The CM class number one problem for curves
Promotores: P. Stevenhagen, A. Enge Co-Promotor: T.C. Streng
- Pinar Kilicer
- 05 July 2016
- Thesis in Leiden Repository
The main subject of this thesis is the CM class number one problem for curves of genus g, in the cases g=2 and g=3. The problem asks for which CM fields of degree 2g with a primitive CM type are the corresponding CM curves of genus g defined over the reflex field. Chapter 1 is an introduction to abelian varieties and complex multiplication theory. We present facts that we will use in later chapters. The results in this chapter are mostly due to Shimura and Taniyama. Chapter 2 is a joint work with Marco Streng, we give a solution to the CM class number one problem for curves of genus 2. Chapter 3 deals with the CM class number one problem for curves of genus 3. We give a partial solution to this problem. We restrict ourselves to the case where the sextic CM field corresponding to such a curve contains an imaginary quadratic subfield. Chapter 4 gives the complete list of sextic CM fields K for which there exist principally polarized simple abelian threefolds that has CM by the maximal order of K with rational field of moduli.