Arithmetic Geometry, D-Modules and Singularities

PI1: Antonio Rojas León
PI2: Luis Narváez Macarro

Abstract: The scientific proposal of this project is structured in three main research lines. There is a close relationship between these three lines, especially in their recent developments, and in our project we explore this relationship via some cross-cutting goals and a working group. 
(A) Arithmetic geometry: elliptic curves, Galois representations and exponential sums. In this line we study several issues that relate Geometry and Number Theory. We work on the study of local cohomological operations in the l-adic theory, deriving new applications to the study of the arithmetic of families of varieties over finite fields and exponential sums. We also study topics related to representations of the Galois group G_Q of the rational number fieldinally, and we will deepen the study of the torsion group of elliptic curves of the form y^2=f(x) in terms of the Galois group of f(x).
(B) Differential operators and D-modules. We will deal with the problem of classifying regular holonomic D-modules and perverse sheaves with preset singularities, and continue the study of Hasse-Schmidt derivations and their applications to the study of rings of differential operators in non-zero characteristic. 
(C) Algebraic and combinatorial tools in the theory of singularities. We work on various problems related to the theory of singularities: extension of valuations to the formal completion of a local ring, the Levi-Zariski algorithm, and the theory of numerical semigroups and roots of equations in Puiseux series.