2024-03-29T09:40:52Zhttps://www.tdx.cat/oai/requestoai:www.tdx.cat:10803/2793142017-08-30T04:33:11Zcom_10803_183col_10803_220
nam a 5i 4500
Arithmetic properties of non-hyperelliptic genus 3 curves
[Barcelona] :
Universitat Politècnica de Catalunya,
2014
Accés lliure
http://hdl.handle.net/10803/279314
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Lorenzo García, Elisa,
autor
1 recurs en línia (119 pàgines)
Tesi
Doctorat
Universitat Politècnica de Catalunya. Departament de Matemàtica Aplicada II
2014
Universitat Politècnica de Catalunya. Departament de Matemàtica Aplicada II
Tesis i dissertacions electròniques
Lario i Loyo, Joan-Carles,
supervisor acadèmic
TDX
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.
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