The PI intends to continue her research into properties of p-adic Galois representations arising from geometry as well as families of such. For isolated representations her main goal is to understand those coming from motivic cohomology as well as the related p-adic regulator maps (and the induced p-adic period maps) into the de Rham and syntomic cohomologies. For families, she plans to study the category of p-adic local systems on varieties over local fields, its subcategories of crystalline and semistable local systems, and the related syntomic cohomologies; the associated category of phi-Gamma-modules and its relation to the differential structure of the variety.
The study of Galois representations coming from geometry in various ways is of fundamental importance in modern number theory, representation theory, and algebraic geometry. The related p-adic methods are also of growing importance in computational number theory and its applications. For example, one of the most fruitful ways to attack problems involving solutions to diophantine equations is by attaching to them some Galois representations. Galois representations are objects of arithmetic nature to which, in turn, one can associate certain complex functions. These functions, conjecturally and in many verified cases, shed light on the geometry of the original diophantine equation and its set of solutions.
