application, we compute the number of genus…

Inside the moduli space of curves of genus 2 with 2 marked points we consider the loci of curves admitting a map to P^1 of degree d totally ramified over the two marked points, for d>= 2. Such loci have codimension two. We compute the class…

We consider families of smooth projective curves of genus 2 with a single point removed and study their integral points. We show that in many such families there is a dense set of fibres for which the integral points can be effectively…

We present an efficient algorithm to compute the Euler factor of a genus 2 curve C/Q at an odd prime p that is of bad reduction for C but of good reduction for the Jacobian of C (a prime of ``almost good'' reduction). Our approach is based…

For a complex abelian variety $A$ with endomorphism ring isomorphic to the maximal order in a quartic CM-field $K$, the Igusa invariants $j_1(A), j_2(A),j_3(A)$ generate an abelian extension of the reflex field of $K$. In this paper we give…

We give an algorithm to compute the conductor for curves of genus 2. It is based on the analysis of 3-torsion of the Jacobian for genus 2 curves over 2-adic fields.

We construct Weil numbers corresponding to genus-2 curves with $p$-rank 1 over the finite field $\F_{p^2}$ of $p^2$ elements. The corresponding curves can be constructed using explicit CM constructions. In one of our algorithms, the group…

This short article concerns a method to obtain effectivity for the search of integral points on certain (sets of) curves of genus 2. More precisely, we wish to illustrate just an example of application of a criterion of Bilu, to derive…