Algebraic Geometry and Arithmetic Geometry: Differential methods, singularities, cohomology and elliptic curves

  PI1: Antonio Rojas León

PI2: Luis Narváez Macarro

Abstract: In this project our goal is to study different problems related to Algebraic Geometry and Number Theory. Specifically, we have divided the work plan in 5 reseach lines: 
(A) Differential structures and cohomological methods in Algebraic Geometry and Singularities. It is divided in two sub-lines, corresponding to the characteristic 0 and positive characteristic cases. 
(B) Extension of valuations to the formal completion of a local ring. This includes proving the existence of tight extensions and searching for examples, in non-excellent rings, in which these tight extensions do not exist. 
(C) Combinatorial methods, Puiseux parametrizations and local methods.  In this line we tudy irreducible equations of integral dependency with coefficients in a ring of power series in several variables and the relationship between singular points and adjacent ideals.
(D) Cohomological methods and arithmetic applications. In this line we apply different cohomological theories (l-adic and p-adic) to deduce results of arithmetic nature about varieties over finite fields.  
(E) Elliptic curves, diophantine equations and numerical semigroups. In this line we use elliptic curves applied to the study of diophantine problems and the inverse Galois problem.