Resumen de Inverse jacobian and related topics for certain superelliptic curves

Anna Somoza Henares

• Given an elliptic curve E over the complex numbers (CC) given by y^2 = x^3 + ax + b, there exists a lattice L in CC such that the group E(CC) of complex points on E is isomorphic to the complex analytic group CC/L. This correspondence between elliptic curves and one-dimensional complex tori is called the Uniformization Theorem, and one can make the inverse map explicit with the Weierstrass p-function, its derivative, and the Eisenstein series.

  Similarly, given an algebraic curve C of genus g, one associates to it a principally polarized abelian variety J(C), the Jacobian of C. Over CC, the Jacobian J(C) is isomorphic to a g-dimensional complex torus CC^g/L for a lattice L of full rank in CC^g.

  This determines a map J from the set M_g of isomorphism classes of algebraic curves of genus g to the set A_g of principally polarized abelian varieties of dimension g, and one may wonder if there exists an explicit inverse to this map, as in the case of elliptic curves. We call this the inverse Jacobian problem.

  This problem has been solved for curves of genus 2 and genus 3. However, for genus g > 3 there is the additional obstruction that not all principally polarized abelian varieties are Jacobians of curves, hence in order to solve the inverse Jacobian problem one needs to study the image by J of M_g in A_g. The problem of describing J(M_g) is known as the Riemann-Schottky problem.

  In this thesis we treat these two problems for two families of superelliptic curves, that is, curves of the form y^k = (x - a_1)·...·(x - a_l). We focus on the family of Picard curves, with (k,l) = (3,4) and genus 3, where we give a more efficient solution, and the family of cyclic plane quintic curves, with (k,l) = (5,5) and genus 6, where we solve both problems. We solve the inverse Jacobian problem from a computational point of view, that is, we provide an algorithm to obtain a model for the curve from the lattice L of its Jacobian. While Picard curves have genus 3, hence there is no obstruction to the inverse Jacobian problem, in the case of CPQ curves we also provide a characterization of the principally polarized abelian varieties that arise as Jacobians of CPQ curves.

  In Chapter 1 we first introduce some background on abelian varieties, Jacobians of curves, and Riemann theta constants, and then we present an inverse Jacobian algorithm for Picard curves.

  This was originally done by Koike and Weng in their paper "Construction of CM Picard curves", but their exposition presents some mistakes that we address and correct here. This chapter is based on joint work with Joan-Carles Lario.

  In Chapter 2 we present an inverse Jacobian algorithm for CPQ curves. We follow a strategy analogous to the one in Chapter 1 for the case of Picard curves.

  In Chapter 3 we address the Riemann-Schottky problem for CPQ curves, that is, we characterize the principally polarized abelian varieties that are Jacobians of CPQ curves. We use a generalization of the classical theory of complex multiplication due to Shimura (see his paper "On analytic families of polarized abelian varieties and automorphic functions") to study how the existence of the automorphism of CPQ curves (x,y) -> (x,e^(2·pi·i/5)y) affects the structure of the Jacobians.

  Finally, in Chapter 4 we present one application for the above algorithms: constructing curves such that their Jacobians have complex multiplication. This has previously been done for genus 2 and genus 3, and here we follow methods presented by Kilicer in her PhD thesis to determine a complete list of CM-fields whose ring of integers occurs as the endomorphism ring over CC of the Jacobian of a CPQ curve defined over the rational numbers.

  In particular, for every field K listed in Chapter 4 we also give a CPQ curve that is numerically close (and conjecturally equal) to a curve C that satisfies End(J(CC)) = O_K.