The use of generalised Jacobians in discrete logarithm based cryptosystems has so far been rather limited since they offer no advantage over traditional discrete logarithm based systems. In this paper we continue the search for possible applications in two directions. Firstly, we investigate pairings on generalised Jacobians and show that these are insecure. Secondly, generalising and extending...

We present three families of pairs of geometrically non-isomorphic curves whose Jacobians are isomorphic to one another as unpolarized abelian varieties. Each family is parametrized by an open subset of P. The first family consists of pairs of genus-2 curves whose equations are given by simple expressions in the parameter; the curves in this family have reducible Jacobians. The second family al...

We begin with a comprehensive discussion of the punctual Hilbert scheme regular two-dimensional local ring in terms Gröbner cells. These schemes are most degenerate fibers Grothendieck-Deligne norm map (the Hilbert-Chow morphism), playing an important role study smooth surfaces. They generally singular, but their cells affine spaces; they admit explicit parametrization due to Conca and Valla. u...

We give algorithms for computing with divisors on projective curves over finite fields, and with their Jacobians, using the algorithmic representation of projective curves developed by Khuri-Makdisi. We show that various desirable operations can be performed efficiently in this setting: decomposing divisors into prime divisors; computing pull-backs and push-forwards of divisors under finite mor...

In an approximation problem with a neural network, a low-output root mean square (rms) error is not always a universal criterion. In this paper, we investigate problems where the Jacobians--first derivative of an output value with respect to an input value--of the approximation model are needed and propose to add a quality criterion on these Jacobians during the learning step. More specifically...

In this paper we study locally symmetric families of curves and jacobians. By a locally symmetric family of jacobians, we mean a family of jacobians parameterized by a locally symmetric variety where the image of the period map is a locally symmetric subvariety of Ag[l], the moduli space of principally polarized abelian varieties of dimension g with a level l structure. A locally symmetric fami...