Anna Somoza Henares

Reconstruction of cyclic plane quintic curves

We consider the problem of computing the equation of a curve with given analytic Jacobian, that is, with a certain period matrix.
In the case of genus one, this can be done by using the classical Weierstrass function, and it is a key step if one wants to write down equations of elliptic curves with complex multiplication (CM). Also in higher genus, the theory of CM gives us all period matrices of principally polarized abelian varieties with CM, among which the periods of the curves whose Jacobian has CM, and computing curve equations is the hardest part.
Beyond the classical case of elliptic curves, efficient solutions to this problem are now known for both genus 2 and genus 3. In this talk I will give a method that deals with the case y^5 = a_5x^5 + ... + a_1x + a_0, inspired by some of the ideas present in the method for the genus-3 family of Picard curves y^3 = b_4x^4 + ... + b_1x + b_0 .