Beato Caro, AntonioRojas León, AntonioRabasco González, Luis María2024-12-202024-12-202024-07-10Rabasco González, L.M. (2024). Sumas exponenciales y funciones zeta. (Trabajo Fin de Grado Inédito). Universidad de Sevilla, Sevilla.https://hdl.handle.net/11441/166078The main goal of this work is the demonstration of Weil’s conjecture in diagonal hypersurfaces. The relevance of the topic that we will deal with goes up to Gauss, who was one of the pioneers of algebraic geometry, passing through many other mathematicians of whom we will talk later. In addition, we will prove other interesting theorems, although not as much as the conjectures. The document is structured into three chapters: In the first chapter we will talk about the paper in which Weil exposed his conjecture to the mathematical world. We will also make a small introduction to these conjecture and Weil’s Zeta function and look at the relationship between the conjectures and the Riemann hypothesis. We will also see the first resolutions of the conjecture and the consequences that they implied. In the second one, we will introduce the multiplicative characters, the Gauss sums and the Jacobi sums, and we will get many properties from these. We will also focus on the Legendre symbol and demonstrate the law of quadratic reciprocity. In fact, we will even see an example of whether a number is a quadratic residue in a finite field or not. Finally we will calculate the number of points that a diagonal hypersurface has in the projective space. In the last chapter we will play a little bit with the Zeta function and get some properties. Furthermore, we will culminate this work with the demonstration of Weil’s conjectures in diagonal hypersurfaces, demonstrate the HasseDavenport relation and demonstrate Gauss’ conjecture in the last entry of his mathematical diary.application/pdf67 p.spaAttribution-NonCommercial-NoDerivatives 4.0 Internationalhttp://creativecommons.org/licenses/by-nc-nd/4.0/info:eu-repo/semantics/bachelorThesisinfo:eu-repo/semantics/openAccess