Related papers: Geometric primality tests using curves of genus 1 …
We consider the problem of checking whether an elliptic curve defined over a given number field has complex multiplication. We study two polynomial time algorithms for this problem, one randomized and the other deterministic. The randomized…
In this paper we develop techniques for computing the relative Brauer group of curves, focusing particularly on the case where the genus is 1. We use these techniques to show that the relative Brauer group may be infinite (for certain…
We give a covariant treatment of the quadratic differential identities satisfied by the P-functions on the Jacobian of smooth hyperelliptic curves of genera 1, 2 and 3.
We find that non-hyperelliptic generalised Howe curves and their twists of genus 5 attain the Hasse-Weil-Serre bound over some finite fields of order p, p^2 or p^3 for a prime p. We are able to decompose their Jacobians completely under…
The information geometry of the 2-manifold of gamma probability density functions provides a framework in which pseudorandom number generators may be evaluated using a neighbourhood of the curve of exponential density functions. The process…
We discuss methods for using the Weil polynomial of an isogeny class of abelian varieties over a finite field to determine properties of the curves (if any) whose Jacobians lie in the isogeny class. Some methods are strong enough to show…
The purpose of this note is to suggest an analogue for genus 2 curves of part of Gross and Zagier's work on elliptic curves. Experimentally, for genus 2 curves with CM by a quartic CM field K, it appears that primes dividing the…
We use the theory of p-curvature of connections to analyze stable vector bundles of rank 2 on curves of genus 2 which pull back to unstable bundles under the Frobenius morphism. We take two approaches, first using explicit formulas for…
We give a new, geometric proof of the section conjecture for fixed points of finite group actions on projective curves of positive genus defined over the field of complex numbers, as well as its natural nilpotent analogue. As a part of our…
We study a natural question in the Iwasawa theory of algebraic curves of genus $>1$. Fix a prime number $p$. Let $X$ be a smooth, projective, geometrically irreducible curve defined over a number field $K$ of genus $g>1$, such that the…
For a class of non-hyperelliptic genus 3 curves C which are 2-fold coverings of elliptic curves E, we give an explicit algebraic description of all birationally non-equivalent genus 2 curves whose Jacobians are degree 2 isogeneous to the…
In this paper, we study combinatorial properties of stable curves. To the dual graph of any nodal curve, it is naturally associated a group, which is the group of components of the N\'eron model of the generalized Jacobian of the curve. We…
We investigate the Gopakumar-Vafa (GV) theory of local curves, namely, the total spaces of rank two vector bundles with canonical determinant on smooth projective curves. Under a certain genericity condition on the rank two bundles, we…
We give the explicit equations for a P^3 x P^3 embedding of the Jacobian of a curve of genus 2, which gives a natural analog for abelian surfaces of the Edwards curve model of elliptic curves. This gives a much more succinct description of…
In this paper we begin to study curves on a weighted projective plane with one trivial weight, ${\mathbb P}(1,m,n)$, by determining the genus of curves of Fermat type. These are curves defined by a ``homogeneous'' polynomial analagous to…
We study the minimal Weierstrass equations for genus 2 curves defined over a ring of integers $\mathcal O_{\mathbb F}$. This is done via reduction theory and Julia invariant of binary sextics. We show that when the binary sextics has extra…
Recently it has been shown that the property of forward-flatness for discrete-time systems, which is a generalization of static feedback linearizability and a special case of a more general concept of flatness, can be checked by two…
We prove the $p$-curvature conjecture for rank two vector bundles with connection on generic curves, by combining deformation techniques for families of varieties and topological arguments.
A survey on algorithms for computing discrete logarithms in Jacobians of curves over finite fields.
This survey is about Galois theory of curves in characteristic p, a topic which has inspired major research in algebraic geometry and number theory and which contains many open questions. We illustrate important phenomena which occur for…