Construction of CM Picard curves

Construction of CM Picard curves

In this article we generalize the CM method for elliptic and hyperelliptic curves to Picard curves. We describe the algorithm in detail and discuss the results of our implementation.

Construction of curves with a Jacobian of given CM-type

Let C be a smooth projective curve of genus g ≥ 1 over a field κ and let JC be the Jacobian of C. Recall that the Jacobian JC is an abelian variety of dimension g whose group of λ-rational points for every field extension λ/κ is isomorphic to the group of λ-rational divisor classes of degree 0 of the curve C. In our talk, the ground field κ will either be finite, a number field or the field of ...

Picard Schemes of Curves 0b92

Edwards curves and CM curves

Edwards curves are a particular form of elliptic curves that admit a fast, unified and complete addition law. Relations between Edwards curves and Montgomery curves have already been described. Our work takes the view of parameterizing elliptic curves given by their j-invariant, a problematic that arises from using curves with complex multiplication, for instance. We add to the catalogue the li...

Picard Group of Moduli of Hyperelliptic Curves

The main subject of this work is the difference between the coarse moduli space and the stack of hyperelliptic curves. We compute their Picard groups, giving explicit description of the generators. We get an application to the non-existence of a tautological family over the coarse moduli space.