Arithmetic properties of non-hyperelliptic genus 3 curves

• Autores: Elisa Lorenzo García
• Directores de la Tesis: Joan Carles Lario Loyo (dir. tes.)
• Lectura: En la Universitat Politècnica de Catalunya (UPC) ( España ) en 2014
• Idioma: inglés
• Tribunal Calificador de la Tesis: Francesc Fité (presid.) , Josep Maria Miret Biosca (secret.) , Anna Rio (voc.)
• Enlaces
  • Tesis en acceso abierto en: TDX
• Resumen

  • This thesis explores the explicit computation of twists of curves. We develope an algorithm for computing the twists of a given curve assuming that its automorphism group is known. And in the particular case in which the curve is non-hyperelliptic we show how to compute equations of the twists. The algorithm is based on a correspondence that we establish beetwen the set of twists and the set of solutions of a certain Galois embedding problem. In general is not known how to compute all the solution of a Galois embedding problem. Throughout the thesis we give some ideas of how to solve these problems. The twists of curves of genus less or equal than 2 are well-known. While the genus 0 and 1 cases go back from long ago, the genus 2 case is due to the work of Cardona and Quer. All the genus 0, 1 or 2 curves are hyperelliptic, however for genus greater than 2 almost all the curves are non-hyperelliptic. As an application to our algorithm we give a classification with equations of the twists of all plane quartic curves, that is, the non-hyperelliptic genus 3 curves, defined over any number field k. The first step for computing such twists is providing a classification of the plane quartic curves defined over a concrete number field k. The starting point for doing this is Henn classification of plane quartic curves with non-trivial automorphism group over the complex numbers. An example of the importance of the study of the set of twists of a curve is that it has been proven to be really useful for a better understanding of the behaviour of the Generalize Sato-Tate conjecture, see the work of Fité, Kedlaya and Sutherland. We show a proof of the Sato-Tate conjecture for the twists of the Fermat and Klein quartics as a corollary of a deep result of Johansson, and we compute the Sato-Tate groups and Sato-Tate distributions of them. Following with the study of the Generalize Sato-Tate conjecture, in the last chapter of this thesis we explore such conjecture for the Fermat hypersurfaces X_{n}^{m}: x_{0}^{m}+...+x_{n+1}^{m} = 0. We explicitly show how to compute the Sato-Tate groups and the Sato-Tate distributions of these Fermat hypersurfaces. We also prove the conjecture over the rational numbers for n=1 and over than the cyclotomic field of mth-roots of the unity if n is greater 1.